Mesoscopic Effects in Metal Grains and Quantum Dots

小金属颗粒或半导体量子点的单粒子特性可以用随机矩阵理论来描述：能级和波函数与实数、复数或四元数的大厄米特矩阵的特征值和特征向量具有相同的统计量，这取决于时间反转对称性和自旋-旋转对称性的存在与否。Kurland， Aleiner和Altshuler提出，这种“随机矩阵极限”中的电子-电子相互作用效应可以用“通用相互作用哈密顿量”来描述，该哈密顿量只有在随机矩阵系综的对称变换下不变的相互作用项。这些相互作用是自平均和非随机的，因此被称为“普遍的”。本研究的第一部分解决了当对称性“部分”破坏时相互作用的作用，例如，在自旋轨道散射存在的情况下，由于太弱而无法在时间内完全随机化电子自旋，从而无法遍历时探索颗粒或点。对于这样的“部分破缺对称”，交互作用的哈密顿量是一个随机量，因为交互作用矩阵元素不再是自平均的。这种相互作用哈密顿量的波动可能会影响金属晶粒中单个电子能级的g因子，或者在临界温度附近超导金属晶粒中超导波动的抑制。本研究还讨论了金属晶粒中的斯通纳不稳定性以及在不稳定性之外形成的铁磁态。如果存在自旋轨道散射，则不稳定性方法和铁磁态本身都显示出有趣的介观现象。在强自旋-轨道耦合下，交换相互作用矩阵元素虽小且违反随机矩阵理论的辛对称性，但它们数量众多，且与自旋-轨道散射强度无关，可触发铁磁态的形成。因此，这种不稳定性为控制计算提供了背景，相互作用如何使系统远离随机矩阵。对于铁磁体，波函数空间部分的随机性通过自旋轨道耦合与磁化耦合，产生各向异性能量的“介观”分量。研究的第二部分涉及量子点的随时间传输。时变运输既有实际意义又有根本意义。实际的兴趣产生了，因为速度是大多数电子设备关注的问题。时间相关输运（以及非线性响应）是一个基本的兴趣，因为电子-电子相互作用在时间相关输运中比在时间无关输运中起着更重要的作用。将进行详细的计算，以深入了解电容充电能量如何影响通过点接触连接到电极的量子点中的相参时相关输运。虽然在这些系统中电荷量子化被解除，但库仑封锁对输运性质仍然很重要。除了在纳米尺度上推进对金属和半导体的基本科学理解外，该研究还将提供培训材料，帮助理论研究生发展成为全面发展的凝聚态物理学家。本科生也将参与研究。研究生还将参加康奈尔大学MRSEC的外展活动。这项理论研究将解决一些与纳米级金属和半导体性质有关的问题。该研究具有非常基本的分支，并且作为该领域的典型，实际应用近在咫尺。除了在纳米尺度上推进对金属和半导体的基本科学理解外，该研究还将提供培训材料，帮助理论研究生发展成为全面发展的凝聚态物理学家。本科生也将参与研究。研究生还将参加康奈尔大学MRSEC的外展活动

Single-particle properties of a small metal grain or semiconductor quantum dot can be described by random-matrix theory: energy levels and wavefunctions have the same statistics as eigenvalues and eigenvectors of a large hermitian matrix with real, complex or quaternion numbers, depending on the presence or absence of time-reversal symmetry and spin-rotation symmetry. Kurland, Aleiner and Altshuler proposed that electron-electron interaction effects in this "random matrix limit" are described by a "universal interaction Hamiltonian," which has only interaction terms that are invariant under the symmetry transformations of the random matrix ensemble. These interactions are self-averaging and non-random, hence the label "universal." The first part of this research addresses the role of interactions when symmetries are "partially" broken, e.g., in the presence of spin-orbit scattering that is too weak to fully randomize the electron spin within the time to ergodically explore the grain or dot. For such "partially broken symmetries," the interaction Hamiltonian is a random quantity, since interaction matrix elements are no longer self-averaging. Such fluctuations of the interaction Hamiltonian may effect g factors of individual electronic levels in metal grains, or the suppression of superconducting fluctuations in a superconducting metal grain near the critical temperature. This research also addresses the Stoner instability in metal grains and the ferromagnetic state formed beyond the instability. Both the approach of the instability and the ferromagnetic state itself display interesting mesoscopic phenomena if spin-orbit scattering is present. With strong spin-orbit coupling, exchange interaction matrix elements are small and violate the symplectic symmetry of random matrix theory, yet, they are many, and trigger the formation of a ferromagnetic state irrespective of the spin-orbit scattering strength. Hence, this instability provides the context for a controlled calculation how interactions may drive systems away from the random matrix. For a ferromagnet, randomness in the spatial part of the wavefunction couples to the magnetization through spin-orbit coupling, giving rise to a "mesoscopic" component of the anisotropy energy.The second part of the research deals with time-dependent transport in quantum dots. Time-dependent transport has both practical and fundamental interest. Practical interest arises because speed is a concern in most electronic devices. Time-dependent transport (as well as nonlinear response) is of fundamental interest because electron-electron interactions play a much more important role for time-dependent transport than for time-independent transport. A detailed calculation will be done that should give insight how the capacitive charging energy affects phase-coherent time-dependent transport in quantum dots connected to electrodes via point contacts. Although charge quantization is lifted in those systems, Coulomb blockade continues to be important for transport properties.Besides advancing the fundamental scientific understanding of metals and semiconductors on the nanometer scale, the research will provide training material to help theory graduate students develop into well-rounded condensed matter physicists. Undergraduates will also participate in the research. Graduate students will also participate in the outreach activities of the Cornell MRSEC.%%%This theoretical research will address a number of issues relating to the properties of nanoscale metals and semiconductors. The research has very fundamental ramifications and, as is typical of this field, practical applications are near.Besides advancing the fundamental scientific understanding of metals and semiconductors on the nanometer scale, the research will provide training material to help theory graduate students develop into well-rounded condensed matter physicists. Undergraduates will also participate in the research. Graduate students will also participate in the outreach activities of the Cornell MRSEC.***