Two-dimensional Potential Function Research Articles

Analytically Modeling the Asymmetric Double Gate Tunnel FET

This paper proposes an analytical model for the asymmetric double-gate (ADG) TFET operation. Two-dimensional Poisson equation is solved based on the depletion approximation [1,2]. The two-dimensional potential function is given by appropriate boundary conditions. Internal electric field is calculated from the potential model and tunnel current is numerically calculated using the Kane’s tunnel current generation model [3,4]. We used the commercial TCAD simulator in order to examine the analytical model [5]. Schematic of ADG-TFET device structure is shown in Fig. 1. It is assumed that p-type source and n-type drain regions are highly doped. Channel region is a p-type with the doping level of Na . tox denotes the gate oxide thickness and tsi denotes the semiconductor film thickness. Hereafter, we call the top gate “front gate” and the bottom gate “back gate”. Potential profile of the channel region is derived from the following two-dimensional Poisson equation based on the depletion approximation. Equation (1) where q denotes the elemental charge and εsi denotes the permittivity of semiconductor. Here, we assume the following potential function with the cubic function of coordinate y [1]. Equation (2) where four factors a(x), b(x), c(x), and d(x) are functions of coordinate x. We introduce the following boundary conditions for the potential in solving eq. (1). Equation (3) where ψf(x) and ψb(x) stand for front and back surface potentials at y=0 and y=t si, respectively. Vfg,fb and Vbg,fb are the front-gate and the back-gate flat-band voltages, respectively. Vfg and Vbg are the front- and back-gate voltages, respectively. εox is the permittivity of gate insulator. Drain current of ADG TFET is numerically calculated by integrating carriers tunneling from the valence band of the source region to the conduction band of the channel region. We can calculate the drain current (IDS ) using the carrier generation rate (G) due to the band-to-band tunnel process. Equation (4) where we use the Kane’s model for the carrier generation rate (G) [3,4]. Equation (5) where |E| (=(Ex 2+Ey 2)1/2 ) is the electric field strength, A and B are constants given byEquation (Si) Equation (Ge) We perform the TCAD simulation [5] in order to examine the validity of the analytical model. In device simulations for Si-based TFETs, we assume the gate material with the work function of 4.1 eV for the front- and the back-gate electrodes. Figure 2 shows the calculated potential profiles at y=0nm, 2.5nm, and 5.0nm for tsi =10nm. Using the potential function revealed in Fig. 2, local electric field values are calculated along with the x direction and the y direction separately. The potential function analytically given and resulting electric fields successfully reproduced those given by TCAD simulations (not shown here). We found a slight difference of potential profile near the surface around the source junction for a relatively low-doping source region because of the depletion in the source region [6]. We also calculated drain current characteristics. Figure 3 demonstrates ID-Vfg characteristics for tsi =10nm and 20nm, where the condition of Lg >2ts is satisfied. In Fig. 5, the model basically reproduces the TCAD result. On the other hand, the model doesn’t well reproduce TCAD-based ID-Vgf characteristics for tsi =30nm and 100nm (not shown here), where Lg <2tsi . This difference stems from the linear approximation of the in-depth electric field in the model. As the condition of Lg >2tsi is important in the future, this is not a crucial issue of the model. In addition, we confirmed the model proposed here is available for Ge-based TFET (not shown here).

Analytically Modeling the Asymmetric Double Gate Tunnel FET

This paper proposes an analytical model for the asymmetric double gate (ADG) TFET. The two-dimensional Poisson equation is solved based on a depletion approximation. The two-dimensional potential function is given by appropriate boundary conditions. Internal electric field is calculated from the potential model and the tunnel current is numerically calculated using Kane’s tunnel current generation model. We use a commercial TCAD simulator to examine the analytical model.

On a class of ideal focusing systems for energy analysis

Every plane electrostatic system produces pair of orthogonal curve families -- field lines and equipotentials, which can be associated with trajectories and lines of constant action of some mechanical system. The pair of opposite parallel line charges has a two-dimensional representation on a plane perpendicular to charge lines. Such electrostatic system is used as trajectories/action generator. Both action and its arbitrary function are orthogonal to particle trajectories simultaneously. This fact can be used to regularize result two-dimensional potential function and to optimize three-dimensional charged particle motion. The method is applied to create charged particle energy analyzer with an ideal focusing in the symmetry plane. The system useful property is that its energy dispersion tends to infinity when the angle between direction to detector and initial velocity tends to π. Also, for some simple examples it was shown that together with the ideal focusing in a plane of symmetry the system has transversal first order focusing for some initial angles.

Structure and conformational dynamics of the cyclopropyl methyl ketone molecule in the ground electronic state

Geometric parameters, vibrational frequencies, conformer energy difference, and barriers to internal rotation for the cyclopropyl methyl ketone (CPMK, c-С 3H 5C(O)CH 3) molecule in the ground electronic state were obtained by performing MP2, MP4(SDQ), CCSD, and CCSD(T) calculations and VFPA. Ab initio one-dimensional and two-dimensional potential functions corresponding to the rotations of the acetyl C(O)CH 3 and methyl CH 3 tops were used to calculate torsion transition energies by the variational method. Two torsion motions were found to be strongly coupled in the region of the near-trans conformer of CPMK.

A two-dimensional potential function for bent hydrogen bonded systems. II-6-hydroxy-2-formylfulvene

A simple and practical method for treating the non-adiabatic proton tunneling in 6-hydroxy-2-formylfulvene was used. A two dimensional potential energy surface function, which couples OH stretching and in-plane bending modes, has been constructed for motion of hydrogen by the aid of quantum mechanical calculations at MP2/6-31G** level for a fixed skeleton geometry. This potential was used for calculation of energy levels from which a tunneling splitting of 181±14 cm −1 was obtained in an excellent agreement with the experimental value of at least 150 cm −1. The calculated barrier height for these results was about 26–27 kJ/mol. This method also assigns a broad band at about 1740 cm −1 to the OH in-plane bending mode. The calculated hydrogen bond strength was estimated to be about 80 kJ/mol.

A two-dimensional double minimum potential function for bent hydrogen bonded systems. I-malonaldehyde

A simple and practical method has been developed for treating the non-adiabatic proton tunneling in malonaldehyde. A two-dimensional potential energy surface function, which couples OH stretching and in-plane bending modes, has been constructed for the motion of hydrogen by the aid of quantum mechanical calculations at MP2/6-31G ∗∗ and B3LYP/6-311++G ∗∗ levels for a fixed skeleton geometry. This potential is used for calculation of energy levels from which a tunneling splitting of 20.8 and 22.8 cm −1 at the MP2 and B3LYP levels, respectively, was obtained in an excellent agreement with the experimental value of 21.58 cm −1. This simple two-dimensional potential function have been employed to study the interaction between the OH in plane bending and the OH stretching motions in bent hydrogen bonded systems. The coupling between OH stretching and bending modes explains the anomalous vibrational behavior of bent hydrogen bonded systems such as the cis enol form of β-diketones.

An experimental and theoretical study of the vibrationally mediated photodissociation of hydroxylamine

We present a detailed investigation of the photodissociation of hydroxylamine following direct single-photon and vibrationally mediated two-photon excitation below 42 000 cm−1. In all cases the lowest dissociation channel [NH2(X̃ 2B1)+OH(X̃ 2Π)] dominates. Single-photon dissociation at 240 nm releases most of the excess energy (20 550 cm−1) into relative translation (53%) and NH2 internal energy (40%, mostly vibrational). OH carries little internal energy (7%), most of it in the form of rotational excitation. Torsional excitation during the dissociation step leads to rotational alignment of the OH fragments and a preferential population of the Π(A″) component of the lambda doublet. Both are lost after isoenergetic two-photon excitation via O–H stretching overtones of NH2OH, also leading to higher internal excitation of the NH2 fragments (∼50%) at the expense of relative translation. At lower total excitation energies the relative translation takes up an increasing fraction of the total excess energy (⩾80% at 5820 cm−1 of excess energy). The results are discussed in terms of ab initio calculations using complete active space second-order perturbation theory with augmented triple-ζ basis sets for the lowest excited singlet states. One- and two-dimensional potential functions explain the OH product state distributions observed in different experiments in terms of the geometry relaxation of NH2OH upon electronic excitation. Crossing between the lowest excitated A′ and A″ singlet states in the Franck–Condon region leads to a barrier of ∼0.5 eV to dissociation in S1, which dominates the photodissociation dynamics.

Derivation of accurate kinetic and potential-energy functions for several simultaneous large-amplitude vibrations: application to acetaldehyde in the S0and T1states

A procedure is presented for deriving kinetic and potential-energy functions from experimental data for an arbitrary number of large-amplitude vibrations. As a starting point, the method requires a high-quality estimate of the spectroscopic data. We use the previously developed technique of spectrum generation from ab initio calculations to obtain the initial kinetic and potential functions. The present technique minimizes the error between the calculated and experimental energy levels by considering both the kinetic and potential terms. The application to the internal rotation of the methyl group in the S0 state of acetaldehyde shows that the technique is capable of yielding agreement to the experimental measurements to within 0.0001 cm–1. The joint methyl torsion and aldehyde wagging modes for the T1 state of acetaldehyde are also considered. From the two-dimensional potential function the barriers to methyl rotation and to inversion of the aldehydic hydrogen are found to be 633.60 and 935.76 cm–1, respectively. The equilibrium conformation obtained form the two-dimensional potential shows one methyl hydrogen almost eclipsing the aldehydic oxygen (torsion angle 186.40°) whereas the aldehydic hydrogen distorts out of the molecular frame by 39.17°. These structural results are found to differ from that obtained using one-dimensional models, but they agree very well with the results of ab initio calculations.

Jet-cooled fluorescence excitation spectra and carbonyl wagging and ring-puckering potential energy functions of cyclobutanone and its 2,2,4,4-d4 isotopomer in the S1(n,π*) electronic excited state

The jet-cooled fluorescence excitation spectra of cyclobutanone and its 2,2,4,4-d4 isotopomer have been recorded in the 305–335 nm region. The electronic band origin of the d0 molecule for the S1(n,π*) state of A2 symmetry occurs at 30 292 cm−1 (30 265 cm−1 for the d4 molecule). The observed spectra consisting of more than 50 bands for each isotopomer involve ν7, ν8, and ν9 (the three A1 ring vibrations) as well as ν20(C=O in-plane wag), ν26 (C=O out-of-plane wag), and ν27 (ring puckering). Five bands associated with the excited vibrational states of ν26 in the S1(n,π*) electronic state were observed for each isotopic species, and these were used to determine the one-dimensional potential energy functions for the C=O out-of-plane wagging. The C=O wagging angle was determined to be 39° and the barrier to inversion is 2149 cm−1 (2188 cm−1 for the deuteride). For the ring-puckering in the S1 state the lowest three vibrational energy spacings were found to be 106, 166, and 185 cm−1 as compared to values of 35, 57, and 65 cm−1 in the S0 ground state. Several ring-puckering potential energy functions with varying degrees of asymmetry are capable of reproducing the observed results. In all cases, however, the v=0 puckering state lies above any barrier to planarity. A two-dimensional potential function which fits the observed data was also determined in terms of the wagging and puckering coordinates.

Two-dimensional inversion potential function for hydrazine determined from rotation—vibration spectra

The two-dimensional potential function for inversion of the amino groups in hydrazine has been obtained from a fit to the available infrared and microwave data. The barriers of 3408 and 2016 cm −1 have been calculated for the symmertric and antisymmetric inversions of both amino groups, respectively. Contrary to earlier assumptions a large potential coupling between the inversions of the amino groups has been found.

Adsorption of hydrogen and deuterium on Ru(001)

The adsorption of hydrogen and deuterium has been studied with high resolution electron energy loss spectroscopy (HREELS). At maximum coverage at 105 K the H adlayer exhibits two dominant loss features at 102 (ν1) and 141 (ν2) meV which show pronounced intensity variations as a function of the primary electron energy (maximum between 5 and 6 eV). The hydrogen is assigned to the threefold coordinated site of the Ru (001) surface leading to a C3v symmetry of the adsorption complex. The angular dependence of both loss peaks shows only a smooth intensity decrease in off-specular directions thus preventing the separation of dipole and impact scattering contributions. Three additional loss peaks with considerably lower intensities are identified with overtone and combination excitations. The anharmonicity coefficients derived from the respective energies are used to fit a two-dimensional potential function for the hydrogen atom located in a plane perpendicular to the surface with first order perturbation theory. The resulting differences in the potential gradient allows the assignment of the ν1 and ν2 fundamentals to the vibrational modes parallel and perpendicular to the surface, respectively. As a function of increasing coverage both ν1 and ν2 shift to higher frequencies with a concomitant decrease in the peak width, the shift of the parallel mode being distinctively larger. At low coverages a third loss peak is observed which is ascribed to a different adsorption state of hydrogen. For deuterium the two fundamentals and the lowest overtone are observed and exhibit reasonably accurately the expected isotopic shift. The adsorption properties are found to be similar.

ChemInform Abstract: THE FAR‐INFRARED SPECTRUM OF TETRAHYDROTHIOPHENE: THE BEND/TWIST VIBRATIONS AND ASSOCIATED BARRIERS

Abstract The far-infrared spectrum of tetrahydrothiophene is reinvestigated with a resolution of 0.12 cm−1 in the region of 50–350 cm−1. In addition to the bend transitions (νb) below 120 cm−1 reported previously, a number of sequences revealed by the improved resolution are observed for the first time and assigned to transitions of 2νb, of the twist (νt), and of difference combinations (νt - νb). Simple one-dimensional modeling of the twist sequence, which is derived from a self-consistent bend-twist energy level diagram, with a quadratic-quartic Hamiltonian suggests a barrier to planarity on the order of 4250 cm−1. A two-dimensional potential function in the dimensionless coordinates is found to be V(qb, qt) = −249.6qb2 + 4.48qb4 − 215.5qt2 + 2.73qt4 + 7.00qb2qt2.

The far-infrared spectrum of tetrahydrothiophene: The bend/twist vibrations and associated barriers

The far-infrared spectrum of tetrahydrothiophene is reinvestigated with a resolution of 0.12 cm −1 in the region of 50–350 cm −1. In addition to the bend transitions ( ν b) below 120 cm −1 reported previously, a number of sequences revealed by the improved resolution are observed for the first time and assigned to transitions of 2 ν b, of the twist ( ν t), and of difference combinations ( ν t - ν b). Simple one-dimensional modeling of the twist sequence, which is derived from a self-consistent bend-twist energy level diagram, with a quadratic-quartic Hamiltonian suggests a barrier to planarity on the order of 4250 cm −1. A two-dimensional potential function in the dimensionless coordinates is found to be V( q b, q t) = −249.6 q b 2 + 4.48 q b 4 − 215.5 q t 2 + 2.73 q t 4 + 7.00 q b 2 q t 2.

Generalized Curvilinear Diffraction Gratings: V. Diffraction catastrophes

This paper is the last in a series [1–4] of five papers on the diffraction and imaging properties of generalized curvilinear diffraction gratings. Here we show how the bifurcation properties of two-dimensional potential functions may be studied via a diffraction grating representation. In particular, we derive the grating functions appropriate to Thom's seven elementary catastrophes. We then present a detailed study of the cusp grating and discuss the behaviour of its point source image diffraction pattern (IDP) in the context of catastrophe theory. In this way, we show how catastrophe theory allows for a simple geometric interpretation of discontinuous type image diffraction patterns.

Principles of three-dimensional magnetohydrostatic axial separation system

A three-dimensional magnetohydtostatic axial system comprises a magnetic field with its gradient set high vertically and low horizontally. The vertical gradient is used for levitation, with its axial rate of change monitored through the horizontal gradient. Useful fields are obtainable with pole faces generating a two-dimensional potential function multiplied by an axial function which obeys the Laplace equation; a tilted two-dimensional profile produces the same apparent effect. Experiment and theory show that the resolving power of the axial system is superior to that of its two-dimensional counterparts.

On relationships among effective potential functions: 2,5-dihydrofuran, s-tetrazine, 1,4-dioxadiene, cyclopentene, and carbon suboxide

Using Van Vleck perturbation theory, a relationship between two-dimensional potential functions in ring-puckering and ring-twisting coordinates and effective one-dimensional ring-puckering coordinates is demonstrated. It is shown that effective one-dimensional potential functions can be derived for 2,5-dihydrofuran and s-tetrazine which have essentially the same eigenvalue separations as the two-dimensional potential functions previously derived. More useful is the fact that this procedure may be reversed and two-dimensional potential surfaces have been derived for 1,4-dioxadiene and cyclopentene. In mass weighted coordinates the potential function for dioxadiene is V (Q1,Q2) =28.49 Q14+24.00 Q12 +2555.2 Q22+294.6 Q12Q22, where V is in cm−1, Q in U1/2 Å. Q1 is a mass weighted ring-puckering coordinate and Q2 is a mass weighted ring-twisting coordinate. The potential function for cyclopentene is V (Q1,Q2) =51.50 Q14−217.4 Q12+2262.4 Q22 −9.10 Q12Q22 yielding a barrier to planarity of 229 cm−1 essentially unchanged from the barrier of 232 cm−1 reported earlier from the one-dimensional potential function. This same formalism has been used to correlate the potential functions for the large amplitude bending mode, ν7, of carbon suboxide in the ground state and several excited states of other modes.

Laminar flow in rectangular channels. Part I: Entry analysis. Part II: Numerical solution for a square channel?

Abstract Laminar incompressible flow in rectangular channels is considered. In Part I, the entry region is evaluated by a boundary layer/potential core analysis. It is shown that the three-dimensional displacement induced potential flow can be described with a pair of two-dimensional potential functions. Second-order boundary layer solutions, with and without surface mass transfer, are determined; an interesting secondary flow reversal is predicted. In Part II, numerical solutions are obtained for the viscous channel equations, which are derived from the asymptotic theory of Part I. A two stream function, velocity, vorticity system, independent of the Reynolds Number, is solved with a combined iterative ADI/point-relaxation numerical procedure. A single calculation applied for all Reynolds numbers, which appears only in the coordinate scaling. The axial flow behavior of Parts I and II are in good agreement in the asymptotic entry region where both analysis apply. Secondary flow reversal is calculated; however, the grid is too crude for quantitative comparisons. Numerical solutions are obtained until fully developed conditions are achieved. Agreement with experimental data is good.

Diffraction and Spectroscopic Studies of the Cobaltic Acid System HCoC2–DCoO2

Cobaltic acid, HCoO2, and deuterated cobaltic acid, DCoO2, have been prepared and studied by x-ray and neutron-diffraction techniques, and by infrared spectroscopy and cold-neutron scattering. The materials crystallize in apparent space group D3d5–R&amp;lt;3¯m of the trigonal system with three molecules in a hexagonal cell of dimensions: HCoO2, a = 2.851(1), c = 13.150(5) Å; DCoO2, a = 2.854(1), c = 13.354(5) Å. The O–O distance, as determined by neutron-powder studies, of HCoCO2 is 2.50(2) Å. On this basis and the fact that there are three O–H–O bonds along the c direction, the O–D–O distance is 2.57(2) Å. Hence, the isotope effect here is as large as in the system HCrO2–DCrO2. Infrared studies at room temperature and at − 195°C over the region 70–4000 cm−1 may be interpreted in terms of an effectively symmetric O–H–O bond and an effectively asymmetric O–D–O bond. Although the neutron inelastic scattering measurements are not inconsistent with this interpretation, there is a band centered at 264 cm−1 in the neutron spectrum for HCoC2, which is not observed in the infrared spectrum. Model calculations, based on a two-dimensional potential function in which the asymmetric stretching frequency is coupled to the symmetric stretching frequency, were unsuccessful owing to the magnitude of the coupling term required to account for the very large isotope effect.

Potential Function for the Inversion Process of Ammonia

The process of inversion of ammonia has been considered as described by two totally symmetric internal coordinates. The potential function has been chosen by theoretical considerations and the values of the parameters which appear in it have been fixed by minimizing the differences between the calculated eigenvalues of the vibrational problem and the known values of the observed frequencies. Two versions of the vibrational problem have been considered; a simplified one (monodimensional), where only the bond angles are allowed to change, and a complete two-dimensional one where the bond lengths were also allowed to change. The potential functions so obtained give a fairly good agreement between calculated and observed frequencies. The two-dimensional potential function has the minimum at a bond length of 1.878 a.u. and at a bond angle of 105°36′ and gives a value of 0.00905 a.u. as the potential barrier to inversion. The bond length for the minimum of energy at the planar configuration is 1.837 a.u. This potential function and the resulting eigenfunctions of the vibrational Hamiltonian have been employed to calculate some spectroscopic constants, such as the quadratic and cubic force constants related to two totally symmetric coordinates, and the differences between the rotational constants of the odd and the even levels. The agreement with the available experimental data is satisfactory. Results of the above calculations are also given for ND3.

The free-surface condition

Two questions regarding steady-state groundwater flow are posed: (1) What restrictions are imposed upon the class of possible two-dimensional potential functions by the single requirement that a (unspecified) free surface exist? (2) What restrictions are imposed upon the class of possible free-surface curves by the single requirement that a two-dimensional potential function exist capable of giving rise to them? Partial answers have been obtained; roughly, the answer to the first question seems to be, ‘Very considerable restriction’; to the second, ‘practically no restriction in principle’.