Algebraic time-decay for the bipolar quantum hydrodynamic model

The initial value problem is considered in the present paper for bipolar quantum hydrodynamic model for semiconductors (QHD) in $\mathbb{R}^3$. We prove that the unique strong solution exists globally in time and tends to the asymptotical state with an algebraic rate as $t\to+\infty$. And, we show that the global solution of linearized bipolar QHD system decays in time at an algebraic decay rate from both above and below. This means in general, we can not get exponential time-decay rate for bipolar QHD system, which is different from the case of unipolar QHD model (where global solutions tend to the equilibrium state at an exponential time-decay rate) and is mainly caused by the nonlinear coupling and cancelation between two carriers. Moreover, it is also shown that the nonlinear dispersion does not affect the long time asymptotic behavior, which by product gives rise to the algebraic time-decay rate of the solution of the bipolar hydrodynamical model in the semiclassical limit.Comment: 23 page

Subsonic steady-states for bipolar hydrodynamic model for semiconductors

In this paper, we study the well-posedness, ill-posedness and uniqueness of the stationary 3-D radial solution to the bipolar isothermal hydrodynamic model for semiconductors. The density of electron is imposed with sonic boundary and interiorly subsonic case and the density of hole is fully subsonic case

Large time behavior of solutions of the bipolar hydrodynamical model for semiconductors

AbstractThe asymptotic behavior of classical solutions of the bipolar hydrodynamical model for semiconductors is considered in the present paper. This system takes the form of Euler–Poisson with electric field and frictional damping added to the momentum equation. The global existence of classical solutions is proven, and the nonlinear diffusive phenomena is observed in large time in the sense that both densities of electron and hole tend to the same unique nonlinear diffusive wave

On the Initial-Boundary Value Problem for the Bipolar Hydrodynamic Model for Semiconductors

AbstractThe global existence and zero relaxation limit results of weak solutions of the initial-boundary value problem to the bipolar hydrodynamic model for semiconductors are established by the theory of compensated compactness. The boundary conditions of weak solutions in the sense of traces are discussed

Approximations of Euler-Maxwell systems by drift-diffusion equations through zero-relaxation limits near non-constant equilibrium

Due to extreme difficulties in numerical simulations of Euler-Maxwell equations, which are caused by the highly complicated structures of the equations, this paper concerns the simplification of Euler-Maxwell system through the zero-relaxation limit towards the drift-diffusion equations with non-constant doping functions. We carry out the global-in-time convergence analysis by establishing uniform estimates of solutions near non-constant equilibrium regarding the relaxation parameter and passing to the limit by using classical compactness arguments. Furthermore, stream function methods are carefully generalized to the non-constant equilibrium case, with which as well as the anti-symmetric structure of the error system and an induction argument, we establish global-in-time error estimates between smooth solutions to the Euler-Maxwell system and those to drift-diffusion system, which are bounded by some power of relaxation parameter

Entropy-Based Moment Closures in Semiconductor Models

We investigate aspects of entropy-based moment closures which are used to simplify kinetic models of particle systems. Closures of this type use variational principles to formally generate balance laws for velocity moments of a kinetic density. These balance laws form a symmetric hyperbolic system of partial differential equations that satisfies an analog of Boltzmann's famous H-Theorem. However, in spite of this elegant structure, practical implementation of entropy-based closures requires that several analytical and computational issues be settled. Our presentation is devoted to the development of electron transport models in semiconductor devices. In this context, balance laws for velocity moments are generally referred to as hydrodynamic models. Such models provide a reasonable alternative to kinetic and Monte Carlo approaches, which are usually expensive, and the well-known drift-diffusion model, which is much simpler but a has a limited range of validity. We first analyze the minimization problem that defines the entropy closure. It is known that there are physically relevant cases for which this problem is ill-posed. Using a dual formulation, we find so-called complementary slackness conditions which give a geometric interpretation of ill-posed cases in terms of the Lagrange multipliers of the minimization problem. Under reasonable assumptions, we show that these cases are rare in a very precise sense. We also develop pertubations of well-posed entropy-based closures, thereby making them useful for modeling systems with heat flux and anisotropic stress. Heat flux has long been known to be an important component of electron transport in semiconductors. However, we also observe that anisotropy in the stress tensor also plays an important role in regions of high electric field. This conclusion is made based on our simulations of two different devices. Finally, we devise a new split scheme for hydrodynamic models. The splitting is based on the balance of forces in the hydrodynamic model that recovers the drift-diffusion equation in the asymptotic limit of small mean-free-path. This scheme removes numerically stiffness and excessive dissipation typically associated with standard shock-capturing schemes in the drift-diffusion limit. In addition, it significantly reduces numerical current oscillations near material junctions