Tsallis non-extensive statistics and multifractal analysis of the dynamics of a fully-depleted MOSFET nano-device

Physica A 533 (2019) 121820

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Tsallis non-extensive statistics and multifractal analysis of the dynamics of a fully-depleted MOSFET nano-device ∗

I.P. Antoniades a , G. Marinos b,e , L.P. Karakatsanis b , , E.G. Pavlos b , S.G. Stavrinides c , D. Tassis d , G.P. Pavlos b,e a

Division of Technology & Sciences, American College of Thessaloniki, Thessaloniki, Greece ‘‘Research Team of Chaos and Complexity’’, Department of Environmental Engineering, Democritus University of Thrace, Xanthi, Greece c School of Science and Technology, International Hellenic University, Thessaloniki, Greece d Physics Department, Aristotle University of Thessaloniki, Thessaloniki, Greece e Department of Electrical Engineering, Democritus University of Thrace, Xanthi, Greece b

highlights • The estimation of Tsallis q-triplet were found to deviate from unity in both cases indicating non-Gaussian Boltzmann–Gibbs statistics. • Our analysis revealed the existence of a non-equilibrium topological phase transition process, taking place when the drain voltage increases.

• We showed the existence of microscopic intermittent turbulence and anomalous diffusion processes at the MOSFET current dynamics.

• We showed that the statistics of the MOSFET current dynamics satisfies the basic principles of Tsallis non-extensive statistical theory.

article

info

Article history: Received 29 January 2019 Received in revised form 25 May 2019 Available online 8 July 2019 Keywords: Random telegraph noise (RTN) Low frequency noise (LFN) Non-extensive statistics Non-gaussian dynamics Anomalous diffusion Tsallis q-triplet Strange dynamics UTBB FD-SOI MOSFET

a b s t r a c t In this study Tsallis non-extensive statistics and multifractal analysis for the physical description and understanding of the Random Telegraph Noise (RTN), observed in UTBB FD-SOI MOSFET current timeseries is presented. Specifically, we estimate the Tsallis q-triplet and q-entropy production, the Hurst exponent, the multifractal singular spectrum and other topological and dynamical characteristics of this MOSFET-current dynamics as a function of the drain voltage. The analysis shows the existence of microscopic intermittent turbulence and anomalous diffusion processes, underlying the noisy timeseries. The relevant results indicate the existence of a self-organized critical behavior manifested by a percolation mechanism and a fractional transport process. The multifractal character of the dynamics was demonstrated and explained by the maximization of the Tsallis q-entropy function. Final, our analysis revealed the existence of a non-equilibrium topological transition process, taking place when the drain voltage increases and satisfies the basic principles of Tsallis non-extensive statistical theory. © 2019 Elsevier B.V. All rights reserved.

∗ Corresponding author. E-mail address: [email protected] (L.P. Karakatsanis). https://doi.org/10.1016/j.physa.2019.121820 0378-4371/© 2019 Elsevier B.V. All rights reserved.

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1. Introduction Random Telegraph Noise (RTN) is a kind of noise demonstrating two distinct levels of voltage or current noisy timeseries. This kind of noise was initially reported in point contact diodes (the well-known detectors, used in AM demodulation circuits), and appears in many small-sized devices [1]. One case of a nanodevice demonstrating such a noise is the ultrathin body and buried box (UTBB), fully depleted, silicon-on-insulator (FD-SOI) MOSFETs, which has been proposed as a promising nanoscale transistor device, appearing in Fig. 1. It has already been reported that these devices demonstrate significant advantages like immunity to short-channel effects [2,3], low threshold voltage and threshold voltage controllability, by utilizing the second gate bias. Moreover, these devices are fully compatible to the standard planar CMOS technology, while they are susceptible to further down-scaling. Since the UTBB FD-SOI MOSFET dimensions are in the area of nanoscale, phenomena related to inherent noise become significant, thus affecting its performance. It is apparent, that noise analysis in such devices is an important issue that must be studied for the proper characterization of the device, in terms of its reliability when operating in various circuit designs. To this direction analysis of the low-frequency noise (LFN) in MOSFET devices appears in [4–8], where a study in the frequency domain and the co-existence of flicker and Lorentzian-like noise was revealed. Averaging of thousands of measurements, as implied in the previously mentioned studies, could hide or diminish the device’s real dynamics or any critical phenomena in noise. Consequently, the necessity to analyze the original signals in the time domain emerges [9]. Although fresh UTBB FD-SOI MOSFETs have a quite normal noise-behavior, stressed transistors demonstrate RTN. It is noted that stressed transistors are ones that have been biased by a relatively high voltage for a significant time duration, creating imperfections in various parts of the device and thus accelerating their aging; in this sense stressed transistors are still operational transistors but behave as aged transistors. In this case as well as in [10] and [11], the studied UTBB FD-SOI MOSFETs were stressed by a gate and drain voltage Vg = Vd = 1.8 V, for a time period of 5000 s. It seems that this stressing introduced energy traps within the gate oxide, leading to complex dynamics that go beyond the usually observed stochastic noise behavior. The outcome of the low-dimensional complex dynamics is the observed RTNtype noise. Although the observed noise level is less than 2% in the specific devices, it is very interesting to study and understand its nature, since it reveals interesting charge transport phenomena that take place at these small scales. To this direction, the deterministic chaotic nature of the complex RTN demonstrated by stressed UTBB FD-SOI MOSFET devices was revealed and further evaluated in [10]. Therein, time domain drain current Id measurements were performed. A typical experimental timeseries (Id versus time) consisted of 105 points. The drain voltage was Vd = 30 mV, and front gate bias was Vg = 0.260 V. Measurements were taken with a sampling frequency fs = 20 kHz. It is noted that, although no similar work had been previously reported, chaotic characteristics of transistor noise in (extremely) low frequencies had been reported in [10]. In [10] the fractal nature of this noise (RTN), appearing as the fluctuation of the drain current, was highlighted. Furthermore, in [10] the complex nature and its deterministic origin of the studied RTN, was again confirmed and studied from a different point of view, that of Critical Phenomena. In specific, in [10] the Method of Critical Fluctuations (MCF) revealed an intermittent criticality for the specific two-trap RTN fluctuations, related to the trap emission/capture process; an apparent analogy to thermal critical systems. Both results described in [10] and [10] fully support the deterministic chaotic nature of this kind of noise (RTN), which in this case appeared to possess a correlation dimension for the resulting chaotic attractor equal to 5, while the dynamics of the system could be fully described in a 7-dimensional space. In the present paper we attempt to further investigate the complex behavior of the UTBB FD-SOI MOSFET current dynamics by looking into the multi-fractal and non-Gaussian character as well as, the non-extensivity of the statistics of the experimentally measured timeseries. For this purpose, we apply Tsallis non-extensive statistical mechanics [12], which is a novel extension of classical Boltzmann–Gibbs statistics for far-from equilibrium dynamics of complex systems. The non-extensive statistics of Tsallis is based on the q-entropy function, the maximization of which can reproduce geometrical, dynamical and statistical characteristics of non-equilibrium complex dynamics. Concerning the application of Tsallis non-extensive statistical theory on the experimental timeseries, we calculated the Tsallis q-triplet (qsen , qrel , qstat ), corresponding to three distinct physical processes: q-entropy production, q-relaxation and the scale invariant meta-equilibrium stationary states in respect. Moreover, in this study the application of Tsallis non-extensive statistical theory was complemented by the estimation of Hurst index and the p-model intermittent turbulence theory. We also used the maximization of the Tsallis q-entropy function for the prediction of the experimentally determined multifractal spectrum f (α ) vs. α , as well as the explanation of the p-modeling results. We also use the generalized dimension spectrum Dq vs. q and the asymmetry of the f (α ) for the description of the geometrical and dynamical dependence of the dynamics on the MOSFET gate voltage. In the following we first present the theoretical background and methodology of our approach, the experimental data analysis and a summary of results. At the end of this study we provide with an extensive discussion of the physical meaning of the results of this study. 2. Theoretical framework 2.1. Non-extensive statistical mechanics Any extension of physical theory is usually related to some special type of mathematics. Non-extensive Tsallis statistical theory is connected to the q-extension of exponential and logarithmic functions and the q-extension of a Fourier transform

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Fig. 1. The UTBB FD-SOI MOSFET nanodevice.

(FT) [13]. The q-extension of mathematics underlying the q-extension of statistics is included in the solution of the non-linear equation dy dx

= yq , (y(0) = 1, q ∈ R)

(1)

Its solution is the q-exponential function exq exq ≡ [1 + (1 − q)x]1/(1−q)

(2)

The q-extension of logarithmic function is the reverse of lnq x ≡

exq

x1−q − 1

(3)

1−q

The q-logarithm satisfies the property lnq (xA xB ) = lnq xA + lnq xB + (1 − q)(lnq xA )(lnq xB )

(4)

According to the pseudo-additive property of the q-logarithm, a generalization of the product and sum as the q-product and q-sum can be introduced in (1) lnq x+lnq y

x ⊗q y ≡ eq

(5)

x ⊕q y ≡ x + y + (1 − q)xy

(6)

Moreover in the context of the q-generalization of the central limit theorem the q-extension of FT can be introduced in Eq. (1) Fq [p](ξ ) ≡

+∞

∫

dxeqixξ [p(x)]

q −1

p(x),

(q ≥ 1)

(7)

−∞

Tsallis, inspired by multi-fractal analysis [13] proposed that the BG entropy SBG = −k

∑

pi ln pi = k⟨ln(1/pi )⟩

(8)

cannot describe all of the complexity of non-linear dynamic systems. BG statistical theory presupposes ergodicity of the underlying dynamics in the system phase space. The complexity of dynamics is far beyond simple ergodic complexity and can be described by non-extensive Tsallis statistics based on the extended concept of q-entropy:

( Sq = k 1 −

N ∑ i=1

) q pi

/(q − 1) = k⟨lnq (1/pi )⟩

(9)

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For a continuous state space, we have

[

∫

Sq = k 1 −

]

[p(x)]q dx /(q − 1)

(10)

For a system of particles and fields with short-range correlations in their immediate neighborhood, the Tsallis q-entropy Sq asymptotically leads to BG entropy (SBG ) corresponding to q = 1. For probabilistically dependent or correlated system A and B, it can be proven that Sq (A + B) = Sq (A) + Sq (B/A) + (1 − q)Sq (A)Sq (B/A) = Sq (B) + Sq (A/B) + (1 − q)Sq (B)Sq (A/B)

(11)

where Sq (A) ≡ Sq pi , Sq (B) ≡ Sq pi , Sq (B/A) and Sq (A/B) are the conditional entropies of systems A, B. When the systems are probabilistically independent, then relation (11) changes to

({ A })

({ B })

Sq (A + B) = Sq (A) + Sq (B) + (1 − q)Sq (A)Sq (B)

(12)

The first part of Sq (A + B) is additive (Sq (A) + Sq (B)) while the second part is multiplicative including long-range correlations supporting the macroscopic ordering phenomena. Zelenyi and Milovanov showed that the Tsallis definition of entropy coincides with the so-called ‘‘kappa’’ distribution for space plasmas and other physical realizations [14]. They also found that the Tsallis entropy formalism can be applied to physical systems with relatively small statistical weights, while for large statistical weights the standard BG statistical mechanism is better. This result means that when the dynamics of a system is attracted in a confined subset of the phase space, then long-range correlations can develop. According to Tsallis, if the correlations are either strictly or asymptotically non-existent, the BG entropy is extensive, whereas Sq for q ̸ = 1 is non-extensive. Oppositely, for strong correlated states BG entropy is non-extensive while for a special q-value Sq is extensive. Non-extensive statistical mechanics includes the q-analog (extensions) of the classical CLT and α -stable distributions corresponding to dynamical statistics of globally correlated systems. The q-extension of CLT leads to the definition of statistical q-parameters of which the most significant is the q-triplet (qsen , qrel , qstat ), where the abbreviations sen, rel, and stat, stand for sensitivity (to the initial conditions), relaxation and stationary (state) in non-extensive statistics respectively [15–17]. These quantities characterize three physical processes: (a) q-entropy production (qsen ), (b) relaxation process (qrel ), (c) equilibrium fluctuations (qstat ). These processes are demonstrated shown in Fig. 2, which shows a schematic representation of the temporal evolution of a complex system’s q-entropy starting from an initial condition to the stationary state, which, in general, can be broken down into three stages: In the first stage, entropy rises, as the complex system moves to occupy larger regions of phase space, and explores more and more microscopic degrees of freedom. After reaching a maximum value of entropy, the system self-organizes, and entropy relaxes until it equilibrates to a steady-state value around which it varies, in general under non-Gaussian statistics. The q-triplet values characterize the attractor set of the dynamics in the phase space of the dynamics and they can change when the dynamics of the system is attracted to another attractor set of the phase space. Eq. (1) for q = 1 corresponds to the case of equilibrium Gaussian (BG) world [18]. In this case, the q-triplet of Tsallis simplifies to qsen = 1, qstat = 1, qrel = 1. 2.2. q-triplet of Tsallis theory 2.2.1. qstat index A long-range-correlated meta-equilibrium non-extensive process can be described by the nonlinear differential equation [18]:

(

d pi Zqstat dEi

)

( )q = −β qstat pi Zqstat stat

(13)

The solution of this equation corresponds to the probability distribution: −β

E

pi = eqstatstat i /Zqstat where βqstat =

1 KTstat

, and Zqstat =

pi ∝ 1 − (1 − q)βqstat Ei

[

(14)

∑

−β qstat Ej

j

eqstat

]1/1−qstat

. Then the probability distribution is given: (15)

for discrete energy states {Ei } and by p(x) ∝ 1 − (1 − q)βqstat x2

[

]1/1−qstat

(16)

for continuous x states of {X }, where the values of the magnitude X correspond to the state points of the phase space. Distributions functions (15) and (16) correspond to the attracting stationary solution of the extended (anomalous) diffusion equation related to the nonlinear dynamics of the system. The stationary solutions P(x) describe the probabilistic character of the dynamics on the attractor set of the phase space. The non-equilibrium dynamics can evolve on distinct attractor sets, depending upon the control parameters, while the qstat exponent can change as the attractor set of the dynamics changes. For the estimation of Tsallis q-Gaussian distributions we use the method described in Ferry et al. [19].

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Fig. 2. Schematic representation of the physical interpretation of Tsallis q-triplet. The red line schematically shows a characteristic time evolution of the q-entropy of a non-equilibrium dynamical system from an initial state to the stationary state. The first part (up to the first vertical striped line) corresponds to initial increase in entropy (q-entropy production) qsen , being the pertinent Tsallis index, the second part (between the first and second striped lines) corresponds to the relaxation process that occurs because of the system’s self-organization (qrel ) and finally the stationary state (qstat ) where mean q-entropy is stabilized and the distribution of system states is stationary and obeys (in general) non-Gaussian statistics.

2.2.2. qsen index Entropy production is related to the general profile of the attractor set of the dynamics. The profile of the attractor can be described by its multi-fractality as well as by its sensitivity to initial conditions. The sensitivity to initial conditions can be expressed as: dξ dt

= λ1 ξ + (λq − λ1 )ξ q

(17)

where ξ is the trajectory deviation in the phase space: ξ ≡ lim∆(x)→0 {∆x(t)\∆x(0)} and ∆x(t) is the distance between neighboring trajectories [20]. The solution of Eq. (6) is given by:

λq λq ξ = 1 − sen + sen e(1−qsen )λ1 t λ1 λ1

[

] 1−q1

sen

(18)

The qsen exponent is related to the multi-fractal profile of the attractor set according to 1 qsen

=

1

αmin

−

1

αmax

(19)

Where αmin , αmax corresponds to zero points of the multi-fractal exponent spectrum f (α ) that is f (αmin ) = f (αmin ) = 0. In order to calculate the spectrum f (α ), we estimate the scaling exponent function τ (q), by using the relation:

Γ (q, ∆t ) =

∑

Pi (∆t )q ≈ (∆t )τ (q) ,

(20)

where Γ (q, ∆t ) is the qth order partition function of the experimental time series z (ti ) and Pi (∆t ) is the probability coarse-grained weight for time segments Λi of time size ∆t of the experimental time series z (ti ) [21]. The experimental estimation of the scaling exponent is used to estimate the generalized dimension spectrum Dq [22] through the relation:

τ (q) = (q − 1) Dq

(21)

By using Dq spectrum we estimate the singularity spectrum f (α) using the Legendre transformation: f (α ) = qα − (q − 1) Dq

(22)

We must note here, that the Tsallis q-entropy number is a special number corresponding to the extremization of Tsallis entropy of the system, while the q’s describe the range of real values of generalized dimension spectrum Dq . It is important to note here that the exponents a of the multifractal spectrum f (α ) correspond to the Holder exponent and reveal the intensity of the topological singularity of the phase space as well as how irregular the physical magnitudes defined in the phase space of the system are. Moreover, the left part of the singularities exponents α corresponding to

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values lower than the value αo where f (α ) is maximized, correspond to the low dimensional regions of the phase space described by the right part of Dq spectrum. For α > αo the spectrum f (α ) corresponds to the high dimensional regions of the phase space, which are related to the left part of the curve Dq of the generalized dimension spectrum. In other words, the high dimensional regions of phase space correspond to higher values of singularity exponents α and smoother fractal topology than the low dimensional regions. The latter correspond to lower values of α , more singular topology and stronger fractal character. In particular, a0 corresponds to the largest fractal dimension. The degree of multi-fractality (or width) is given by:

∆α = αmax − αmin

(23)

and the degree of asymmetry A be estimated by the relation: A=

αo − αmin αmax − αo

(24)

For A > 1, the singular exponents a < a0 are more dominant (left skewness). In this case, the low dimensional regions corresponding to the right part of Dq spectrum are also dominant over the high dimensional regions. Low dimensional regions are related with small fluctuations in the physical measurements and a strong fractal character as the singular exponents are lower and the discontinuities are stronger too. The opposite scenario happens, when A < 1 (right skewness). This implies the dominance of high dimensional regions of phase space corresponding to the left part of the Dq spectrum. The values of singular exponents α > α0 are more dominant and the evolution of the physical magnitudes is smoother. 2.3. qrel index Thermodynamic fluctuation–dissipation theory is based on the Einstein original diffusion theory (Brownian motion theory). Diffusion is a physical mechanism for extremization of entropy. If denote the deviation of entropy from its equilibrium value, then the probability of a proposed fluctuation is given by: P ≈ exp(DS/k)

(25)

The Einstein–Smoluchowski theory of Brownian motion was extended to the general FP diffusion theory of nonequilibrium processes. The potential of FP equation may include many meta-equilibrium stationary states near or far away from thermodynamical equilibrium. Macroscopically, relaxation to the equilibrium stationary state of some dynamical observable related to system evolution in the phase space can be described by the form of general form: dΩ

1 ∼ (26) =− Ω τ where Ω (t ) ≡ [O (t ) − O (∞)]/[O (0) − O (∞)] describes the relaxation of the macroscopic observable towards dt

its stationary state value. The non-extensive generalization of fluctuation–dissipation theory is related to the general correlated anomalous diffusion processes [18]. The equilibrium relaxation process is transformed to the meta-equilibrium non-extensive relaxation process according to: dΩ dt

=−

1 Tqrel

Ω qrel

(27)

the solution of this equation is given by: −t /τ Ω (t) ∼ = eqrel rel

(28)

The autocorrelation function C (t) or the mutual information I(t) can be used as candidate observables Ω (t) for estimation of qrel . However, in contrast to the linear profile of the correlation function, the mutual information includes the nonlinearity of the underlying dynamics and it is proposed as a more faithful index of the relaxation process and the estimation of the Tsallis exponent qrel . 2.4. Fractal topology, transport processes and Hurst exponent Strong non-linear dynamics produces fractal–multifractal structuring of the phase space. The geometry and topology of the fractal has self-similar character and creates power law behavior and scale invariance. Moreover, the fractal structure of phase space creates non-ergodicity and multi-scale spatio-temporal long-range correlations of the distributed system dynamics. The Hausdorff fractal dimension D is smaller of the dimension of the embedding Euclidean space. In addition to the fractal dimension D, the connectivity index θ , which describes the shape and the anomalous transport processes on the fractal set. θ is defined as: dθ = (2 + θ) /2

(29)

Where dθ is the minimal Hausdorff dimension of the minimal geodesic line for all possible homeomorphisms that transform the fractal set F into a new fractal F’ [23]. The geodesic line on a self-similar fractal set is a self-affine fractal

I.P. Antoniades, G. Marinos, L.P. Karakatsanis et al. / Physica A 533 (2019) 121820

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curve whose own fractal dimension is equal to dθ is a topological invariant quantity of the fractal set, it may take different values for fractals with the same Hausdorff dimension and plays a crucial role in many dynamical phenomena on the fractal set. From the fractal dimension D of the fractal set and the connectivity index θ we can define the spectral dimension dS as follows: dS = 2D/(2 + θ)

(30)

dS is a hybrid parameter of the fractal set and represents the density of states for vibrational excitations in the fractal network, called fractions [24]. The mean square displacement of a random walker on the fractal set is given by

⟨

⟩ ∆x2 ∼ t 2/(2+θ ) = t 1/dθ = t µ

(31)

where µ is defined as the inverse of the Hausdorff dimension of the geodesic lines on the fractal (µ ≡ 1/dθ ). We can also relate µ to the connectivity θ by:

µ = 2/(2 + θ) = dS /D

(32)

The value of the connectivity index θ determines the type of diffusion process on the fractal set as follows: if θ < 0, then µ > 1 and the transport process on the fractal is characterized by super-diffusion, i.e. is faster than diffusion on a Euclidean space (normal diffusion, where θ = 0). Super-diffusion corresponds to a value of dθ , the fractal dimensions the geodesic line, smaller than 1. Topologically, this corresponds to a case where the fractal does not contain a large proportion of holes (tree-like structure), in which case the random walker may reach a distant point faster and thus transport is accelerated (flights). If θ > 0, then µ < 1 and the transport process is characterized by sub-diffusion, i.e. slower than diffusion in a Euclidean space. Topologically, this corresponds to a situation when the fractal set contains a large proportion of holes around which the random walker loops (trapping regions) and thus transport is slowed down. We have 0 < µ < 2. The exponent µ can also be related to the structure function Sq corresponding to the first moment (q = 1) of the differences between points on a random walk. More specifically, the structure function Sq of a time series Xn (n = 1, 2, . . .) is defined as: Sq = |Xn+τ − Xn |q

⟨

⟩ n

∼ τ qHq

(33)

where q > 0, τ is the lag time and ⟨· · ·⟩n is the average over all time steps n of the series. Hq is the generalized Hurst exponent. The historical (basic) definition of the Hurst exponent H corresponds to the Hq spectrum value for q = 1. Combining (15) with (18) we conclude that:

µ = 2H

(34)

When the Hurst exponent spectrum Hq in Eq. (33) is a function of q, then the timeseries corresponds to multifractal intermittent turbulence dynamics. If it is a constant, it corresponds to monofractal dynamics. If H = 1/2, the dynamics is Euclidean (θ = 0). From the condition 0 < µ < 2 above, it follows that 0 < H < 1. For H > 1/2(θ < 0), the random walk process is persistent (super-diffusion) and for H < 1/2, the random walk process is anti-persistent (sub-diffusion). 2.5. P-model According to Meneveau and Sreenivasan (1987), the p-model is a one dimensional model version of a cascade model of eddies, each breaking down into two new ones according to a generalized two scale Cantor set with l1 = l2 = 21 . The p-model was introduced to account for the occurrence of intermittence in fully developed turbulence. 2.6. Prediction of multi-fractal and intermittent turbulence structure by tsallis theory As shown by Arimitsu and Arimitsu [18,25,26] and Arimitsu et al. [27] the multi-fractal singular spectrum f (α ) can be theoretically estimated as the extreme of the Tsallis entropy functional Sq . More specifically, by extremizing the Tsallis q-entropy the probability density function P(α )dα is the probability of set of points in phase space with singularity measure between α and (α + dα ) is given as the q−exponential function P(α ) = Zq−1 [1 − (1 − q)

(α − α0 )2 2X ln 2

1

(35)

] 1 −q

where the partition function Zq is given by Zq =

√

2X /[(1 − q) ln 2]B

(

1

,

2

) (36)

2 1−q

and B(a, b) is the Beta function. The partition function Zq and the quantities X and q can be estimated using the following equations

√ 2X =

[√

]

√

a20 + (1 − q)2 − (1 − q) / b

b = (1 − 2−(1−q) )/[(1 − q) ln 2]

⎫ ⎬ ⎭

.

(37)

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The spectrum f (a) can be expressed using the relations f (a) = F (a) − (d − 1) and P(a) ≈ lnd−F (a) yielding f (a) = D0 + log2 [1 − (1 − q)

(a − ao )2 2X ln 2

]/(1 − q)−1 ,

(38)

where a0 corresponds to the q — expectation (mean) value of a, expressed as 2

⟨(a − a0 ) ⟩q = (

∫

∫ daP(a) (a − a0 ) )/ daP(a)q . q

q

(39)

And Do is the generalized dimension Dq for q = 1. 2.7. Singular Value Decomposition (SVD) The SVD method is an appropriate tool for discriminating discrete dynamical components contained in the original timeseries. Nonlinear dynamics can lead to self-organization, which can give rise asymptotic dynamics to low-dimensional attracting sets in the phase space. According to embedding theory experimental timeseries can be used for the embedding of dynamics in the reconstructed phase space by using the method of delays. The SVD method can be used in distinguishing chaotic dynamics as well as to discriminate distinct dynamical components included in the experimental signal. In this study the embedding theory of Takens [28] and the theory of SVD analysis [29] can be used for the discrimination of deterministic and noisy (stochastic) components included in the observed signals, as well as for the discrimination of distinct dynamical components. We used the realization of the Monte Carlo Singular Spectrum Analysis (Growth, 2015) based on the embedding theory. 3. Experimental data analysis and results 3.1. Device and measurement methodology As already mentioned in the Introduction, the device (DUT), tested for its properties in the frame of noise analysis, was an ultrathin body and buried-box, fully-depleted, silicon-on-insulator (UTBB FD-SOI) MOSFET nanodevice (Fig. 1), which was fabricated by STMicroelectronics in France. The MOSFET’s channel width was W = 0.5 µm and its channel length L = 30 nm, while its silicon film thickness was tSi = 7 nm [15]. The equivalent front gate oxide thickness (EOT) was tox = 1.55 nm (TiN/HfSiON stack) and the box thickness 25 nm. For the demonstrated noise to be studied and evaluated, time-domain, drain-current Id measurements were conducted at room temperature, using an Agilent BT1500 semiconductor analyzer (with the option B1530WGFMU), in front gate mode (back gate bias Vbg = 0 V). The measured drain-current Id timeseries was consisted of 105 points (samples) and the preferred sampling frequency, for the proper registration and representation of the timeseries dynamics, was selected to be 20 kHz. In this case, like in [15] and [11] the studied noise was demonstrated by stressed samples of the considered double-trap nanoscale UTBBFDSOI MOSFET, i.e. MOSFETs that had been technically aged. 3.2. Description of experimental time series In Fig. 3 the measured timeseries of the drain current Id is shown for bias conditions: Vd = 30 mV, Vbg = 0 V in three different cases for (a) Vg = 260 mV, (b) Vg = 450 mV and (c) Vg = 600 mV; it is noted that the gate bias voltage Vg was considered as the control parameter. Looking at Fig. 3, it is apparent that the noise demonstrated by the stressed UTBB FD-SOI) MOSFET nanodevice appears to have a rather distinct level structure, thus appearing to be a Random Telegraph Noise (RTN) of some form. The original experimental timeseries for different Vg values, contains distinct dynamical components, as it has been shown in a previous study of the series [10]. For this reason, and in order to extract and study the distinct dynamical components of the drain current timeseries, SVD analysis of all three timeseries was applied, as described above. For the Theiler embedding of each of the time series we used the proper delay times τ (at which mutual information drops to approximately 0.1). In this study we restricted our non-extensive statistical theory analysis to the first principal component which contains the main dynamics. (See Fig. 4.) 3.3. Tsallis q-triplet estimation 3.3.1. Tsallis qsen estimation Fig. 5 presents the singular spectrum function f (α ) vs. α estimated experimentally for the three timeseries (blue line). The green line corresponds to the theoretically predicted f (α ) in accordance with the previously described method developed by Arimitsu and Arimitsu [18,26]. The red line corresponds to the theoretically predicted f (α ) function by using the p-model method according to intermittent turbulence theory, as was previously described. It is evident that Tsallis q-entropy principle can accurately describe the experimental curve and the p-model predicted curve. This means

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Fig. 3. The measured drain-current timeseries for three value of the control parameter: (a) Vg = 0.260 V, (b) Vg = 0.450 V and (c) Vg = 0.600 V.

Fig. 4. The first SVD principal component of the original timeseries: (a) Vg = 0.260 V, (b) Vg = 0.450 V and (c) Vg = 0.600 V.

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Fig. 5. Multifractal spectrum ′ for the first SVD principal component of the original timeseries: (a) Vg = 0.260 V, (b) Vg = 0.450 V and (c) Vg = 0.600 V. The blue line is the experimentally estimated singular spectrum f (α ) and the green dotted line is the predicted spectrum by using Tsallis q-entropy principle. The red line corresponds to the predicted spectrum by using p-model method.

Table 1 Values of characteristic properties of the multifractal exponent spectra for the different time series. Vg (V)

αmin

αmax

∆α

A

0.260 0.450 0.600

0.834 ± 0.007 0.853 ± 0.015 0.816 ± 0.011

1.239 ± 0.029 1.176 ± 0.019 1.169 ± 0.022

0.405 ± 0.030 0.323 ± 0.024 0.354 ± 0.024

0.799 ± 0.051 0.931 ± 0.056 1.220 ± 0.073

that the multifractal structure of the intermittent turbulence of the dynamics of the MOSFET electric current flow can be reproduced and physically described by the extremization of Tsallis q-entropy function. In Table 1 we summarize the characteristic properties of the multifractal analysis include in the singular spectrum function f (α ) as shown in Fig. 5 and described previously. The first two columns show the minimum and maximum values of the singular exponents a in respect for the three values of the control parameter Vg . αmin and αmax correspond to the zeros of the singular spectrum function f (α ). The third column shows the width of the f (α ) spectrum ∆α = αmax − αmin and the fourth column shows the skewness A of f (α ) estimated by using Eq. (24). Fig. 6(a, b, c, d) depicts the respective quantities which are plotted vs. V g . As we observe, the values of αmax decrease as Vg increases. The values of amin also show a tendency to decrease (if one compares their values between at Vg = 0.260 and Vg = 0.600) although at Vg = 0.450

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Fig. 6. Plots of the characteristic properties of the multifractal exponent spectra for the different time series (a) αmin , (b) αmax , (c) ∆α = αmax − αmin , (d) A.

αmin slightly increases. This suggests that the topology of the phase space of MOSFET dynamics mirrored in the first SVD component on the overall becomes more singular, according to Holder exponent theory, as the drain voltage and current increase. Although the topology becomes more singular, the degree of multi-fractality profile depends clearly on the control parameter, however not monotonically, as ∆α decreases between values of Vg = 0.260 and Vg = 0.450 and increases between Vg = 0.450 and Vg = 0.600. On the other hand, A clearly increases with Vg which means that the percentage of singular exponents lower than αo (which corresponds to the maximum value of f (α )) increases. Therefore, the increase of A strengthens the proposition that the topology of phase space becomes more and more anomalous and singular. The qsen parameter of the Tsallis q-triplet can be estimated from amin and amax by using Eq. (19). Comparison between the obtained values of qsen for the different time series is discussed later in this section. Fig. 7 presents the generalized dimension spectrum Dq vs. q for the three timeseries. The red line to Vg = 0.260 V, the blue line to Vg = 0.450 V and the green line to Vg = 0.600 V. We can clearly observe that at negative values of q the generalized dimensions Dq decrease monotonically as Vg increases. For positive values of q, we observe a similar decrease of the dimensions between Vg = 0.260 V and Vg = 0.600 V timeseries. Table 2 presents the characteristic quantities of the generalized dimension spectrum Dq vs.qas a function of the control parameter Vg , i.e. the dimensions at very low and very high values of parameter q, D−∞ and D∞ . The respective quantities D−∞ , D∞ are plotted vs. Vg in Fig. 8. We can see that both of the dimensions decrease with Vg which means that the overall average dimension of the multifractal phase space decreases as voltage increases. This reduction of dimensionality of the multifractal space is in agreement with the reduction of the singular exponent values presented in Table 1 and Fig. 6, as the reduction of dimensionality strengthens the anomalous character of the phase space topology. Table 3 shows the relaxation lag times estimated by the mutual information time evolution for each of the different timeseries. We show three characteristic lag times: τe is the time for the mutual information to drop to 1/e of the initial value, τ0.l is the time to drop to 10% of the initial value and τ0 is the time to drop to the stationary state. The dependence of the relaxation times on Vg is also shown graphically in Fig. 9. We can see that the relaxation times increase with Vg ,

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Fig. 7. Generalized dimension spectrum Dq vs. q for the first SVD principal component of the original timeseries: (a) Vg = 0.260 V, (b) Vg = 0.450 V and (c) Vg = 0.600 V.

Table 2 Characteristic quantities of the Dq v s. q spectrum for each of the time series. Vg (V)

D−∞

D∞

∆ Dq

0.260 0.450 0.600

1.20 ± 0.01 1.14 ± 0.01 1.12 ± 0.01

0.88 ± 0.005 0.89 ± 0.005 0.85 ± 0.005

0.32 ± 0.011 0.24 ± 0.011 0.27 ± 0.011

Table 3 Relaxation lag times obtained from characteristic values of mutual information of the different timeseries. Vg (V)

τe

τ0.l

τ0

0.260 0.450 0.600

9 15 62

21 25 387

100 600 2000

slower between Vg = 0.260 V and Vg = 0.450 V and much more steeply between Vg = 0.450 V and Vg = 0.600 V. This is in agreement with the previous results for the singular spectrum f (α ) and generalized spectrum Dq , because as the dimensionality is reduced the system becomes more self-organized and long-range correlations in the dynamics are strengthened which results in a slower relaxation process. In Fig. 10 we present the best lnq I(τ ) vs. τ fitting of the mutual information function for three different timeseries of Vg . The linear part of these curves describes the q-exponential power law corresponding to the q-relaxation process of the system dynamics. With the red circles, we emphasize the power law fitting where used for the estimation of qrel index, according to the relation qrel = s−s 1 , where s is the slope of power law fitting. The results showed that the qrel index was found to be qrel = 1.15 ± 0.03 for the Vg = 0.260 V timeseries (Fig. 10a). For the Vg = 0.450 V timeseries the qrel index was found to be qrel = 3.08 ± 0.04 (Fig. 10b), and for the Vg = 0.600 V timeseries, the qrel index was found to be qrel = 3.58 ± 0.04 (Fig. 10c). Therefore, for all timeseries, Tsallis parameter qrel remains different than one. This reveals a non-Gaussian relaxation process of the system to its meta-equilibrium state. In Fig. 11, we present the results concerning Tsallis statistics for the three timeseries of Vg . In particular, in the left column of Fig. 11 we present the best linear correlation (red line) between lnq [p(xi )] (open blue circles) and (xi )2 . For the timeseries of Vg = 0.260, the best fitting was found for the value of qstat = 1.26 ± 0.06, with correlation coefficient cc = 0.9611, (Fig. 8a). This value was used to estimate the q-Gaussian distribution presented in Fig. 8b by the solid red line, where the difference between the q-Gaussian and the Gaussian PDF (green line) in long tails is clearly pictured, in a log[p(xi )] vs. xi graph. The open blue circles correspond to the experimental de-trended timeseries. For the timeseries of Vg = 0.450, the best fitting was found for the value of qstat = 1.46 ± 0.05, with correlation coefficient cc = 0.9801, (Fig. 8c). This q-Gaussian distribution is depicted in Fig. 8d by the solid red line in a log[p(xi )] vs.xi graph. For the timeseries corresponding to Vg = 0.600 V, the best fitting was found for the value of qstat = 1.62 ± 0.03 with correlation coefficient

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Fig. 8. (a) D−∞ (b) D∞ and (c) ∆Dq for the different timeseries.

cc = 0.9933 (Fig. 8e). This value, corresponding to a q-Gaussian distribution, is presented in Fig. 8f by the solid red line in a log[p(xi )] vs. xi graph. Table 4 summarizes the q-triplet values for different values of Vg also shown graphically in Fig. 12. As we can see, qstat steadily increases with Vg , while qrel increases only slightly from Vg = 0.260 to Vg = 0.450, but more intensely from Vg = 0.450 to Vg = 0.600. The qsen increases clearly between Vg = 0.260 and Vg = 0.600, although it decreases between Vg = 0.260 and Vg = 0.450. The increase of the qstat parameter suggests a clear departure from the Gaussian character of the dynamics as the non-extensivity of the statistics is strengthened. Comparing the values of the q-triplet parameters between Vg = 0.260 V and Vg = 0.600 V, it becomes clear that there is a strong phase transition that transforms the dynamics revealing a strong intermittent multi-scale and long-range correlated turbulence character. 3.4. Tsallis q-entropy estimation Fig. 13 shows the calculated values of Tsallis q-entropy for the SVD first principal component of the three timeseries.

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Fig. 9.

The mutual information relaxation lag times vs. the control parameter Vg according to Table 3. (a) τe , (b) τ0.l and (c)τ0 .

Table 4 Tsallis q-triplet values for the different timeseries. Vg (V)

qstat

qrel

qsen

0.260 0.450 0.600

1.29 ± 0.06 1.46 ± 0.05 1.62 ± 0.03

1.15 ± 0.03 3.08 ± 0.04 3.58 ± 0.04

−1.55 ± 0.25 −2.11 ± 0.29 −1.7 ± 0.23

The value of q-entropy corresponds to the meta-equilibrium stationary states. As shown in Fig. 13, the q-entropy clearly decreases with increasing Vg . This suggests that the meta-equilibrium stationary state becomes more organized as Vg increases which means that effectively fewer degrees of freedom are available to the dynamics as the entropy decreases. This is in accordance with the basic theoretical framework of Tsallis theory [12] and our previous results concerning the qstat and qrel parameters which are found to increase as Vg increases: as qstat increases the long-range correlations become stronger causing the lowering of the entropy. On the other hand, the increase in qrel is the direct result of the slower relaxation rate caused by the decrease in the available degrees of freedom as Vg increases. 3.5. Hurst exponent estimation In Fig. 14 we present the calculated values of the Hurst exponent corresponding to the SVD first principal component of the timeseries vs. Vg . The values of the Hurst index are very high (near 1) for all values of Vg . Since H is much higher than 1/2, the random walk related to MOSFET noisy electric charge transport represents a persistent anomalous super-diffusion process. Given that the statistics of the dynamics is non-Gaussian, we conclude that the transport process is a fractional dynamical phenomenon. In the Discussion section we explain the profile of Hurst exponent as the result of a percolation process.

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Fig. 10. The qrel parameter estimated for the three time series by using the mutual information vs. time curves: (a) Vg = 0.260 V, (b) Vg = 0.450 V and (c) Vg = 0.600 V.

4. Summary of data analysis 4.1. Tsallis q-entropy estimation

• The estimated values of Tsallis q-entropy S q decrease with increasing control parameter Vg . This fact is evidence for the strengthening of the self-ordering and self-organized character of the system as Vg increases. 4.2. Hurst exponent estimation

• The value of the Hurst exponent H was found to be almost constant for all three series at ∼0.986. This corresponds to a negative connectivity index θ revealing a strong super-diffusive process of the electric current flow inside the MOSFET. 4.3. Multifractal analysis — qsen parameter

• For the three timeseries, we have clearly observed the multifractal character of the phase-space dynamics. The qsen parameter was nearly constant-within error- (qsen = −2.11 to −1.55) as the control parameter changes. However, the multifractal characteristics described by αmax and the skewness parameter A, reveal a topological phase transition process as the control parameter increases. The skewness parameter changes clearly between Vg = 0.260 and Vg = 0.450 as well as between Vg = 0.450 and Vg = 0.600 showing a multiple topological phase transition process. • The fact that αmax decreases and skewness A increases with Vg indicates that the topology of the phase space becomes more singular as Vg increases, because the lower values of the singular exponents α (Holder exponents) become more pronounced.

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Fig. 11. Index qstat for the three timeseries of Vg . Figures in the left column shows linear correlation lnq [p(xi )] vs. (xi )2 , while figures in the right column shows log[p(xi )] vs. xi , with q-Gaussian function that fits p(xi ) for each timeseries. The values of qstat in Table 4 used to estimate the q-Gaussian distribution presented (red line), where the difference between the q-Gaussian and the Gaussian PDF (green line) in long tails is clearly pictured.

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Fig. 12. Plots of Tsallis q-triplet values for the different timeseries. (a) qstat , (b) qrel and (c) qsen .

• The experimental f (α ) spectral function is accurately reproduced by the intermittent turbulence p-model theory. • In accordance with the previous two points, the generalized dimensions Dq of the phase space dynamics decrease for all values of q, which is evidence that system dynamics becomes, on the average, more singular at all scales in phase space. Especially, the value of D−∞ (corresponding to the regions of large dimensions in phase space) and the value of D∞ (corresponding to the regions of small dimensions in phase space) show a clear decrease as Vg increases.

• Moreover, the increasing of the asymmetry in the skewness of the spectrum f (α ) towards the lowest α ’s (as the parameter A increases) clearly reveals that the regions of phase space with low dimension (D∞ ) become more abundant relative to the high-dimension regions (D−∞ ).

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Fig. 13. Tsallis q-entropy as a function of the control parameter Vg .

Fig. 14. Hurst exponent as a function of the control parameter Vg .

• The total profile of Dq vs. q curves provide further evidence of topological phase transition process as Vg increases. • The extremization of Tsallis q-entropy can faithfully explain the intermittent spatio-temporal turbulence structure of the MOSFET current flow. 4.4. Relaxation process — qrel parameter

• The characteristic relaxation times τe , τ0.1 and τ0 , clearly increase as Vg increases. In particular, we see a very sharp increase in all τ ’s between Vg = 0.450 V and Vg = 0.600 V. This suggests that the degrees of freedom of the system dynamics decrease as the Vg increases. This is in accordance with the lowering of the generalized dimension spectrum Dq lowering discussed above.

• The mutual information vs. time profile reveals a clear region with power law relaxation in accordance with the Tsallis q-relaxation process.

• The qrel parameter also increases with Vg in accordance with the q-exponential function profile (2) corresponding to the q-relaxation process. 4.5. Meta-equilibrium stationary states — qstat parameter

• The probability distribution function clearly reveals non-Gaussian character for all values of Vg . In all cases, Tsallis q-exponential probability function was found, especially for Vg = 0.450 V and Vg = 0.600 V. • The estimation of qstat parameter corresponding to the Tsallis probability distribution was larger than 1 and increases as Vg increases.

• The above results provide evidence for the strengthening of meta-equilibrium non-extensive statistics as well as the existence of a topological phase transition process as Vg increases. 5. Discussion In the following, we try to provide a deeper theoretical interpretation of the results presented in the previous section concerning: (a) the non-equilibrium thermodynamic character of the MOSFET dynamics, (b) the anomalous diffusion

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character of the dynamics, (c) multi-fractal structuring of the phase-space and the intermittent turbulent character of the dynamics and (d) the non-extensive statistics of the dynamics. More specifically, this study has shown strong non-extensive character of the drain current dynamics of a previously stressed MOSFET, which is a result of strongly non-linear deterministic dynamics, as was also shown in other studies [10]. The non-extensive character described by the Tsallis q-triplet parameters is related to strange dynamics, anomalous diffusion process, fractional dynamics mirrored in fractional Langevin and Fokker–Planck equations [30–34]. All these theoretical concepts describe the complex and self-organized character of the MOSFET drain current dynamics and are related to the fractal topology of the phase-space [35]. Also, the Tsallis non-extensive character of the system dynamics can be related to percolation self-organized states included in the multi-fractal structure of the phase-space, which create long-ranged and multi-scale spatio-temporal correlations [36–45]. This phenomenon indicates the existence of intermittent multi-fractal turbulence states in the nano-scale spatio-temporal current distribution inside the transistor. This spatially distributed intermittent turbulent dynamics is mirrored in the experimental timeseries representing the local measurement of the drain current at the transistor output. Apart from the above, our results show a topological change of the attractor of the MOSFET current dynamics as the control parameter (drain Voltage Vg ) grows. This is supported by the significant change of the multi-fractal spectra and the Tsallis q-triplet parameter values with Vg . Moreover, the topological change is clearly towards a more singular fractal structure of the attractor which is an implication that the system is moving towards more ordered stationary states. The latter implication is further supported by the fact that the values of the maximal Tsallis q-entropy function decrease with Vg . (At thermodynamic equilibrium the statistics of the system dynamics is described by the maximization of the Boltzmann–Gibbs entropy function, whereas at non-equilibrium state the statistics of the dynamics and corresponding multi-fractal structuring of the dynamical phase-space is produced by the maximization of the Tsallis q-entropy function). This means that, as Vg increases, the system transforms into more self-organized states, which is consistent with the fact that the Tsallis qrel and qstat parameters were also found to increase with Vg . The decrease of the q-entropy corresponds to a more ordered state of the system, i.e. fewer available degrees of freedom, stronger long-ranged multi-scale spatio-temporal correlations and a slower relaxation process. Non-linear dynamics can create fractal structuring of the phase-space and global correlations in the non-linear distributed system. For non-extensive dynamical systems, the entire phase space is dynamically not entirely occupied (the system is not ergodic) but only a scale-free-like part of it is visited yielding a long-standing multi-fractal-like occupation. The fractal or multi-fractal structuring of the phase space makes the effective number Weff of possible states, namely those whose probability is non-zero, to be smaller than the total number of states W. The fractal or multi-fractal structuring of the phase space can be highly complicated including trapping and flight of the dynamics creating strange dynamics and an anomalous diffusion process. Anomalous diffusion can be described by a fractional Fokker–Planck–Kolmogorov equation [46]:

∂βP ∂α 1 ∂ 2α = (AP) + (BP) β α ∂t ∂ (−ξ ) 2 ∂ (−ξ )2α

(40)

where P(x, t) is the probability density of the state x at time t. The critical exponents α and β correspond to the fractal dimension of the spatio-temporal non-Gaussian distributions. The parameters A, B are accordingly defined as velocity and diffusion rate constant parameters in respect [46]. The above fractional equation describes an anomalous transport process in the fractal topology of the phase space, i.e. a fractal random walk process, where the mean-square displacement is given by the equation:

⟨

⟩ ∆x2 = 2Dt µ

(41)

where µ = 2H = β/α , the ratio of the spatial and temporal fractal coefficients β and α in respect [30]. For normal diffusion, the random walk occurs on the normal Euclidean space and µ = 1, yielding the well-known result according to which the mean-square displacement is directly proportional to time. On the other hand, the anomalous diffusion process is related to an intermittent turbulence process described by the Hurst exponent H, which is related to the coefficient µ simply by µ = 2H. We conclude that H = β/2α

(42)

For classical diffusion process β = α =1 and H = 1/2. As our experimental result for H was found to be much higher than 21 for all three values of the control parameter Vg , we understand that MOSFET dynamics is characterized by an anomalous diffusion random-walk process described by a fractional Fokker–Planck–Kolmogorov equation. Moreover, the experimental value of H was found to be much higher than 1/2, which implies a super-diffusive (persistent) fractional transport process of the charge carriers inside the MOSFET. According to non-equilibrium renormalization group critical theory, the transition of the system from one free energy minimum to another is accompanied with a reduction of dimensionality, increase of self-organization and development of long-ranged spatio-temporal correlations in the current distribution inside the transistor. Evidence for this concept is provided by our experimental results and specifically (a) the lowering of the generalized dimension spectrum Dq and (b) the increase of the qstat , qrel and skewness A parameters. As the non-extensivity and self-organization becomes stronger,

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Fig. 15. Schematic representation of the meta-stable steady states of free energy vs. control parameter.

the meta-equilibrium stationary states must correspond to free energy local minima at higher values of free energy. This is shown schematically in Fig. 15. As dimensionality of the system decreases, the relaxation process becomes slower and slower because the available degrees of freedom for the energy to disperse are fewer. The results of this study provide evidence for an anomalous diffusion process related with multifractal intermittent turbulence structure in the effective flow of electric charges in the MOSFET. Milovanov & Rasmussen [16] and Milovanov 2010 [17] provided a model explaining anomalous flows in disordered solids by using percolation theory and SelfOrganized-Criticality (SOC) on the dynamics of the electric charge motion inside the solid. In their study they found that for a percolating random walk on a fractal geometry the Hurst index is raising from a value of 0.5 in infinite embedding dimension of space down to a value of 1 as the embedding dimension approaches 1. The values of the Hurst index for all the timeseries in this study were found to be very close to 1 (around 0.98) which is justified by the fact that the geometry of space for the current flow is (nearly) one-dimensional since the MOSFET channel is long and thin (Fig. 1). In conclusion, our results confirm the richness of the deterministic strange dynamics of the stressed UTBB FD-SOI MOSFET drain current noisy signal. As described in previous works [10,10], the strange dynamics are attributed to the possible charge exchange between the gate oxide and the semi-conducting channel. More precisely, the aged (or previously stressed) MOSFET is assumed in [10] to have developed structural defects inside the gate material that act as traps of electrons flowing in the conducting channel. What is revealed in this paper is the fact that the interchange between trapping and release of electrons is essentially a self-organized critical nonlinear phenomenon causing the ‘‘avalanche’’ effect that produces the current bursts in the timeseries. The existence of more than one trap and the interactions among the trapped electrons are further assumed to be responsible for the multi-fractal character of the dynamics and the non-Gaussian statistics of the noisy signal. Further investigation of the physical mechanism behind the deterministic complex behavior of the noisy behavior arising in stressed UTBB FD-SOI MOSFET dynamics is required, as well as theoretical modeling and simulation of the physical system in order to fully understand the underlying principles. This is intended to be part of future work. Acknowledgments The authors would like to thank Prof. G. Ghibaudo for providing the measurements and Prof. C. Dimitriadis for fruitful discussions. References [1] [2] [3] [4]

P.H. Young, Electronic Communication Techniques, fourth ed., Prentice Hall, 1999. J.G. Fossum, V.P. Trivedi, K. Wu, Extremely scaled fully depleted SOI CMOS, in: Proc. IEEE Int. SOI Conf., Oct. 2002, pp. 135–136. T. Skotnicki, Competitive SOC with UTBBSOI, in; Proc. IEEE Int. SOI Conf., Oct. 2011, pp. 1–61. C.G. Theodorou, E.G. Ioannidis, F. Andrieu, T. Poiroux, O. Faynot, C.A. Dimitriadis, G. Ghibaudo, Low-frequency noise sources in advanced UTBB FD-SOI MOSFETs, IEEE Trans. Electron Devices 61 (4) (2014) 1161–1167. [5] E. Simoen, M. Aoulaiche, S.D. dos Santos, J.A. Martino, V. Strobel, B. Cretu, J.-M. Routoure, R. Carin, A.L. Rodríguez, J.A.J. Tejada, C. Claeys, Low-frequency noise studies on fully depleted UTBOX silicon-on-insulator nMOSFETs: Challenges and opportunities, ECS J. Solid-State Sci. Technol. 2 (2013) Q205–Q210. [6] L. Gerrer, S.M. Amoroso, R. Hussin, A. Asenov, RTN distribution comparison for bulk, FDSOI and FinFETsdevices, Microelectron. Reliab. 54 (2014) 1749–1752.

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