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Physical Review

Absorption of Light by Atoms in Solids

David L. Dexter

Published Jan 1, 1956• 352 citations
Publisher

Abstract

The interactions of an atom with its neighbors in an idealized solid are taken into account explicitly in treating the absorption of light, as contrasted with the usual introduction of an "effective" field and mass. In the discussion of the magnitude of the absorption coefficient, two cases are treated: one, the absorption by an impurity atom, in which case we are led to an equation similar to Smakula's; and two, the absorption by one of the atoms of the perfect crystal. The computations are based on a simple idealized model whose validity is discussed for existing systems.

DOI
10.1103/physrev.101.48

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The interactions of an atom with its neighbors in an idealized solid are taken into account explicitly in treating the absorption of light, as contrasted with the usual introduction of an "effective" field and mass. In the discussion of the magnitude of the absorption coefficient, two cases are treated: one, the absorption by an impurity atom, in which case we are led to an equation similar to Smakula's; and two, the absorption by one of the atoms of the perfect crystal. The computations are based on a simple idealized model whose validity is discussed for existing systems.
We treat the interaction between an electromagnetic radiation bath and a material system in the semi-classical picture: the radiation is treated according to the Maxwell equations, but the process of emission/absorption is described through the concept of a photon; on the other hand, the material system is modelled according to quantum mechanics, assuming a discrete two-level energy spectrum. This very simple model allows one to elaborate the elementary theory of the absorption/emission coefficients, both for the spontaneous and for the stimulated processes which will be described within the electric dipole approximation. Selection rules for allowed transitions are eventually obtained and the nature of high-order transition is briefly outlined. Finally, a qualitative description of the light amplification by stimulated emission of radiation is presented.
Quantum optical theory of absorption properties of interacting atoms is developed. The concept of local absorptance is introduced as a derivative of the logarithm of intensity with respect to the distance in the vicinity of a given spatial point and a moment of time. The intensity is represented by the quantum and statistically averaged normal product of creation and annihilation operators of the electromagnetic field. The development of an analytical method of the estimation for the kinetic and optical parameters for the system is proposed here. The calculation method of the absorption coefficient includes thermal atomic motion, Doppler effect, and the short-range interaction between atoms. The absorption coefficient explicitly takes into account the quantum nature of the optical field. The ability of the system to absorb or emit quanta is quantitatively expressed through the special form of interaction integrals. The specific form of integrals results from the structure of the quantum brackets. The interplay between the collective (virtual photon exchange) and binary (optically induced inter-particle bonding) processes determines the system behavior. The spectral profile of the local absorption coefficient for different atomic densities and time intervals is simulated for realistic parameters.
The interaction of light with nanometer-sized structures is at the core of nano-optics. It is obvious that as the particles become smaller and smaller the laws of quantum mechanics will become apparent in their interaction with light. In this limit, continuous scattering and absorption of light will be supplemented or replaced by resonant interactions if the photon energy hits the energy difference of discrete internal (electronic) energy levels. In atoms, molecules and nanoparticles, like semiconductor nanocrystals and other “quantum confined” systems, these resonances are found at optical frequencies. Due to the resonant character, the light–matter interaction can often be approximated by treating the quantum system as an effective two-level system, i.e. by considering only those two (electronic) levels whose difference in energy is close to the interacting photon energy ħ ω 0 . In this chapter we consider single-quantum systems that are fixed in space, either by deposition to a surface or by being embedded into a solid matrix. The material to be covered should familiarize the reader with single-photon emitters and with concepts developed in the field of quantum optics. While various theoretical aspects related to the fields emitted by a quantum system have been discussed in Chapter 8, the current chapter focuses more on the nature of the quantum system itself. We adopt a rather practical perspective since more rigorous accounts can be found elsewhere (see e.g. [1–4]).
Introduction In most of the discussion presented so far in this book, the quantum character of atoms and electrons has been ignored. The Ising spin models have been an exception, but since the Ising Hamiltonian is diagonal (in the absence of a transverse magnetic field), all energy eigenvalues are known and the Monte Carlo sampling can be carried out just as in the case of classical statistical mechanics. Furthermore, the physical properties are in accord with the third law of thermodynamics for Ising-type Hamiltonians (e.g. entropy S and specific heat vanish for temperature T → 0, etc.) in contrast to the other truly classical models dealt with in previous chapters (e.g. classical Heisenberg spin models, classical fluids and solids, etc.) which have many unphysical low temperature properties. A case in point is a classical solid for which the specific heat follows the Dulong–Petit law, C = 3 Nk B , as T → 0, and the entropy has unphysical behavior since S → –∞. Also, thermal expansion coefficients tend to non-vanishing constants for T → 0 while the third law implies that they must be zero. While the position and momentum of a particle can be specified precisely in classical mechanics, and hence the groundstate of a solid is a perfectly rigid crystal lattice (motionless particles localized at the lattice points), in reality the Heisenberg uncertainty principle forbids such a perfect rigid crystal, even at T → 0, due to zero point motions which ‘smear out’ the particles over some region around these lattice points. This delocalization of quantum-mechanical particles increases as the atomic mass is reduced; therefore, these quantum effects are most pronounced for light atoms like hydrogen in metals, or liquid helium. Spectacular phenomena like superfluidity are a consequence of the quantum nature of the particles and have no classical counterpart at all. Even for heavier atoms, which do not show superfluidity because the fluid–solid transition intervenes before a transition from normal fluid to superfluid could occur, there are genuine effects of quantum nature. Examples include the isotope effects (remember that in classical statistical mechanics the kinetic energy part of the Boltzmann factor cancels out from all averages, and thus in thermal equilibrium no property depends explicitly on the mass of the particles).