Mesoscopic Effects in Metal Grains and Quantum Dots

Cornell University, Ithaca NY

Investigators

Abstract

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. ***