Quantum Physics of Electron Statistics, Ballistic Transport, and Photonics in Semiconductor Nanostructures Debdeep Jena (
[email protected]) Departments of ECE and MSE, Cornell University 1. Introduction This article discusses the quantum physics of electron and hole statistics in the bands of semiconductors, the quantum mechanical transport of the electron and hole states in the bands, and optical transitions between bands. The unique point of view presented here is a unified picture and single expressions for the carrier statistics, transport, and optical transitions for electrons and holes in nanostructures all dimensions - ranging from bulk 3d, to 2d quantum wells, to 1d quantum wires. The focus in the early parts is on nanostructures of dimensions d = 1, 2, 3 that allow transport, and the 0d quantum dot case is discussed for photonics. 2. Electron Energies in Semiconductors 2 2 Electrons in free space have continuous values of allowed energies E(k) = h¯ k , where h¯ = h/2p is the 2me h reduced Planck’s constant, me the rest mass of an electron, and k = 2p/l is the wavevector. hk¯ = l = p is the momentum of the free electron by the de Broglie relation of wave-particle duality. A periodic potential V(x + a) = V(x) in the real space of a crystal, when included in the Schrodinger equation, is found to 2 2 split the continuous energy spectrum electron energies E(k) = h¯ k into bands of energies E (k), separated 2me m th by energy gaps. The m allowed energy band is labeled Em(k). The states of definite energy Em(k) have ikx real-space wavefunctions yk(x) = e uk(x), where uk(x + a) = uk(k), called Bloch functions.