Theoretical derivation of 1/f noise in quantum chaos

It was recently conjectured that 1/f noise is a fundamental characteristic of spectral fluctuations in chaotic quantum systems. This conjecture is based on the power spectrum behavior of the excitation energy fluctuations, which is different for chaotic and integrable systems. Using random matrix theory, we derive theoretical expressions that explain without free parameters the universal behavior of the excitation energy fluctuations power spectrum. The theory gives excellent agreement with numerical calculations and reproduces to a good approximation the 1/f (1/f(2)) power law characteristic of chaotic (integrable) systems. Moreover, the theoretical results are valid for semiclassical systems as well.     

1/f alpha noise in spectral fluctuations of quantum systems

The power law 1/f(alpha) in the power spectrum characterizes the fluctuating observables of many complex natural systems. Considering the energy levels of a quantum system as a discrete time series where the energy plays the role of time, the level fluctuations can be characterized by the power spectrum. Using a family of quantum billiards, we analyze the order-to-chaos transition in terms of this power spectrum. A power law 1/f(alpha) is found at all the transition stages, and it is shown that the exponent alpha is related to the chaotic component of the classical phase space of the quantum system.     

Quantum 1/f noise associated with ionized impurity scattering and electron-phonon scattering in condensed matter

Scattering of charged particles is accompanied by the emission of soft photons. Handel's theory of 1/f noise, based on the infrared quasi-divergent coupling of the system to the electromagnetic field, indicates that the current associated with a beam of scattered particles will exhibit 1/f noise. His derivation is valid in a vacuum. Here we extend his results and obtain the fluctuation spectrum for the fluctuations in cross-section and for the scattering rates w _kk′ in k-space, using the Born approximation. Next we consider mobility fluctuations due to these scattering rates, employing the relaxation time solutions of the Boltzmann transport equation, valid in non-degenerate semiconductors. Explicit results are obtained for the mobility-fluctuation noise caused by ionized impurity scattering, acoustic phonon scattering, optical phonon scattering, polar optical phonon scattering, and intervalley scattering. We derive Hooge's law, and the Hooge parameters for the above-mentioned processes are obtained in detail. This is then applied to n-type silicon and n-type gallium arsenide; the overall Hooge parameter, which is a weighted average of the partial α-parameters, is computed as a function of temperature and compared with experiment. For silicon, good agreement is obtained with available data. As a byproduct we also find the mobilities as function of temperature for these materials. Excellent agreement with the well-known experimental data is observed.

We still note that this is the first theoretical derivation of Hooge's law and that the magnitude of the noise is obtained in detail without adjustable parameters. We believe that quantum 1/f noise gives the limiting value of 1/f noise that can be observed.     

Burst and 1/f noise in light-emitting diodes with quantum dots

We study the low-frequency electric noise characteristics of light-emitting diodes with InAs quantum dots in a GaInAs layer. Burst noise having the character of random telegraph signal (RTS) is found against the 1/f noise background in the noise voltage of some specimens. A procedure based on the standard theory of signal detection against the noise background is proposed for a separate study of these noise components. It is found that Hooge’s empirical relation applied to p-n diodes for the first time by Kleinpenning is also applicable to 1/f noise in quantum-dot diodes. The current dependences of statistical characteristics of the 1/f and RTS noise components are compared to show that the physical origins of RTS noise and 1/f noise in the studied specimens are different. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 49, No. 5, pp. 437–447, May 2006.     

1/f noise in varistors

An experimental investigation was made of low-frequency fluctuations of the open-circuit voltage of mass-produced Varistors made of silicon carbide. It was concluded that the main charge transport mechanism in varistors may be the Frenkel ionization of impurities.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 11–15, November, 1980.     

纤锌矿GaN/ZnO 量子阱中的界面声子及其电声相互作用

根据介电连续模型和单轴晶体模型研究了纤锌矿量子阱中的界面声子模及其电声子相互作用的Fröhlich哈密顿。我们计算和讨论了纤锌矿GaN/ZnO单量子阱中的界面声子的色散关系和电声相互作用的耦合强度。色散曲线充分体现了纤锌矿晶体的各向异性;四支界面声子模出现在两个能量区域中,分别是：[ , ]和[ , ]。界面声子模出现消失的现象,光学声子模之间存在能量交迭区域。我们的结果也阐述了纤锌矿GaN/ZnO单量子阱中每支声子模与电子相互作用的对称性和耦合强度。     

A theory of 1/f noise

Letu() be an absolutely integrable function and define the random process where thet _i are Poisson arrivals and thes _i, are identically distributed nonnegative random variables. Under routine independence assumptions, one may then calculate a formula for the spectrum ofn(t), S _n(), in terms of the probability density ofs, p_s(). If any probability density p_s() having the property p_s() I for small is substituted into this formula, the calculated S_n() is such that S_n() 1 for small . However, this is not a spectrum of a well-defined random process; here, it is termed alimit spectrum. If a probability density having the property p_s() for small , where > 0, is substituted into the formula instead, a spectrum is calculated which is indeed the spectrum of a well-defined random process. Also, if the latter p_s is suitably close to the former p_s, then the spectrum in the second case approximates, to an arbitrary, degree of accuracy, the limit spectrum. It is shown how one may thereby have 1/f noise with low-frequency turnover, and also strict 1/f ^1– noise (the latter spectrum being integrable for > 0). Suitable examples are given. Actually, u() may be itself a random process, and the theory is developed on this basis.     

Quantum chaos and 1/f noise

It is shown that the energy spectrum fluctuations of quantum systems can be formally considered as a discrete time series. The power spectrum behavior of such a signal for different systems suggests the following conjecture: The energy spectra of chaotic quantum systems are characterized by 1/f noise.     

The so-called 1/Δ f noise

A sinusoidal ac current of frequeencyf _c generates an excess noise in many solidstate conductors. The power spectrum of this noise is calculated starting from the autocorrelation function of the stochastic process. Besides a 1/Δf term the spectrum includes another contribution varying with 1/(f _c+f). In the low frequency range there is white noise.     

Fluctuation theorems and 1/f noise from a simple matrix

Here we present a model for a small system combined with an explicit entropy bath that iscomparably small. The dynamics of the model is defined by a simple matrix, M. Each row ofM corresponds to a macrostate of the system, e.g. net alignment, while the elements in therow represent microstates. The constant number of elements in each row ensures constantentropy, which allows reversible fluctuations, similar to information theory where aconstant number of bits allows reversible computations. Many elements in M come from themicrostates of the system, but many others come from the bath. Bypassing the bath statesyields fluctuations that exhibit standard white noise; whereas with bath states the powerspectral density varies as S(f) ∝ 1 /f overa wide range of frequencies, f. Thus, the explicit entropy bath is the mechanismof 1/f noisein this model. Both forms of the model match Crooks’ fluctuation theorem exactly,indicating that the theorem applies not only to infinite reservoirs, but also tofinite-sized baths. The model is used to analyze measurements of 1/f-like noise from asub-micron tunnel junction.     

金属薄膜电迁移1/f噪声与1/f2噪声统一模型

应用晶粒边界自由体积的概念建立了能够统一描述金属薄膜1f噪声与1f2噪声的模型.该模型表明,结构完整的多晶金属薄膜产生的电噪声为1f噪声,当金属薄膜受到电迁移损伤而形成空洞时就会引入1f2噪声的成分.在电迁移应力实验中,观察到金属薄膜1fγ噪声在空洞成核前γ约为10,一旦发生空洞成核,即突增至16以上,这一规律与本模型的预测相符合 关键词： 金属薄膜 1fγ噪声 电迁移 自由体积     

On the interpretation of 1/f noise

We propose a model of 1/f noise based on a random walk in a random potential. Numerical support for the model is given, and physical applicability discussed.     

Modeling 1/f noise using moving averages

A model of 1/f noise is considered, based on moving averages of ordern. The coefficientsα _k defining the model are calculated numerically using Seidel iteration which turns out to converge rapidly. The convergence is independent ofn which seems to be caused by the fact that the nonlinear problem solved is defined by a self-similar matrix. The coefficientsα _k appear to approach, with indefinitely growingn, valuesα _k =1/√k and thus the model has a kind of Fourier invariance. Physical interpretation of the invariance is suggested as well as of coefficientsα _k describing long-range correlations. Fractal sets of dimensiond=2.5 are proposed to play certain role in explaining the latter. This work was partly supported by the Scientific Grant Agency of the Ministry of Education of Slovak Republic (VEGA) under the Grant No. 1/3143/96 and by the Slovak Literary Fund.