Search query: The quantum hydrodynamic model for semiconductor devices,

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abstract: The classical hydrodynamic equations can be extended to include quantum effects by incorporating the first quantum corrections. These quantum corrections are $O( \hbar ^2 The full three dimensional quantum hydrodynamic (QHD) model is derived for the first time by a moment expansion of the Wigner Boltzmann equation. The QHD conservation laws have the same form as the classical hydrodynamic equations, but the energy density and stress tensor have additional quantum terms. These quantum terms allow particles to tunnel through potential barriers and to build up in potential wells.The three dimensional QHD transport equations are mathematically classified as having two Schrodinger modes, two hyperbolic modes, and one parabolic mode. The one dimensional steady state QHD equations are discretized in conservation form using the second upwind method.Simulations of a resonant tunneling diode are presented that show charge buildup in the quantum well and negative differential resistance (NDR) in the current v.
author_names: Carl L Gardner
title: The Quantum Hydrodynamic Model for Semiconductor Devices
venue: SIAM J. Appl. Math.
year: 1994

abstract: A "smooth" quantum hydrodynamic (QHD) model for semiconductor devices is derived by a Chapman Enskog expansion of the Wigner Boltzmann equation which can handle in a mathematically rigorous way the discontinuities in the classical potential energy which occur at heterojunction barriers in quantum semiconductor devices. A dispersive quantum contribution to the heat flux term in the QHD model is introduced.
author_names: Carl L Gardner

Christian A Ringhofer
title: The Chapman Enskog Expansion and the Quantum Hydrodynamic Model for Semiconductor Devices
venue: VLSI Design
year: 2000

abstract: Abstract A finite element method for numerical simulation of the transient quantum hydrodynamic model for semiconductor devices is presented. This model treats electron flow in a semiconductor device in the same manner as the classical hydrodynamic model, but the energy density and stress tensor include additional quantum terms, which allow particles to tunnel through potential barriers and to build up in potential wells. The finite element method under consideration is based on use of a mixed method for the approximation of the electric field and a shock capturing Runge Kutta discontinuous Galerkin method for the quantum hydrodynamic conservation laws. Numerical simulations of a resonant tunneling diode are presented, which show charge buildup in the quantum well and negative differential resistance and a hysteresis in the current voltage curve. A comparison between the present method and a finite difference method is given.
author_names: Zhangxin Chen
title: A finite element method for the quantum hydrodynamic model for semiconductor devices
year: 1996

abstract: Abstract In this article, fifth order finite volume multi resolution weighted essentially non oscillatory (MR WENO) scheme is developed for solving one dimensional non linear viscous quantum hydrodynamical model for semiconductor devices with constant temperature. This non linear model consists of three equations, two equations for the current density and electron density, and third equation is the Poisson equation. The first two equations include the viscous and quantum correction terms. Further, numerical technique is used to obtain the solution due to high non linearity of considered model. The multi resolution technique is used to reduce the computational cost of high resolution numerical scheme. In the procedure of derivation of the MR WENO scheme unequal central spatial stencils are used and linear weights can be chosen any positive numbers with only restriction that their total sum is one. Various numerical test problems are considered to check the validity and accuracy of the derived numerical scheme. Further, the results obtained from considered numerical scheme are compared with those of kinetic flux vector splitting numerical scheme.
author_names: Tauqeer Ahmed

Asad Rehman

Ashiq Ali

Shamsul Qamar
title: A high order multi resolution WENO numerical scheme for solving viscous quantum hydrodynamic model for semiconductor devices
year: 2021

abstract: Abstract Quantum transport effects including electron or hole tunneling through potential barriers and buildup in quantum wells are important in predicting the performance of ultra small semiconductor devices. These effects can be incorporated into the hydrodynamic description of charge propagation in the semiconductor device. A new extension of the classical hydrodynamic model to include quantum transport effects was derived. This "smooth" quantum hydrodynamic (QHD) model is derived specifically to handle in a mathematically rigorous way the discontinuities in the classical potential energy which occur at heterojunction barriers in quantum semiconductor devices. The smooth QHD model makes the barriers partially transparent to the particle flow and provides the mechanism for particle tunneling in the QHD model. Smooth quantum hydrodynamic model simulations of the resonant tunneling diode were presented which exhibit enhanced negative differential resistance (NDR) when compared to simulations using the original QHD model. At both 300 K and 77 K, the smooth QHD simulations predict significant NDR even when the original QHD model simulations predict no NDR.
author_names: Carl L Gardner
title: The Quantum Hydrodynamic Model for Semiconductor Devices: Theory and Computations
year: 1998

abstract: Abstract An extension of the classical hydrodynamic model for semiconductor devices to include quantum transport effects is reviewed. This "smooth" quantum hydrodynamic (QHD) model is derived specifically to handle in a mathematically rigorous way the discontinuities in the classical potential energy which occur at heterojunction barriers in quantum semiconductor devices. A conservative upwind discretization of the one dimensional (1D) steady state smooth QHD equations is outlined. Smooth QHD model simulations of the resonant tunneling diode are presented which exhibit enhanced negative differential resistance when compared with simulations using the original O( 2 QHD model.
author_names: Carl L Gardner

Christian A Ringhofer
title: Numerical simulation of the smooth quantum hydrodynamic model for semiconductor devices
year: 2000

abstract: Abstract The "smooth" quantum hydrodynamic (QHD) model is an extension of the classical hydrodynamic model for semiconductor devices which can handle in a mathematically rigorous way the discontinuities in the classical potential energy which occur at heterojunction barriers in quantum semiconductor devices. Smooth QHD model simulations of the resonant tunneling diode exhibit negative differential resistance the experimental signal for quantum resonance effects. Resonant tunneling is analyzed in fluid dynamical terms from the point of view of the smooth QHD transport equations.
author_names: Carl L Gardner

Christain Ringhofer
title: Resonant tunneling in the smooth quantum hydrodynamic model for semiconductor devices
year: 2000

abstract: Abstract We study a hybrid model linking the quantum hydrodynamics equation with classical hydrodynamics deriving the transmission conditions between the two PDE systems modeling the quantum and the classical dynamics. These conditions are derived under the assumption of constant scaled temperature and assuming a current jump across the interface between the classical and the quantum region. In Section 2.2 we shall give a heuristic and physically plausible explanation of this assumption. Because of this fact we produce not a weak solution on the whole device domain, but a piecewise smooth solution.
author_names: Federica Di Michele

Pierangelo Marcati

Bruno Rubino
title: Steady states and interface transmission conditions for heterogeneous quantum classical 1 D hydrodynamic model of semiconductor devices
year: 2013

abstract: Abstract In this article, one dimensional viscous quantum hydrodynamical model of semiconductor devices is numerically investigated. The model treats the propagation of electrons in a semiconductor device as the flow of a charged compressible fluid. It plays an important role in predicting the behavior of electron flow in semiconductor devices. The nonlinear viscous quantum hydrodynamic models contain Euler type equations for density and current, viscous and quantum correction terms, and a Poisson equation for electrostatic potential. Due to high nonlinearity of model equations, numerical solution techniques are applied to obtain their solutions. The proposed numerical scheme is a splitting scheme based on the kinetic flux vector splitting (KFVS) method for the hyperbolic step, and a semi implicit Runge Kutta method for the relaxation step. The KFVS method is based on the direct splitting of macroscopic flux functions of the system on the cell interfaces. The second order accuracy of the scheme is achieved by using MUSCL type initial reconstruction and Runge Kutta time stepping method. Several case studies are considered. For validation, the results of current scheme are compared with those obtained from the splitting scheme based on the NT central scheme. The effects of various parameters such as device length, viscosities, different doping and voltage are analyzed. The accuracy, efficiency and simplicity of the proposed KFVS scheme validates its generic applicability to the given model equations.
author_names: Ubaid Ahmed Nisar

Waqas Ashraf

Shamsul Qamar
title: Application of kinetic flux vector splitting scheme for solving viscous quantum hydrodynamical model of semiconductor devices
year: 2018

abstract: In this article, one dimensional viscous quantum hydro dynamical model of semiconductor devices is numerically investigated. The model treats the propagation of electrons in a semiconductor device as the flow of a charged compressible fluid. It plays an important role in predicting the behaviour of electron flow in semiconductor devices. The nonlinear viscous quantum hydrodynamic models contain Euler type equations for density and current, viscous and quantum correction terms, and a Poisson equation for electrostatic potential. Due to high nonlinearity of model equations, numerical solution techniques are applied to obtain their solutions. The proposed numerical scheme is a splitting scheme based on the kinetic flux vector splitting (KFVS) method for the hyperbolic step, and a semi implicit Runge Kutta method for the relaxation step. The KFVS method is based on the direct splitting of macroscopic flux functions of the system on the cell interfaces. The second order accuracy of the scheme is achieved by using MUSCL type initial reconstruction and Runge Kutta time stepping method. Several case studies are considered. For validation, the results of current scheme are compared with those obtained from the splitting scheme based on the NT central scheme. The effects of various parameters such as device length, viscosities, different doping and voltage are analyzed. The accuracy, efficiency and simplicity of the proposed KFVS scheme validates its generic applicability to the given model equation
author_names: Waqas Ashrafb

Shamsul Qamarc

Ubaid Ahmed Nisara
title: Application of Kinetic Flux Vector Splitting Scheme for Solving Viscous Quantum Hydrodynamical Model of Semiconductor Devices
year: 2018