All measurements were done at room temperature. Schematic of the noise measurement system; R — carbon soot resistor; red dashed boxes — Faraday cages. The Raman spectrum of the soot Fig. V exhibits peaks which stand for the local violation of the Ohm law. This deviation from the linear response theory seems to be driven by microscopic nonlinearities manifesting preferentially at some voltages, as in the case of the two dominant noise peaks located at 0.
As shown in Fig. The two dominant noise peaks and their voltage location are indicated by arrows. If one assumes that EPC would act the microscopic source of the noise peaks at Kohn anomalies in our carbonic material, the difference in the peak intensities could be tentatively assigned to the stronger coupling at the K point 44 , 51 , In search for a quantitative support of this hypothesis, we resorted to Piscanec et al.
With the Raman frequencies from Fig. This relation allows the calculation of the matrix element ratio from the noise peak intensities. The key factor in obtaining this result was the equation 1 deduced by Piscanec et al. It is thus necessary to find other ways to go further. For instance, except for constant, a similar equation as 1 was deduced for two-dimensional silicene and germanene 53 , therefore it may be exploited to further investigate whether the procedure described above would apply to these materials, two.
Or, it was exactly this aspect which proved to be very difficult to solve for decades.
Another qualitative one will be given later in this work for the case of graphene. The two variables feature a similar structure. Standing for the local deviations from the linear dissipation law, the two noise peaks can now be ascribed to a nonlinear behavior of the matrix element at Kohn anomalies. With this background idea, Teitler and Osborne 45 calculated the fluctuation spectrum of the energy dissipated in a resistor biased at a dc voltage V.
Moreover, the nonlinearity-dispersion interplay appears as the only factor which controls the behavior of the frequency exponent in the equation which results for it from the above analysis:. This new formula for the frequency exponent is very general in its simplicity. In this work, the authors reported heating-induced nonlinearities in the I V characteristic of a gold film. The noise spectrum of this film was measured at a voltage bias 0.
Similar effects were observed by Eberhard and Horn 59 , 60 in a silver film.
They have found that the voltage exponent deviates from 2 when the sample starts heating at voltages higher than about 0. Therefore, although nonlinearity is a common factor in these works, its possible effect on noise mechanism was not investigated. As shown in this work, electron-phonon coupling is the source of the nonlinearity responsible for the noise structure presented in Fig.
To verify this hypothesis, in Fig. For comparison purposes, the absolute noise intensity is also presented. One notes that at this scale the faint wavy shape of the frequency exponent presented in Fig. It results that at least qualitatively the two equations are closely related. In this respect, the results presented in Fig. It turns out that phonons are implied in the noise mechanism even when the sample is temperature scanned. At microscopic level, the balance nonlinearity-dispersion in equation 2 can be understood by the same approach Akimenko, Verkin and Yanson 36 used to explain the noise structure in sodium point contacts.
Sensu stricto , for a given voltage, the exact values of n and d in the equation 2 are dictated by the shape of both Fermi surface and the phonon branch at their intersection. If not impossible, an in situ measurement seems to be extremely difficult. A possible simpler way to find n would be to inspect I-V characteristics, but at very low or low voltages local deviations in it could be difficult to observe, even if the measurement is done at low temperature and the first or second derivative were used.
For the determination of dispersion one has to resort to specific methods of lattice dynamics investigation. All these measurements-related aspects warrant further investigations. Due to the high uncertainties in the determination of both n and d , testing the validity of equation 2 is difficult. Nevertheless, if one considers the unique property of the dispersion exponent d to change its sign at a Kohn singularity, Eq.
Therefore, for a given value of the nonlinearity parameter n , due to the sign changes of d one might expect a transition around 1 in the frequency exponent at each anomaly. The necessary condition for noise intensity to increase with the voltage at Kohn anomalies see Fig. From Eq. On the other hand, we attributed the noise increase at a Kohn anomaly to a strengthening of the EPC. This prediction is confirmed by the data in Fig. In the particular case of a Kohn anomaly, the strengthening-weakening of the EPC at it may result in the observation of the phonon spectrum in the electron conductivity fluctuations on Fermi surface.
So far, our discussion was limited to the noise mechanism at Kohn anomalies. However, Fig. We should admit that it is very tempting to consider that it may be due to some spurious or random effect, such as temperature fluctuations, for instance. In search for the thermal heating as a possible source of nonlinearity in resistor, we calculated the resistance at each voltage point in the I-V characteristic Fig.
This indicates that heating-induced nonlinearity in our resistor is negligible. Next, the temperature of the resistor was calculated at different voltages. At the highest voltage across the resistor terminals 0.
Journal of the Optical Society of America A
At the voltage of the first Kohn anomaly 0. Such insignificant increase in the temperature cannot explain almost an order of magnitude noise increase at the first Kohn anomaly, for instance. Therefore, Joule heating can be hardly considered as the source of the local nonlinear manifestation reported in Fig.
As shown recently 65 , to keep this equation dimensionally correct, deviations of the voltage exponent from 2 must attract deviations from 1 in the frequency exponent. This is exactly what Eq. So far, we have shown that this is valid at Kohn anomalies only. By contrast, this is due to an out-of-plane optical phonon mode ZO To clarify the origin of the peaks, we further compared our noise data with those of Back et al. One mentions that only peaks having clear correspondent in the spectral exponent have been selected for comparison with the phonon energies.
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Comparison between the noise data black dots - absolute spectral density; red dots - frequency exponent and the Raman spectrum of graphene blue curve - reproduced adapted with permission from ref. Heller et al. Theory of graphene Raman scattering. ACS Nano. Very surprising in this correlation is the fact that, except maybe for the peak 5, all the weak noise peaks correlate with the out-of-plane phonons or combinations of these phonons with some in-plane modes. Spectroscopic observation of these phonons is notoriously difficult 73 , 74 in graphene because, in sharp contrast with the in-plane phonons, the electrons interact very weakly with these phonons This wavenumber range is equivalent to — meV, which partially covers the voltage range where some less intense noise peaks are located.
Such a Raman spectrum, which is due to Bernard and coworkers 74 , is compared 40 in Fig. To this purpose, Fig. Also, the correlation extends to the noise peak 2. As for the noise peaks 1and 2, a comparison not shown in Fig. These results indicate that the origin of weak noise structure is in the weak interaction between electrons and the out-of-plane phonon modes. The correlations presented above stand for a strong argument that, as in the case of Kohn anomalies, the origin of the peaks in both noise intensity and frequency exponent is in the electron-phonon coupling.
On the experimental side, Zhang et al. Such a G band phonon energy renormalization by the injected electrons has been observed experimentally both in mono- 83 , 84 and bilayer graphene As predicted, they consist of two singularities minima located at Fermi energies equaling the half of the G band phonon energy, an approximate W shape with respect to the Dirac point. The correlation between the noise intensity and the frequency exponent presented in Fig.
This is why we have investigated whether such correlations are visible in other physical systems. For instance, such detailed temperature dependences have been reported by Xiong et al. Again, it is a matter of evidence that the evolutions of the two parameters are correlated in time. The presence of such correlated structures asks for a common microscopic source.
In this respect, although the foregoing examples support our finding, they give no hint on the origin of this correlation. However, at a first glance, how EPC would be able to modify the slope of the spectrum at low frequency does is not evident whatsoever. Nevertheless, in our view, such a manifestation of the EPC on the slope of the spectrum would be possible if the visible, low-frequency part of the spectrum extends into the thermal noise of the resistor till phonon frequencies.
Our finding offers the plausible physical justification for this empirical procedure. A simple procedure was presented to calculate the ratio of the electron-phonon matrix element at the anomalies from the noise data.
Suggestions have been made on how to extend this result to other physical systems, such as silicene and germanene and even MOS transistor. A new, general and simple formula was found for the frequency exponent, whose value is determined by the nonlinearity-dispersion balance. This formula revealed that, for constant dispersion, the deviations of the frequency exponent from 1 are the signatures of nonlinearity.
The same conclusion we arrived at by dimensional considerations in the Hooge formula. We have shown that nonlinearity and dispersion are hidden in the DDH formula which describes the temperature effect on the spectral exponent. It resulted that the two equations are related and both have in common phonon specific parameters. Exploiting the properties of the dispersion exponent at the phonon kink, this formula predicted transitions sublinear-supralinear in the frequency exponent at Kohn anomaly.
This prediction was confirmed experimentally at both Kohn anomalies.
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It has been found that the dependence of the frequency exponent on voltage is featuring the same structure as the one observed in the noise intensity in the whole voltage range. Less intense noise peaks correlated very well with the out-of-plane phonon energies. It has been shown that the whole structure in noise intensity and spectral exponent is the image of the phonon spectrum. It turned out that the source of nonlinearity is in the electron-phonon coupling, which controls both the noise intensity and the slope of the spectrum.
This observation represents the long sought physical background for the Hooge empirical approach. Also, the violation of equipartition is another inherent consequence of our finding. Johnson, J. The Schottky effect in low frequency circuits. Bernamont, J. Leipzig 7 , 71— Fluctuations in the resistance of thin films.
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Carruthers, T. Bias-dependent structure in excess noise in GaAs Schottky tunnel junctions. Mihaila, M. Yanson, I. Electrical fluctuations in normal metal point-contacts. Akimenko, A. Point-contact noise spectroscopy of phonons in metals. Low Temp. Planat, M. Plana, M. System of phonon spectroscopy. Back, J. Hammig, M. Suppression of interface-induced noise by the control of electron-phonon interaction. IEEE Trans. Its publishing company, IOP Publishing, is a world leader in professional scientific communications. Get permission to re-use this article.
Create citation alert. Journal RSS feed. Sign up for new issue notifications. Provided the incident wave packet is incoherent on a nonvanishing band width, the phase of the reflected wave exhibits intermittency with "bursts" characterized by large phase and "laminar" regimes with small phases.