Publication: Métodos espectrales aplicados a problemas de transporte de carga en dispositivos semiconductores
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2013-07
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2013-07-12
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Abstract
El objetivo de esta tesis doctoral es proponer métodos numéricos eficientes para la resolución de modelos que describen la dinámica de los portadores en dos tipo de dispositivos semiconductores: una estructura de múltiples pozos cuánticos (MQW) y un transistor de efecto de campo (MESFET). Para estudiar los procesos dinámicos en un dispositivo de MQW, consideramos un modelo hidrodinámico bidimensional propuesto por Sherman, Abrarov y Sipe [1], el cual describe la dinámica de las cargas producida mediante inyección óptica. Dicho modelo está descrito por un sistema de ecuaciones diferenciales en derivadas parciales hiperbólicas acopladas y fuertemente no lineales. Para resolver numéricamente este modelo, usamos métodos espectrales, los cuales han demostrado una alta eficacia en la resolución de este tipo de ecuaciones diferenciales que involucran fenómenos ondulatorios. El esquema numérico que hemos desarrollado es estable, robusto y permite calcular soluciones para tiempos largos. De hecho, la solución del modelo hidrodinámico bidimensional nos permite observar que la dinámica de las cargas tiene un comportamiento predominantemente unidimensional. Este hecho nos ha motivado a formular una versión unidimensional del modelo, en principio más sencilla, pero que conserva las propiedades físicas esenciales del modelo original. Con este modelo unidimensional hemos reproducido, fielmente, los resultados obtenidos por el modelo bidimensional completo. Además, para tiempos largos, hemos derivado una solución asintótica que concuerda con los resultados numéricos. Por otro lado, hemos extendido el modelo analizado incluyendo los efectos de la aplicación de un campo magnético y lo hemos resuelto numéricamente con nuestro algoritmo. Para estudiar la dinámica de las cargas en un dispostivo MESFET, consideramos un modelo de Drift-Diffusion (DD) [43, 74]. La geometría y las condiciones de contorno discontinuas, típicas de estos dispositivos, hacen muy difícil su solución numérica usando métodos numéricos tradicionales. Una alternativa eficiente para la solución de este tipo de problemas es el método sin malla de Funciones de Base Radial (RBF). El método RBF global posee precisión espectral, pero tiene la desventaja de que el sistema lineal resultante se convierte en mal condicionado cuando se incrementa el número de nodos o disminuye el parámetro de forma. El esquema numérico que proponemos en esta tesis se basa en el uso de aproximaciones con RBF locales en lugar de las globales. El método RBF local carece de la precisión espectral del método RBF global, pero permite eludir los problemas de mal condicionamiento. Los resultados obtenidos usando RBF locales son muy acordes con los que se obtienen con el método global. ---------------------------------------------------------------------------------------------------------------------------------------------------------------
The aim of this Ph.D. thesis is to propose efficient numerical methods for solving the models that describe the dynamics of the carriers in two kinds of semiconductor devices: a multiple quantum wells structure (MQW) and a field effect transistor (MESFET). To study the dynamic processes in a MQW device, we consider a 2D hydrodynamic model proposed by Sherman, Abrarov and Sipe [1], which describes the dynamics of the charges produced by optical injection. The model consist of nonlinear hyperbolic partial differential equations strongly coupled. To numerically solve this model, we use spectral methods, which have shown a high efficiency in the solution of this type of differential equations involving wave phenomena. The numerical scheme based in spectral methods is stable and robust. Furthermore, it can be used to compute the solutions for long times, and it can be extended to analyze the effect of applying a magnetic field. The solution of the 2D hydrodynamic model allows us to observe that the dynamics of charges have a predominantly unidimensional behavior. This fact motivates us to formulate a one-dimensional version of it, which is simple enough, but contains the main physics of the original one. With this 1D model we reproduce, closely, the results obtained from the solution of the 2D model. For long times, we derive an asymptotic solution which is in close agreement with the numerical results. To study the dynamics of the charges in a MESFET device, we consider the Drift-Diffusion Model (DD) [43, 74]. The discontinuous boundary conditions and the complex geometry typical of these devices, makes very challenging its numerical solution using traditional numerical methods. An efficient alternative for the solution of this type of problems is the use of Radial Basis Functions (RBF) meshless method. The global RBF method has spectral accuracy but has the disadvantage that the resulting linear system becomes ill-conditioned when the number of nodes increases or the shape parameter decreases. The numerical scheme that we propose in this thesis is based on using the local RBF approximation instead of the global version. The local RBF method lacks the spectral accuracy of the global RBF method, but avoids the ill-conditioning problem. The results obtained using local RBF are very consistent with those obtained with the global method.
The aim of this Ph.D. thesis is to propose efficient numerical methods for solving the models that describe the dynamics of the carriers in two kinds of semiconductor devices: a multiple quantum wells structure (MQW) and a field effect transistor (MESFET). To study the dynamic processes in a MQW device, we consider a 2D hydrodynamic model proposed by Sherman, Abrarov and Sipe [1], which describes the dynamics of the charges produced by optical injection. The model consist of nonlinear hyperbolic partial differential equations strongly coupled. To numerically solve this model, we use spectral methods, which have shown a high efficiency in the solution of this type of differential equations involving wave phenomena. The numerical scheme based in spectral methods is stable and robust. Furthermore, it can be used to compute the solutions for long times, and it can be extended to analyze the effect of applying a magnetic field. The solution of the 2D hydrodynamic model allows us to observe that the dynamics of charges have a predominantly unidimensional behavior. This fact motivates us to formulate a one-dimensional version of it, which is simple enough, but contains the main physics of the original one. With this 1D model we reproduce, closely, the results obtained from the solution of the 2D model. For long times, we derive an asymptotic solution which is in close agreement with the numerical results. To study the dynamics of the charges in a MESFET device, we consider the Drift-Diffusion Model (DD) [43, 74]. The discontinuous boundary conditions and the complex geometry typical of these devices, makes very challenging its numerical solution using traditional numerical methods. An efficient alternative for the solution of this type of problems is the use of Radial Basis Functions (RBF) meshless method. The global RBF method has spectral accuracy but has the disadvantage that the resulting linear system becomes ill-conditioned when the number of nodes increases or the shape parameter decreases. The numerical scheme that we propose in this thesis is based on using the local RBF approximation instead of the global version. The local RBF method lacks the spectral accuracy of the global RBF method, but avoids the ill-conditioning problem. The results obtained using local RBF are very consistent with those obtained with the global method.
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Keywords
Dispositivos semiconductores, Procesos dinámicos, Transporte de carga, Métodos numéricos