Evaluation of alignment target designs for Cu and low-K dual damascene processes

Mukherjee-Roy, Moitreyee; Singh, Navab; Mehta, Sohan Singh; Chik, Wai Meng; Sim, Chin Tiong; Cheong, Francis

doi:10.1117/12.482800

Cited by 4 publications

(3 citation statements)

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“…, Rosen-Morse [26], pseudo-harmonic [27], squared tangent well [28], hyperbolic [29], position-dependent mass Schrödinger equation [30,31], infinite circular well [32], hyperbolic double-well (DW) potential [33], etc. Recently, entropic measures were successfully engaged to understand trapping and oscillation of a particle within symmetric, asymmetric DW potential [34,35], confined quantum harmonic oscillator [36], CHA [9,10], etc.…”

Section: Introductionmentioning

confidence: 99%

Information-entropic measures for non-zero l states of confined hydrogen-like ions

Mukherjee

Roy

2018

Eur. Phys. J. D

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Rényi entropy (R), Tsallis entropy (T ), Shannon entropy (S), and Onicescu energy (E) are studied in a spherically confined H atom (CHA), in conjugate space, with special emphasis on non-zero l states. This work is a continuation of our recently published work [1]. Representative calculations are done by employing exact analytical wave functions in r space. Accurate p space-wave functions are generated numerically by performing Fourier transform on respective r-space counterparts.Further, these are extended for H-isoelectronic series by applying the scaling relations. R, T are evaluated by choosing the order of entropic moments (α, β) as ( 3 5 , 3) in r and p spaces. Detailed, systematic results of all these measures with respect to variations of confinement radius r c are offered here for arbitrary n, l quantum numbers. For a given n, at small r c , R α r , T α r , S r collapse with rise of l, attain a minimum, then again grow up. Growth in r c shifts the point of inflection towards higher l values. An increase in Z enhances localization of a particular state. Several other new interesting inferences are uncovered. Comparison with literature results (available only for S in 2p, 3d states), offers excellent agreement. Confinement of an atom or molecule inside an impenetrable cavity was first studied in the fourth decade of twentieth century [2]. Progress of research on such quantum systems was reviewed several times [2-5] recording their importance in both fundamental physics and chemistry as well as in various engineering branches. They have relevance in many different physical situations, e.g., atoms under plasma environment, impurities in crystal lattice and semiconductor materials, trapping of atoms/molecules in zeolite cages or inside an endohedral cobweb of fullerenes, quantum wells, quantum wires, quantum dots [6] and so forth. Furthermore, such models were designed to mimic the high pressure environment inside the core of planets. Also, they have contemporary significance in interpreting various astrophysical phenomena [7] and many other interesting areas.Theoretical study of a Hydrogen atom within an infinite spherical cavity was first published in 1937 [8]. Over the years, this simple confined hydrogen atom (CHA) model has served as a precursor to improve our understanding about the consequences of confinement in atomic electronic structure. In last decade, a CHA under the influence of various restricted environment has been extensively followed. Majority of these investigations include trapping of H atom either in a spherical box of penetrable, impenetrable walls or inside a hard box of different geometrical shape and size [5,[9][10][11][12]. In the realm of atomic physics, CHA provides us with many attractive physical and chemical properties. Numerous theoretical methods like perturbation theory, Padé approximation, WKB method, Hypervirial theorem, power-series solution, Lie algebra, Lagrange-mesh method, asymptotic iteration method, generalized pseudo-spectral (GPS) method were invoked for their prop...

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Section: Introductionmentioning

confidence: 99%

Information-entropic measures for non-zero l states of confined hydrogen-like ions

Mukherjee

Roy

2018

Eur. Phys. J. D

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“…In recent years, S is examined in a number of systems, such as, Pöschl-Teller [44], Rosen-Morse [45], pseudo-harmonic [46], squared tangent well [47], hyperbolic [48], position-dependent mass Schrödinger equation [49,50], infinite circular well [51], hyperbolic double-well (DW) potential [52], etc. Recently, some of these measures have been found to be quite efficient and useful to explain the oscillation and localization-delocalization behavior of a particle in symmetric and asymmetric DW potential [53,54], as well as in a confined 1D quantum harmonic oscillator [14]. IE quantifies the spatial delocalization of single-particle density of a system in several complimentary ways.…”

Section: Introductionmentioning

confidence: 99%

Information‐Entropic Measures in Confined Isotropic Harmonic Oscillator

Mukherjee

Roy

2018

Advcd Theory and Sims

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Information based uncertainty measures like Rényi entropy (R), Shannon entropy (S) and Onicescu energy (E) (in both position and momentum space) are employed to understand the influence of radial confinement in isotropic harmonic oscillator. The transformation of Hamiltonian in to a dimensionless form gives an idea of the composite effect of oscillation frequency (ω) and confinement radius (r c ). For a given quantum state, accurate results are provided by applying respective exact analytical wave function in r space. The p-space wave functions are produced from Fourier transforms of radial functions. Pilot calculations are done taking order of entropic moments (α, β)as ( 3 5 , 3) in r and p spaces. A detailed, systematic analysis is performed for confined harmonic oscillator (CHO) with respect to state indices n r , l, and r c . It has been found that, CHO acts as a bridge between particle in a spherical box (PISB) and free isotropic harmonic oscillator (IHO). At smaller r c , E r increases and R α r , S r decrease with rise of n r . At moderate r c , there exists an interaction between two competing factors: (i) radial confinement (localization) and (ii) accumulation of radial nodes with growth of n r (delocalization). Most of these results are reported here for the first time, revealing many new interesting features.

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“…Further, it was realized that, while traditional uncertainty relation and I were unable to explain such dual effects, measures like S and E were quite successful. In an analogous study [26], oscillation of a particle between larger and smaller wells were followed through information analysis in an asymmetric DW potential, given by, V (x) = αx 4 − βx 2 + γx. In this case, it was possible to frame some simple rules to predict quasi-degeneracy that occurs only for some characteristic values of the parameters present in the potential.…”

Section: Introductionmentioning

confidence: 99%

Information entropic measures of a quantum harmonic oscillator in symmetric and asymmetric confinement within an impenetrable box

2016

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Information-based uncertainty measures like Shannon entropy, Onicescu energy and Fisher information (in position and momentum space) are employed to understand the effect of symmetric and asymmetric confinement in a quantum harmonic oscillator. In symmetric case, a wide range of confinement length (x c ) has been considered, whereas asymmetric confinement is followed by shifting the minimum of potential from origin keeping box length and boundary fixed. Eigenvalues and eigenvectors for these systems are obtained quite accurately via an imaginary time propagation scheme. One finds that, in symmetric confinement, after a certain characteristic x c , all these properties converge to respective values of free harmonic oscillator. In asymmetric situation, excited-state energies always pass through a maximum. For this potential, the classical turning-point decreases, whereas well depth increases with the strength of asymmetry. Study of these uncertainty measures reveals that, localization increases with an increase of asymmetric parameter.

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Evaluation of alignment target designs for Cu and low-K dual damascene processes

Cited by 4 publications

References 0 publications

Information-entropic measures for non-zero l states of confined hydrogen-like ions

Information-entropic measures for non-zero l states of confined hydrogen-like ions

Information‐Entropic Measures in Confined Isotropic Harmonic Oscillator

Information entropic measures of a quantum harmonic oscillator in symmetric and asymmetric confinement within an impenetrable box

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