Nonlinear optics in semiconductors

Semiconductors show many nonlinear optical effects. For photon energies near to the bandgap energy, they show strong saturable absorption and nonlinear refraction associated with changes in optical properties that result from optically creating electron-hole pairs in the material.

In the simplest view, the carriers fill states in the conduction and valence bands, blocking them by Pauli exclusion from participating in further absorption processes. Therefore, there is some “absorption saturation” as a result. Those changes in the optical absorption spectrum result in changes in refractive index; through the Kramer-Kronig relations, any change in absorption at some photon energy leads to changes in refractive index at all photon energies, especially those close to the regions of large absorption spectrum changes.

David Miller’s research discovered these nonlinear refractive effects near to the bandgap of semiconductors, which led in particular to further research in optical bistability and optics for logic in digital systems.

Such simple “band filling” models of absorption saturation and nonlinear refraction are often a good first approach, and are relatively good models for narrow bap semiconductors like InSb. For wider bandgap semiconductors, such as GaAs, the optical absorption of semiconductors near to the bandgap energy is very strongly influenced by so-called “excitonic” effects, which enhance both the absorption saturation and the related nonlinear refractive phenomena. Such excitonic effects are particularly strong in quantum well materials.

The absorption saturation effects, especially those enhanced by excitonic effects, can be used as saturable absorbers for mode-locking lasers. David Miller’s research has included several examples of this application.

Excitons – a short introduction

The absorption of photons in semiconductors for photon energies near the bandgap energy is not creating fully “free” electrons and holes; rather it is creating electron-hole pairs that “see” each other through their electrostatic “Coulomb” attraction. The resulting states into which we are creating the electron-hole pair are actually “exciton” states, which, for the excitonic effects in most of the semiconductors used in optoelectronics, can be analogous to the states of the hydrogen atom, both bound and unbound (“ionized”), but based on the effective masses of the electron and hole rather than the actual masses of the electron and the proton in the hydrogen atom. Because the effective masses can be small (e.g., ~ 1/10 of the free electron mass) and because the dielectric constant in semiconductors is often quite large (of the order of 10), and it appears squared in the binding energy formula, the binding energies of these excitons can in the scale of 10 meV rather than the scale of 10eV for electrons in atoms, 3 orders of magnitude smaller. Correspondingly, their effective Bohr radius is more on the scale of 10 nm rather than 0.1 nm.

(Excitons that can be described in this effective mass model are known as Wannier excitons. Another class of excitons that are much more tightly bound, and can be found, for example, in wider gap materials, are known as Frenkel excitons. Though the properties of Frenkel excitons are still in many ways analogous to hydrogenic or atomic systems, they are tightly bound, with sizes that can be comparable to or smaller than a unit cell, effective mass models do not work, and other approaches are needed.)

In this excitonic model, the strength of the corresponding optical absorption is proportional to the probability of finding the electron and hole in the same place in the final state in which we are creating them. (Note that, unlike for the hydrogen atom, in the absorption we are discussing here, this process is creating excitons, not raising them from one exciton state to another. That is not the typical process we are considering with hydrogen atoms, though in principle it exists in the vacuum for the creation of positronium atoms – states of an electron-positron pair – though that process requires two gamma ray photons to complete it.) That probability is particularly large for the s-like bound states of the exciton. This excitonic process leads to the appearance of absorption “peaks” just below the nominal bandgap energy of the semiconductor, where the electron-hole pair is created in “bound” states that therefore have less energy than the “free” electrons and holes would require (which requires photon energies above the bandgap energy). This excitonic process also greatly enhances the optical absorption in the spectral region just above the bandgap energy, even when we are not creating the electron-hole pairs in bound states, a phenomenon known as Sommerfeld enhancement; in a classical analogy, we are creating the electron-hole pair in “hyperbolic” orbits, like those of a comet round a star, but where the electron and hole “spend longer” closer together than we would expect for totally free electrons and holes.

D. S. Chemla, D. A. B. Miller, P. W. Smith, A. C. Gossard and W. Wiegmann, “Room Temperature Excitonic Nonlinear Absorption and Refraction in GaAs/AlGaAs Multiple Quantum Well Structures,” IEEE J. Quantum Electron. QE‑20, 265‑275 (1984).

S. Schmitt-Rink, D. S. Chemla and D. A. B. Miller, “Theory of Transient Excitonic Optical Nonlinearities in Semiconductor Quantum-Well Structures,” Phys. Rev. B32, 6601‑6609 (1985).

S. Schmitt-Rink, D. A. B. Miller, and D. S. Chemla, “Theory of the Linear and Nonlinear Optical Properties of Semiconductor Microcrystallites,” Phys. Rev. B35, 8113‑8125 (1987).