Calculation of distribution functions by exploiting the stability of the steady state

Rees, H.D.

doi:10.1016/0022-3697(69)90018-3

Cited by 257 publications

(36 citation statements)

References 7 publications

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“…To determine the occurrence of scattering events without the difficulty of solving integral equations for each event, we use the imaginary selfscattering technique. 60,61 A homogeneous Poisson process of scattering is simulated with the rate parameter Γ 0 , in which a fictitious self-scattering is included such that the total scattering rate together with the self-scattering is Γ 0 . If an electron undergoes a 'self-scattering' event, its wavevector immediately before and after the scattering is unchanged.…”

Section: B Intrinsic Scattering Mechanismsmentioning

confidence: 99%

Spin and energy relaxation in germanium studied by spin-polarized direct-gap photoluminescence

et al. 2013

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Spin orientation of photoexcited carriers and their energy relaxation is investigated in bulk Ge by studying spin-polarized recombination across the direct band gap. The control over parameters such as doping and lattice temperature is shown to yield high polarization degree, namely larger than 40%, as well as a fine-tuning of the angular momentum of the emitted light with a complete reversal between right-and left-handed circular polarization. By combining the measurement of the optical polarization state of band-edge luminescence and Monte Carlo simulations of carrier dynamics, we show that these very rich and complex phenomena are the result of the electron thermalization and cooling in the multi-valley conduction band of Ge. The circular polarization of the direct-gap radiative recombination is indeed affected by energy relaxation of hot electrons via the X valleys and the Coulomb interaction with extrinsic carriers. Finally, thermal activation of unpolarized L valley electrons accounts for the luminescence depolarization in the high temperature regime.

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Section: B Intrinsic Scattering Mechanismsmentioning

confidence: 99%

Spin and energy relaxation in germanium studied by spin-polarized direct-gap photoluminescence

et al. 2013

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“…This last expression is known as Chambers-Rees path integral. Rees [29] innovation is the introduction of the fictitious scattering term. Ignoring the transient term, one can find the solution of the distribution function using the following iterative procedure that is obtained by time discretization, i.e.…”

Section: Monte Carlo and Path-integral Methodsmentioning

confidence: 99%

“…If there is no external driving field leading to a change of k between scattering events (for example in ultrafast photo-excitation experiments with no applied bias), the time dependence vanishes, and the integral is trivially evaluated. As noted in the previous section, in the general case where this simplification is not possible, it is expedient to introduce the so called self-scattering method [29], in which one introduces fictitious scattering mechanism whose scattering rate always adjusts itself in such a way that the total (self-scattering plus real scattering) rate is a constant in time…”

Section: Bulk Monte Carlo Methodsmentioning

confidence: 99%

Monte Carlo Device Simulations

2017

Handbook of Optoelectronic Device Modeling and Simulation

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Applications of Monte Carlo Method in Science and Engineering 386 materials are given to completely specify the collision integral in the BTE. A discussion followed with the presentation of a generic flow-chart for implementing bulk Monte Carlo code is presented. Note that bulk Monte Carlo approach is suitable for the characterization of materials, but in order to study behavior of semiconductor devices coupling of the Monte Carlo transport kernel with a Poisson equation solver which gives the self-consistent field that moves the carriers around is needed. Important ingredients in describing particle-based device simulators are the particle-mesh coupling, treatment of the Ohmic contacts and calculation of the current. A generic flowchart of a particle-based device simulator is provided. The prospects of the Monte Carlo method for the solution of the Boltzmann transport equation, in the context of device simulations of nanoscale structures and of solar cells and power devices, are discussed at the end of the book chapter.2. Importance of MC particle-based device simulations 2.1 Industry trends and the need for modeling and simulation As semiconductor feature sizes shrink into the nanometer scale regime, even conventional device behavior becomes increasingly complicated as new physical phenomena at short dimensions occur, and limitations in material properties are reached [1]. In addition to the problems related to the understanding of actual operation of ultra-small devices, the reduced feature sizes require more complicated and time-consuming manufacturing processes. This fact signifies that a pure trial-and-error approach to device optimization will become impossible since it is both too time consuming and too expensive. Since computers are considerably cheaper resources, simulation is becoming an indispensable tool for the device engineer. Besides offering the possibility to test hypothetical devices which have not (or could not) yet been manufactured, simulation offers unique insight into device behavior by allowing the observation of phenomena that can not be measured on real devices. Computational Electronics [2,3,4] in this context refers to the physical simulation of semiconductor devices in terms of charge transport and the corresponding electrical behavior. It is related to, but usually separate from process simulation, which deals with various physical processes such as material growth, oxidation, impurity diffusion, etching, and metal deposition inherent in device fabrication [5] leading to integrated circuits. Device simulation can be thought of as one component of technology for computer-aided design (TCAD), which provides a basis for device modeling, which deals with compact behavioral models for devices and sub-circuits relevant for circuit simulation in commercial packages such as SPICE [6]. The relationship between various simulation design steps that have to be followed to achieve certain customer need is illustrated in Figure 1.

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“…Rees [29] innovation is the introduction of the fictitious scattering term. Ignoring the transient term, one can find the solution of the distribution function using the following iterative procedure that is obtained by time discretization, i.e.…”

Section: Monte Carlo and Path-integral Methodsmentioning

confidence: 99%

“…If there is no external driving field leading to a change of k between scattering events (for example in ultrafast photoexcitation experiments with no applied bias), the time dependence vanishes, and the integral is trivially evaluated. As noted in the previous section, in the general case where this simplification is not possible, it is expedient to introduce the so called self-scattering method [29], in which one introduces fictitious scattering mechanism whose scattering rate always adjusts itself in such a way that the total (self-scattering plus real scattering) rate is a constant in time…”

Section: Bulk Monte Carlo Methodsmentioning

confidence: 99%

Monte Carlo Device Simulations

Raleva¹,

Shaik

Hathwar

et al. 2011

Applications of Monte Carlo Method in Science and Engineering

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As semiconductor devices are scaled into nanoscale regime, first velocity saturation starts to limit the carrier mobility due to pronounced intervalley scattering, and when the device dimensions are scaled to 100 nm and below, velocity overshoot (which is a positive effect) starts to dominate the device behavior leading to larger ON-state currents. Alongside with the developments in the semiconductor nanotechnology, in recent years there has been significant progress in physical based modeling of semiconductor devices. First, for devices for which gradual channel approximation can not be used due to the two-dimensional nature of the electrostatic potential and the electric fields driving the carriers from source to drain, drift-diffusion models have been exploited. These models are valid, in general, for large devices in which the fields are not that high so that there is no degradation of the mobility due to the electric field. The validity of the drift-diffusion models can be extended to take into account the velocity saturation effect with the introduction of field-dependent mobility and diffusion coefficients. When velocity overshoot becomes important, drift diffusion model is no longer valid and hydrodynamic model must be used. The hydrodynamic model has been the workhorse for technology development and several high-end commercial device simulators have appeared including Silvaco, Synopsys, Crosslight, etc. The advantages of the hydrodynamic model are that it allows quick simulation runs but the problem is that the amount of the velocity overshoot depends upon the choice of the energy relaxation time. The smaller is the device, the larger is the deviation when using the same set of energy relaxation times. A standard way in calculating the energy relaxation times is to use bulk Monte Carlo simulations. However, the energy relaxation times are material, device geometry and doping dependent parameters, so their determination ahead of time is not possible. To avoid the problem of the proper choice of the energy relaxation times, a direct solution of the Boltzmann Transport Equation (BTE) using the Monte Carlo method is the best method of choice. That is why the focus of this review paper is on explaining basic Monte Carlo device simulator and then the focus will be shifted on the inclusion of various higher order effects that explain particular physical phenomena or processes.The Monte Carlo book chapter is organized as follows. First, the idea behind the Monte Carlo technique is outlined by revoking the path integral method for the solution of the BTE. This approach naturally leads to the free-flight-scatter sequence that is used in solving the BTE using the Monte Carlo method. Various scattering mechanisms relevant for different materials are given to completely specify the collision integral in the BTE. A discussion followed with the presentation of a generic flow-chart for implementing bulk Monte Carlo code is presented. Note that bulk Monte Carlo approach is suitable for the characterization of ma...

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Calculation of distribution functions by exploiting the stability of the steady state

Cited by 257 publications

References 7 publications

Spin and energy relaxation in germanium studied by spin-polarized direct-gap photoluminescence

Spin and energy relaxation in germanium studied by spin-polarized direct-gap photoluminescence

Monte Carlo Device Simulations

Monte Carlo Device Simulations

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