Density-gradient Equations Research Articles

Compact charge model for Si gate-all-around nMOSCAPs with cylindrical cross-sections considering the density-gradient equation

A compact charge model for Si gate-all-around n-type metal-oxide-semiconductor capacitors (nMOSCAPs) with cylindrical cross-sections including the quantum confinement effect is presented. The density-gradient equation with a penetrating boundary condition is integrated to consider the quantum confinement effect. From the integrated equation, an expression for the surface potential is derived, and a compact charge model is presented. To calculate the charge–voltage characteristics, parameter modeling for terms related to quantum correction is done. The results from the model for various radii show excellent agreement with numerical simulations.

Compact Charge Modeling of Double-Gate MOSFETs Considering the Density-Gradient Equation

A compact charge model for double-gate metal–oxide–semiconductor field-effect transistors with the quantum confinement effect is presented. In addition to the Poisson equation, the density-gradient equation with a realistic boundary condition is considered to include the quantum confinement effect. The coupled governing equations are rigorously integrated. Contribution of the density-gradient equation is clearly identified. Based on the resultant integrated equation, a compact charge model is proposed. Expressions for model parameters are found. Numerical examples for various double-gate MOS structures are shown.

Sensitivity of static noise margins to random dopant variations in 6-T SRAM cells

The random dopant induced fluctuations of static noise margins (SNM) in 6-T SRAM cells are analyzed by using the formalism of doping sensitivity functions, which show how sensitive the SNM are to variations of the doping concentration at different locations inside the cell. The technique presented in this article is based on a full circuit perturbation theory at the level of each device transport model. It provides important information for the design and optimization of SNM and can capture correlation effects of doping fluctuations inside the same semiconductor device and between more devices. The bias points and the magnitude of random dopant induced fluctuations are computed by solving the Poisson, current continuity, and density–gradient equations for all the devices self-consistently (mixed-mode simulation). Simulation results for a well scaled SRAM cell with 30 nm channel length transistors show that the most sensitive regions to doping fluctuations extend for approximately 10 nm below the oxide/semiconductor interface and are located in the middle of the conduction channels for both the p-channel and n-channel transistors. It is apparent that random dopant induced fluctuations can significantly impinge on the yield and reliability of SRAM circuits and constitute a fundamental limit for further scaling unless these devices are properly optimized.

Convergent finite element discretizations of the density gradient equation for quantum semiconductors

We study nonlinear finite element discretizations for the density gradient equation in the quantum drift diffusion model. In particular, we give a finite element description of the so-called nonlinear scheme introduced by Ancona. We prove the existence of discrete solutions and provide a consistency and convergence analysis, which yields the optimal order of convergence for both discretizations. The performance of both schemes is compared numerically, in particular, with respect to the influence of approximate vacuum boundary conditions.

An accelerated monotone iterative method for the quantum-corrected energy transport model

A non-stationary monotone iterative method is proposed and analyzed for the quantum-corrected energy transport model in nanoscale semiconductor device simulation. For the density-gradient equations, it is analytically and numerically shown that the convergence rate of the method is optimal in the sense of Gummel’s decoupling iteration. This is a globally convergent method in the sense that the initial guess can be taken as a lower or an upper solution which is independent of applied voltages. The method integrates the monotone parameters, grid sizes, and Scharfetter–Gummel fitting in an adaptive and automatic way to treat the singularly perturbed nature of the model that incurs boundary, junction, and quantum potential layers in the device.

Quantum corrections: a multilevel solver for the density-gradient equation

Miniaturisation of integrated circuits continues to shrink device lengths to such an extent that quantum tunnelling and confinement effects change the behaviour of MOSFET devices. In this paper, we present a methodology by which to model the gate region of an n-Metal Oxide Semiconductor (MOS) device using a simplified version of the density-gradient equations. The resulting singularly perturbed ODEs are solved using an adaptive wavelet collocation method that adapts dynamically to the boundary layer. Our results are shown to be in good agreement with those from a direct numerical solution of the Schrodinger-Poisson system.

A quantum corrected energy-transport model for nanoscale semiconductor devices

An energy transport model coupled with the density gradient method as quantum mechanical corrections is proposed and numerically investigated. This new model is comprehensive in both physical and mathematical aspects. It is capable of describing hot electron transport as well as significant quantum mechanical effects for advanced devices with dimensions comparable to the de Broglie wave-length. The model is completely self-adjoint for all state variables and hence provides many appealing mathematical features such as global convergence, fast iterative solution, and highly parallelizable. Numerical simulations on diode and MOSFET with the gate length down to 34 nm using this model have been performed and compared with that using the classical transport model. It is shown that the I– V characteristics of this short-channel device is significantly corrected by the density-gradient equations with current drive reduced by up to 60% comparing with that of the classical model along. Moreover, a 2D quantum layer, which is only a fraction of the length scale of inversion layer, is also effectively captured by this new model with very fine mesh near the interface produced by an adaptive finite element method.

Scattering from Body Thickness Fluctuations in Double Gate MOSFETs: An ab initio Monte Carlo Simulation Study

In this paper we present an ab initio approach for including body thickness variation effects in 3D Monte Carlo simulation studies through the real space trajectories of Monte Carlo particles moving and scattering in the presence of a quantum potential obtained from the solution of the density gradient equation by Ancona and Iafrate (1989). This approach is used to study the mobility degradation associated with body thickness variation in large self-averaging devices and the transport and performance variation in small devices introduced by the unique body thickness pattern in each one of them.

Intrinsic parameter fluctuations in nanometre scale thin-body SOI devices introduced by interface roughness

An investigation is presented into intrinsic parameter fluctuations in thin-body SOI MOSFETs due to local variations in body thickness as a result of interface roughness. A series of well scaled devices from 15 nm channel length down to 5 nm are investigated using three-dimensional drift-diffusion simulations which include the density gradient equation to account for quantum mechanical effects. It is shown that the intrinsic parameter variations are enhanced by the quantum mechanical confinement within the channel of the device. A comparison with parameter fluctuations due to discrete random doping in the source and drain is also presented.

Discretization Scheme for the Density-Gradient Equation and Effect of Boundary Conditions

In recent years, the density gradient theory (DG) (M.G. Ancona and H.F. Tiersten 1987, Phys. Rev. B. 35(15): 7959–7965, M.G. Ancona 1990, Phys. Rev. B. 42: 1222) has been established as a viable alternative to the solution of the Schrodinger equation for solving problems such as charge density distribution in MOS inversion layers and MOS tunneling (M.G. Ancona 1998, J. Tech. CAD(11), M.G. Ancona et al. 2000, IEEE Trans. Electron Devices 47: 1449). Primary advantages of the DG method over the Schrodinger method are flexibility in extending to multi-dimension and easiness in incorporating into the conventional drift-diffusion or hydrodynamic solver (C.S. Rafferty et al. 1998, Proc. SISPAD, p. 137, A. Wettstein et al. 2001, IEEE Trans. Elec. Dev. 48: 279). However, the DG term that represents the quantum effects is a singular perturbation term and requires special care for discretization (X. Wang 2001, Master's thesis, University of Massachusetts, Amherst). In this work, we examine the validity of the linear vs. the nonlinear discretization scheme and the effect of boundary conditions on the scheme used.

Finite-Difference Schemes for the Density-Gradient Equations

Two new finite-difference schemes are derived for solving the density-gradient equations describing quantum transport in semiconductors that improve on a simple linear discretization. The first scheme is also linear but includes additional terms that serve to maintain conservation to second order. The second scheme is a nonlinear exponential-fitting method that is more complicated but also more efficient than the linear schemes. The methods are evaluated numerically using device examples involving both quantum confinement and tunneling and are shown to perform quite well allowing for a substantial easing in the mesh refinement especially in tunneling problems.

Binding in pair potentials of liquid simple metals from nonlocality in electronic kinetic energy.

The paper presents an explicit expression for the pair potential in liquid simple metals from low-order density-gradient theory when the superposition of single-center displaced charges is employed. Numerical results are presented for the gradient expansion pair interaction in liquid Na and Be. The low-order density-gradient equation for the pair potential is presented.