【作者】 阎占元；

【导师】 刘建新；

【作者基本信息】 华北电力大学（河北） ， 电工理论与新技术， 2007， 博士

【摘要】 随着微电子学和纳米技术的飞速发展,集成电路的基本器件已经到了介观的尺度,电路的量子效应凸显。科学研究和生产实践中急需建立关于介观电路的量子理论。人们对基本介观电路的量子化通常是和经典简谐振子的量子化对比实现的,并在此基础上对基本介观电路做了深入的研究。但是,只有考虑了电路中电荷应取分立值的量子理论才是更全面的量子理论。在介观电路的全量子理论的框架下,本文研究了一些基本介观电路的量子效应,给出了处理电路中电阻、电源、耦合相的方法,并应用于研究实际的介观器件,本文还首次研究了介观电路的稳定性性质。考虑电路中电荷应取分立值的事实,给出了介观RLC电路系统的差分薛定谔方程。利用么正变换的方法,可以把电阻和电源耦合项消掉,在p?表象中,系统的薛定谔方程变为马丢方程的形式,从而实现了介观RLC电路的求解。并在WKBJ近似下计算了稳态电路的能谱和电流的量子涨落。通过合适选取系统哈密顿的形式,不仅能够实现介观含源RLC电路的量子化,给出该电路的瞬态薛定谔方程,还可以解决电阻给薛定谔方程的求解带来的困难。在p?表象中,薛定谔方程变为标准的马丢方程,利用WKBJ方法求得了系统的能级和本征态,并计算了电流在各本征态中的量子涨落。对介观电容电感耦合电路,写出了系统的拉格朗日和哈密顿。通过变换消去耦合相,系统薛定谔方程变为两个谐振子方程,完成了该电路的量子化过程,进而研究了该电路中由电荷分立取值引起的库仑阻塞效应。利用WKBJ方法对薛定谔方程求解,对电流的量子涨落及其相互关系进行了研究。介观电子谐振腔是一个新制备的介观装置,引起了人们的广泛关注。在这一装置的等效哈密顿基础上,利用本文研究基本介观电路的方法实现了该系统的量子化。该系统的差分薛定谔方程在电流的表象中可以变为马丢方程的形式,从而系统最基本的量子性质——能级、本征态和电流的量子涨落可以在WKBJ近似下完成。在基本介观电路的量子化过程中,如果考虑电荷取分立值这一基本事实,系统的薛定谔方程变为马丢方程。马丢方程是典型的微分方程,它的稳定性对介观电路的影响还是一个全新的课题。本文在二级近似下,研究了介观电子谐振腔和含源介观RLC电路的稳定性问题,计算出了这两个电路的稳定区和非稳定区分布情况。介观电路的全量子理论可以求解基本介观电路的能级和波函数,同时可以研究电路中电流的量子涨落,电流的量子涨落导致电路中普遍存在量子噪声,这些介观电路的最基本的量子性质值得人们深入研究,对于进一步设计微小电路、降低量子噪声具有指导意义。更多还原

【Abstract】 With the rapid development in microelectronics and nanotechnology, the electronic components in integrated circuits have reached to a mesoscopic scale, and the quantum effects in the circuits become more and more important. A quantum theory to mesoscopic circuits is demanded eagerly in the experiment and manufacture. Comparing with the quantization of a harmony, the quantum effects in the fundamental mesoscopic circuits have been studied recently. However, the charge in the circuits is discrete, this character is seldom considered. A profound quantum theory about the mesoscopic circuits should consider the discreteness of the charge. In the framework of an improved quantum theory, the quantum effects in mesoscopic circuits are studied. The methods to deal with resistance phase, source phase, couple phase are proposed and applied to a mesoscopic device.The nonlear characters of mesoscopic circuits are firstly studied.Based on the discrete character of charge, the finite-differenced Schr?dinger equation of mesoscopic RLC circuit is achieved. The couple phase in the Schr?dinger equation could be eliminated with a unitary translation, and then the finite-differenced Schr?dinger equation becomes a standard Mathieu equation in p? -representation. Using the WKBJ method, Schr?dinger equation is solved, the stable energy spectrum and wave functions of the system are obtained, the average of currents and square of the current are calculated.With an appropriate formation of Hamilton, the quantization of a transient RLC mesoscopic circuit with source is performed, not only the transient finite-differenced Schr?dinger equation is achieved, but also the resistance phase problem in the equation is skillfully solved. With a unitary transformation, the finite-differenced Schr?dinger equation becomes a standard Mathieu equation in p? -representation. Using the WKBJ method, the energy spectrum and the wave functions of the system are obtained, the average of currents and square of the current are calculated.The Lagrange and Hamilton of the mesoscopic inductance and capacity coupling circuit are achieved. The couple phase in the Schr?dinger equation could be eliminated with a translation, and then the circuit is quantilazed with the finite-differenced Schr?dinger equation divided into two harmony equations. The Coulomb blockade effect, which is caused by the discreteness of electric charges, is studied. With the WKBJ method, Schr?dinger equations are solved, the currents quantum fluctuations and relationship of the two circuits are studied.The open electron resonator is a mesoscopic device that has attracted considerable attention due to its remarkable behavior-conductance oscillations. Based on anequivalent Hamilton of the device, the open electron resonator system is quantizated with the methods proposed by us. With a presentation transformation, the Schr?dinger equation becomes a standard Mathieu equation, and then the basal quantum character such as energy spectrum and wave functions of the system are obtained. Using WKBJ method, the average of currents and square of the current are calculated.In the case of the discreteness of the charge in mesoscopic circuits, the finite-differenced Schr?dinger equation is a Mathieu function in p? -representation, Mathieu equation is a typical nonlinear equation, and its stability character would be a new subject for the mesoscopic circuits. We firstly studied the stability of the open electron resonator and RLC circuits with a source, the stability and instability regions are discussed in a second approximate.The application of improved quantum theory to mesoscopic circuits can give the energy spectrum and the wave functions of the system. Furthermore the current fluctuation could be calculated, which is a ubiquitous quantum noise in mesoscopic circuits. The deep researching works about these basal quantum characters would be benefit to the control of quantum noise and design of mesoscopic circuits.更多还原