Transport Through Mesoscopic Semiconductor Microstructures: Making Semiclassics Quantitative

Since the mid-1980s, people have been able to fabricate mesoscopic semiconductor devices ofhigh electronic mobility with the great progress in modern science, material preparation andmicro-machining technology. As the devices are increasingly miniaturized due to Moore’s law,research on the transport in mesoscopic Systems has evolved into a rapidly progressing andexciting interdisciplinary field in condensed matter physics, which not only has importantsignificance of the basis research, but provides further physical basis to the development for newprinciples and structures solid state electronic devices and optoelectronic devices. On this basis,research on the electronic transport phenomena such as quantum effect, ballistic transport,conductance fluctuations and coulomb blockade, seeking to break through the physical limits ofthe existing solid state devices, and preparation of the quantum devices, provides important basicidea.Mesoscopic systems build a bridge between the microscopic particles which are unobservableand immeasurable by the naked eye such as atomic electrons on the one side and themacroscopic objects at the length scale upon which Boltzmann transport theory has come to beapplied on the other side. Mesoscopic semiconductor devices, whose dimensions areintermediate between a few nanometers and hundreds micrometers, exhibit both classical andquantum signatures. At very low temperature, the phase relaxation length of the electrons in thestructure is retained over micron scales, which is beyond the dimension of the microstructure,and the coherence of the electron wavefunctions may extend over the whole system. Thusmesoscopic behavior calls for new theoretical methods which combine statistical concepts andassumptions on the one hand with tools to treat coherent quantum mechanics on the other hand.The physical properties of electrons in the system are dominated by quantum mechanics, whichmakes it possible to observe and measure some important quantum behaviors.We know the wave-like nature of electrons will be very obvious when the scale of the devicesand the phase-relaxation length of electrons are analogous, and we should consider theinterference effect caused by different classical paths. Due to the quantum interference effect ofelectrons, transport in mesoscopic system is quite different from that of macroscopic system,which makes the physical properties of mesoscopic semiconductor microstructures differentfrom that of macroscopic conductor. According to the uncertainty principle, it is known that theelectrons cannot have definite momentums and positions at the same time in such small scale. Thus we should develop the electronic transport theory based on Schr dinger equation to dealwith small scale semiconductor devices.Examples of systems whose behaviors can be classified generally as mesoscopic are found invarious fields of physics: nuclear scattering processes, strongly perturbed Rydberg atoms,polyatomic molecules, clusters,“quantum corrals”, acoustic waves, microwaves and opticalradiation in cavities, and electrons in small metallic particles or semiconductors of reduceddimensionality. The latter, mesoscopic semiconductor devices, will be the focus of application ofthe theoretical concepts presented in this paper. The emergence of quantum interference in thesestructures has given rise to a variety of surprising effects: universal conductance fluctuations indisordered samples, quantized conductance, persistent currents in rings, weak localization,Aharonov–Bohm effects, and so on. These quantum phenomena constitute the heart ofmesoscopic physics.Mesoscopic physics has emerged as a new, interdisciplinary field combining concepts ofatomic, molecular, cluster, and condensed-matter physics. On the one hand mesoscopic systemsrepresent an important class of electronic devices in the rapidly growing fields of micro-andnano-physics. On the other hand mesoscopic physics has posed conceptually new questions totheory. Initially, disordered metals were the focus of interest in mesoscopic physics. The adventof high-mobility semiconductor heterostructures, the basis of the physics of two-dimensionalelectron gases, and the advances in lithographic techniques have allowed the confinement ofelectrons in nanostructures of controllable geometry. These rather clean systems, where impurityscattering is strongly reduced, have been termed ballistic since scattering comes from specularreflection on the boundary. The wide ranges of experimentally accessible systems–metal andsemiconductor, disordered and ballistic, normal and superconducting, have made mesoscopicphysics an interface between apparently different theoretical approaches.There are, on the one hand, methods which have been especially designed to deal with randompotentials in disordered metals. Traditionally, diagrammatic perturbation theory in a randompotential has been a very useful tool. During the last decade, powerful nonperturbative methods,in particular the supersymmetry method, have attracted considerable interest and have beenapplied to a large number of different problems in mesoscopics. On the other hand, quantuminterference effects in these small, low-dimensional electronic systems have led to various fancyphysical phenomena. Moreover, they often show features of coherent quantum mechanicscombined with statistical properties and classical chaos. Quantum chaos, as a fancy discipline,devotes itself to the relation between classical and quantum mechanics; in particular, the questionof how classical chaotic behavior is reflected on the level of the corresponding quantum system. These systems are usually too complex to treat starting from microscopic models. Approachesdealing with quantum chaos have been directed towards mesoscopic physics, since thesemethods appear promising for combining statistical concepts with quantum coherence. Hencemesoscopics has developed into a prominent field of application of quantum chaos. Moreover,phase coherent ballistic nanostructures can be regarded as ideal laboratories for investigatingchaos in quantum systems.The study of electronic transport through small conductors is one of the most prominentresearch areas in mesoscopic physics. Billiards has traditionally served as prominent modelsystems in the field of classical and quantum chaos: they combine conceptual simplicity–themodel of a free particle in a box–with complexity with regard to the character of the classicaldynamics and to features of the spectra and wavefunctions. Hence, the possibility of realizingsuch quantum billiards in microstructures has been fascinating and has opened a whole branch ofresearch. Investigations on closed billiards have revealed information on the statistics of energylevels and pronounced enhancements of wave functions near unstable periodic orbits. Recently,the study of quantum transport through open billiards has received considerable attention due toadvances in the fabrication of semiconductor structures which led to the experimental realizationof phase coherent scattering devices. The aim of investigations on the electronic transportproperties in mesoscopic systems, on one hand, is to reveal the new effects of the quantumstructures we have known and investigate their physical mechanism and principle, on the otherhand, is to provide physical models and theory basis for the design of the devices based on thequantum structures.Semiclassical approximations are among the most useful tools in describing and analyzingballistic transport in mesoscopic systems. Semiclassical theory, applied to the description ofmesoscopic systems, is a way of handling quantum mechanics problems by a simplified pathintegral formalism, with a focus on the particle’s motion of classical mechanics inside thebilliards along trajectories with a different number of bounces at the boundary, with the purposeof bridging the gap between quantum mechanics and its classical limit. Moreover, on afundamental level, semiclassical approaches allow one to build a link between classicaldynamics of the particle’s motion in the billiards and the quantum leads in a very direct way:each classical trajectory carries an amplitude that reflects its geometric stability and a phasewhich contains the classical action and accounts for quantum interference effect.The thesis is divided into four chapters. The first chapter is introduction, in which we brieflyintroduce the background and the current situation of the development of the research field athome and abroad, and we introduce the relevant knowledge of mesoscopic system down to the last detail. In the next chapter, we give an introduction to the general method of investigating thetransport through mesoscopic semiconductor microstructures-Two-dimensional quantumbilliards has extensively served as model systems in the study of transmission throughmesoscopic microstructures and we present the dynamics of two-dimensional quantum billiardsin detail. In the third chapter, we present the semi-classical theory for the transport throughtwo-dimensional quantum billiards on the basis of Landuer fomula within the framework of theFraunhofer diffraction effect at the lead openings. Due to the fact that the width of the leads iscomparable to the electron de Broglie wavelength, the diffraction effect at the lead openingsshould be taken into account. Considering the diffractive scattering effect at the lead openings,we modify the transmission fomula and calculate the transmission probability to invesgate thecorrespondence between classical theory and quantum theory. Moreover, we invesgate thetransmission through two-dimensional quantum billiards in different fields to reveal the neweffects and we analyse the reasons for the transport through mesoscopic semiconductormicrostructures. In the last chapter are the conclusions of our study and the outlook on the futureapplications.