Mathematical model of Boltzmann’s sigmoidal equation applicable to the set-up of the RF-magnetron co-sputtering in thin films deposition of 
                $$\hbox {Ba}_{{x}}\hbox {Sr}_{1-{x}} \hbox {TiO}_{3}$$
                
                    
                                    
                        
                            
                                Ba
                                x
                            
                            
                                Sr

Reséndiz-Muñoz, Juan; Corona-Rivera, Miguel Á.; Fernández-Muñoz, J. L.; Zapata‐Torres, M.; Márquez-Herrera, A.; Ovando‐Medina, Víctor M.

doi:10.1007/s12034-017-1441-x

Cited by 17 publications

(10 citation statements)

References 7 publications

(9 reference statements)

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“…Furthermore, if the development changes from continuous to discontinuous at the inflection point, the inflection point may correspond to the critical stage. The sigmoidal mathematical model (proposed by Boltzmann in 1879) was based on the following sigmoidal logistic Equation (1) [10]:…”

Section: Mathematical Modelmentioning

confidence: 99%

“…Equation 1has been used to describe behaviors exhibited when a certain factor triggers a transition (reversible or not) from a steady state to another one [10].…”

Section: Mathematical Modelmentioning

confidence: 99%

“…The effects of the stoichiometry defined by x and 1 − x parameters into the BST chemical formula have been investigated in some physical properties such as: Resistive switching [5,6], resistivity (ρ) in crystalline and amorphous phases [7][8][9], thickness [10], and band gap energy (E g ) [11][12][13].…”

Section: Introductionmentioning

confidence: 99%

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Temperature–Power Simultaneous Effect on Physical Properties of BaxSr1−x TiO3 Thin Films Deposited by RF–Magnetron Cosputtering for 0 ≤ x ≤1

et al. 2018

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The combined effect on the variation of the in-situ deposition temperature and the variation of the applied power on the deposition rate (DR), gap energy (Eg), and resistivity (ρ) in barium strontium titanate thin films, deposited into RF (radio frequency)–magnetron cosputtering equipment, are presented in this research. The simultaneous action of two magnetrons (BaTiO3 and SrTiO3) is explained using the first and second derivative of Boltzmann’s sigmoidal equation. This found that a deposition gradient is a very novel analysis. Using the color-code lines built through MATLAB® and analyzing the trend information, taking into account the influence of the calculated “x” parameter, by means of the Boltzmann’s sigmoidal equation fit, we propose a method to set up an RF–magnetron cosputtering system to predict the DR(x,T), Eg(x,T), and ρ(x,T) values of BaxSr1−xTiO3 solid solutions with 0 ≤ x ≤ 1 for amorphous and crystalline phases. This method can be a versatile tool to optimize the deposition process with, or without, in situ deposition temperature.

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Section: Mathematical Modelmentioning

confidence: 99%

“…Equation 1has been used to describe behaviors exhibited when a certain factor triggers a transition (reversible or not) from a steady state to another one [10].…”

Section: Mathematical Modelmentioning

confidence: 99%

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Temperature–Power Simultaneous Effect on Physical Properties of BaxSr1−x TiO3 Thin Films Deposited by RF–Magnetron Cosputtering for 0 ≤ x ≤1

et al. 2018

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“…The x and 1−x as a function of the r.f. power data can be coupled to the Boltzmann equation, which allows for simpler analysis [23].…”

Section: Resultsmentioning

confidence: 99%

Local hardening of Raman phonons in Ba_xSr_1−xTiO₃ thin films deposited by r.f. sputtering

Zelaya-Ángel

Meléndez‐Lira

Reséndiz-Muñoz

et al. 2020

Mater. Res. Express

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Thin films of nanometric order of a solid solution of barium-strontium titanate Ba x Sr 1−x TiO 3 (BSTO), in the entire 0x1 range, were deposited at 495°C on quartz by RF reactive cosputtering. The composition was controlled through the RF power (P) applied to the targets. X ray diffractograms (XRD) reveal that all the samples show the cubic perovskite crystalline structure. The percentage of elements (x) in the films were evaluated by usingof electron dispersion spectroscopy, also through the change of the (110) lattice interplanar spacing determination of BSTO by means of XRD patterns. Experimental data points of concentration of Ba (x) and Sr (1−x) as a function of P describe a symmetric sigmoidal curve. The average crystal size for the whole composition is around 20 nm as estimated from XRD data. Raman measurements in the 100-900 cm −1 range show that optic phonons experience local hardening in the x interval 0.6-0.8. Around this region of x it is, in general, expected in BSTO a crystalline cubic-tetragonal phase transition when is prepared at low substrates temperatures. Distortion of the unit cell is observed in that interval, which probably originates the local hardening of phonons.

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“…Also, whether the transition from continuous to discontinuous at inflection point, the inflection point will lead to the critical phase. To model this pattern, Boltzmann have suggested equation (Reséndiz-Muñoz et al (2017)) was based on the sigmoidal equation of logistics:…”

Section: Sigmoidal-boltzmann Equationmentioning

confidence: 99%

Mathematical model of Boltzmann’s sigmoidal equation applicable to the spreading of the coronavirus (Covid-19) waves

Aferni¹,

Guettari²,

Tajouri³

2020

Environ Sci Pollut Res

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Currently, investigations are intensively conducted on modeling, forecasting, and studying the dynamic spread of coronavirus (Covid-19) new pandemic. In the present work, the sigmoidal-Boltzmann mathematical model was applied to study the Covid-19 spread in 15 different countries. The cumulative number of infected persons I has been accurately fitted by the sigmoidal-Boltzmann equation (SBE), giving rise to different epidemiological parameters such as the pandemic peak t p , the maximum number of infected persons I max , and the time of the epidemic stabilization t ∞. The time constant relative to the sigmoid Δt (called also the slope factor) was revealed to be the determining parameter which influences all the epidemiological parameters. Empirical laws between the different parameters allowed us to propose a modified sigmoidal-Boltzmann equation describing the spread of the pandemic. The expression of the spread speed V p was further determined as a function of the sigmoid parameters. This made it possible to assess the maximum speed of spread of the virus V pmax and to trace the speed profile in each country. In addition, for countries undergoing a second pandemic wave, the cumulative number of infected people I has been successfully adjusted by a double sigmoidal-Boltzmann equation (DSBE) allowing the comparison between the two waves. Finally, the comparison between the maximum virus spread of two waves V p max 1 and V p max 2 showed that the intensity of the second wave of Covid-19 is low compared to the first for all the countries studied.

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Mathematical model of Boltzmann’s sigmoidal equation applicable to the set-up of the RF-magnetron co-sputtering in thin films deposition of $$\hbox {Ba}_{{x}}\hbox {Sr}_{1-{x}} \hbox {TiO}_{3}$$ Ba x Sr

Cited by 17 publications

References 7 publications

Temperature–Power Simultaneous Effect on Physical Properties of BaxSr1−x TiO3 Thin Films Deposited by RF–Magnetron Cosputtering for 0 ≤ x ≤1

Temperature–Power Simultaneous Effect on Physical Properties of BaxSr1−x TiO3 Thin Films Deposited by RF–Magnetron Cosputtering for 0 ≤ x ≤1

Local hardening of Raman phonons in Ba_xSr_1−xTiO₃ thin films deposited by r.f. sputtering

Mathematical model of Boltzmann’s sigmoidal equation applicable to the spreading of the coronavirus (Covid-19) waves

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Mathematical model of Boltzmann’s sigmoidal equation applicable to the set-up of the RF-magnetron co-sputtering in thin films deposition of $$\hbox {Ba}_{{x}}\hbox {Sr}_{1-{x}} \hbox {TiO}_{3}$$ Ba x Sr

Cited by 17 publications

References 7 publications

Temperature–Power Simultaneous Effect on Physical Properties of BaxSr1−x TiO3 Thin Films Deposited by RF–Magnetron Cosputtering for 0 ≤ x ≤1

Temperature–Power Simultaneous Effect on Physical Properties of BaxSr1−x TiO3 Thin Films Deposited by RF–Magnetron Cosputtering for 0 ≤ x ≤1

Local hardening of Raman phonons in BaxSr1−xTiO3 thin films deposited by r.f. sputtering

Mathematical model of Boltzmann’s sigmoidal equation applicable to the spreading of the coronavirus (Covid-19) waves

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Product

Resources

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Local hardening of Raman phonons in Ba_xSr_1−xTiO₃ thin films deposited by r.f. sputtering