A 1D coupled Schrödinger drift-diffusion model including collisions

We consider a one-dimensional coupled stationary Schroedinger drift-diffusion model for quantum semiconductor device simulations. The device domain is decomposed into a part with large quantum effects (quantum zone) and a part where quantum effects are negligible (classical zone). We give boundary conditions at the classic-quantum interface which are current preserving. Collisions within the quantum zone are introduced via a Pauli master equation. To illustrate the validity we apply the model to three resonant tunneling diodes

Can Wigner distribution functions with collisions satisfy complete positivity and energy conservation?

Altres ajuts: Acord transformatiu CRUE-CSICTo avoid the computational burden of many-body quantum simulation, the interaction of an electron with a photon (phonon) is typically accounted for by disregarding the explicit simulation of the photon (phonon) degree of freedom and just modeling its effect on the electron dynamics. For quantum models developed from the (reduced) density matrix or its Wigner-Weyl transformation, the modeling of collisions may violate complete positivity (precluding the typical probabilistic interpretation). In this paper, we show that such quantum transport models can also strongly violate the energy conservation in the electron-photon (electron-phonon) interactions. After comparing collisions models to exact results for an electron interacting with a photon, we conclude that there is no fundamental restriction that prevents a collision model developed within the (reduced) density matrix or Wigner formalisms to satisfy simultaneously complete positivity and energy conservation. However, at the practical level, the development of such satisfactory collision model seems very complicated. Collision models with an explicit knowledge of the microscopic state ascribed to each electron seems recommendable (Bohmian conditional wavefunction), since they allow to model collisions of each electron individually in a controlled way satisfying both complete positivity and energy conservation

무한한 상관길이를 가지는 위스너 수송 방정식의 결정론적 해

학위논문(박사) -- 서울대학교대학원 : 공과대학 전기·정보공학부, 2022. 8. 최우영.We propose a new discrete formulation of the Wigner transport equation (WTE) with infinite correlation length of potentials. Since the maximum correlation length is not limited to a finite value, there is no uncertainty in the simulation results, and Wigner-Weyl transformation is unitary in our expression. For general and efficient simulation, the WTE is solved self-consistently with the Poisson equation through the finite volume method and the fully coupled Newton-Raphson scheme. By applying the proposed model to resonant tunneling diodes and double gate MOSFET, transient and steady-state simulation results including scattering effects are shown.본 연구에서는 무한한 상관 길이를 가지는 위그너 수송 방정식의 새로운 수치해석적 풀이법을 제시하였다. 최대 상관 길이가 한정된 값으로 제한되지 않기 때문에, 시뮬레이션 결과에 불확실성이 발생하지 않으며, 제안된 표현법에서는 Wigner-Weyl 변환이 unitary하다. 일반적이고 효율적인 시뮬레이션을 위해, 위그너 수송 방정식을 푸아송 방정식과 유한 체적법과 뉴턴-랩슨 방식을 통해 self-consistent하게 풀었다. 제안된 모델을 resonant tunneling diode와 double gate MOSFET에 적용하여, 산란효과를 고려한 동적 그리고 정적 시뮬레이션 결과를 보여주었다.Chapter 1 Introduction 1 1-1 Various models for device simulation 1 1-2 Numerical problems in solving WTE 5 Chapter 2 Simulation methods 10 2-1 WTE with infinite correlation length 10 2-2 Numerical Methods 13 2-3 Multi-dimensional Simulation Methods 23 Chapter 3 Simulation methods 26 2-1 Simulation results according to the correlation length 26 2-2 Simulation for resonant tunneling diode 30 2-3 Simulation for double gate MOSFET 51 Chapter 4 Conclusion 70 Appendix 72 A-1 Numerical integration method of the nonlocal potential terms 72 A-2 2D electron density and electric potential results 75 A-3 Wigner function for each subband 78 References 85 Abstract 92박

Dual current anomalies and quantum transport within extended reservoir simulations

Quantum transport simulations are rapidly evolving and now encompass well-controlled tensor network techniques for many-body transport. One powerful approach combines matrix product states with extended reservoirs. In this method, continuous reservoirs are represented by explicit, discretized counterparts where a chemical potential or temperature drop is maintained by relaxation. Currents are strongly influenced by relaxation when it is very weak or strong, resulting in a simulation-analog of Kramers' turnover in solution-phase chemical reactions. At intermediate relaxation, the intrinsic conductance-that given by the Landauer or Meir-Wingreen expressions-moderates the current. We demonstrate that strong impurity scattering (i.e., a small steady-state current) reveals anomalous transport regimes within this methodology at weak-to-moderate and moderate-to-strong relaxation. The former is due to virtual transitions and the latter to unphysical broadening of the populated density of states. The Kramers' turnover analog thus has five standard transport regimes, further constraining the parameters that lead to the intrinsic conductance. In particular, a relaxation strength proportional to the reservoir level spacing-the commonly assumed strategy-can prevent convergence to the continuum limit. This underscores that the turnover profiles enable identification of simulation parameters that achieve proper physical behavior.Comment: 16 pages, 5 figure

Current coupling of drift-diffusion models and dissipative Schrödinger--Poisson systems: Dissipative hybrid models

A 1D coupled drift-diffusion dissipative Schrödinger model (hybrid model), which is capable to describe the transport of electrons and holes in semi-conductor devices in a non-equilibrium situation, is mathematically analyzed. The device domain is split into a part where the transport is well-described by the drift-diffusion equations (classical zone) and a part where a quantum description via a dissipative Schrödinger system (quantum zone) is used. Both system are coupled such that the continuity of the current densities is guaranteed. The electrostatic potential is self-consistently determined by Poisson's equation on the whole device. We show that the hybrid model is well-posed, prove existence of solutions and show their uniform boundedness provided the distribution function satisfy a so-called balance condition. The current densities are different from zero in the non-equilibrium case and uniformly bounded

Macroscopic modeling of quantum effects in semiconductor devices

This dissertation explores the use of macroscopic quantum hydrodynamic (QHD) models as tools for investigating the transport of charge carriers in semiconductor devices in the regime where quantum effects are important. Chapter 1 provides a panoramic view of the field of carrier transport modeling in semiconductors. The essential differences between classical and quantum transport is brought out and a brief outline is given of the derivation of successively less detailed models from the fundamental starting points of the Boltzmann transport equation (BTE) for classical transport and the quantum distribution function (Wigner function, density matrix) based methods for quantum transport. A mention is made of the various quantum hydrodynamic models without going into the details of their derivation and applicability. Chapter 2 brings into focus the area of quantum hydrodynamic modeling of carrier transport. A detailed derivation using the method of moments is presented for each of the popular quantum hydrodynamic models currently being explored in the literature, namely the density-gradient method and the smooth quantum potential model. A summary is made of their limitations and these limitations are then shown as arising out of particular assumptions made in their derivations that could hamper their applicable regimes. Chapter 3 presents an analysis of the boundary layers near interfaces obtained in density-gradient theory. An integral equation for the density near such interfaces is obtained and this is used to analytically compare the DG solution with the solutions from one-electron quantum mechanics in non-degenerate conditions. Confinement in simple potential wells is then discussed using the macroscopic equations. Chapter 4 discusses the derivation of macroscopic equations to describe quantum mechanical tunneling through large barrier potentials. Using the approximate solutions of the Schr?dinger equation it is analytically shown that the density profile inside the barrier satisfies a second order differential equation, very similar to the Schr?dinger equation for a carrier at a suitably chosen average energy. Use of this is made to derive a consistent macroscopic treatment of tunneling transport in the insulating barrier. Chapter 5, the final chapter, summarizes the major contributions of this dissertation and concludes it with several suggestions for future research directions that can stem from this work