THE SOFT X-RAY AND EXTREME ULTRAVIOLET REGIONS OF THE ELECTROMAGNETIC SPECTRUM

One of the last regions of the electromagnetic spectrum to be developed is that between ultraviolet and x-ray radiation, generally shown as a dark region in charts of the spectrum. It is a region where there are a large number of atomic resonances, leading to absorption of radiation in very short distances, typically measured in nanometers (nm) or micrometers (microns, µm), in all materials. This has historically inhibited the pursuit and exploration of the region. On the other hand, these same resonances provide mechanisms for both elemental (C, N, O, etc.) and chemical (Si, SiO2, TiSi2) identification, creating opportunities for advances in both science and technology. Furthermore, because the wavelengths are relatively short, it becomes possible both to see smaller structures as in microscopy, and to write smaller patterns as in lithography. To exploit these opportunities requires advances in relevant technologies, for instance in materials science and nanofabrication. These in turn lead to new scientific understandings, perhaps through surface science, chemistry, and physics, providing feedback to the enabling technologies. Development of the extreme ultraviolet and soft x-ray spectral regions is presently in a period of rapid growth and interchange among science and technology. Figure 1.1 shows that portion of the electromagnetic spectrum extending from the infrared to the x-ray region, with wavelengths across the top and photon energies along the bottom.

In our studies of radiation from charged particles moving at velocities approaching that of light, a number of interesting phenomena are observed, such as the searchlight effect wherein radiation from the charged particle is constrained to a very narrow forward radiation cone. Furthermore, the calculation of detailed angular radiation patterns, in the frame of reference moving with the charged particle, and wavelength distributions are readily accomplished. The results can then be transformed back to the laboratory, or observer, frame of reference. For instance, the calculation of undulator radiation reduces to use of the well-known formula for so-called dipole radiation from a simple oscillating electron. With this approach we need solve Maxwell's equations for only the simplest radiating system, a small amplitude oscillating electron. This approach is not only simple to follow, but gives valuable physical insights to the radiation process and the parameters that characterize it.

In order to relate calculations in one frame of reference to those in another frame of reference when the relative speed between the two approaches that of light, we must make use of the Lorentz space-time transformations, which provide relationships between spatial and temporal scales in the two frames of reference, and are consistent both with Einstein's postulates of special relativity and with all known experiments (see Ref. 2 for a discussion of the Lorentz transformations and their reduction to Galilean transformations as v/c → 0).

This book is intended to provide an introduction to the physics and applications of soft x-rays and extreme ultraviolet (EUV) radiation. These short wavelengths are located within the electromagnetic spectrum between the ultraviolet, which we commonly associate with sunburn, and harder x-rays, which we often associate with medical and dental imaging. The soft x-ray/EUV region of the spectrum has been slow to develop because of the myriad atomic resonances and concomitant short absorption lengths in all materials, typically of order one micrometer or less. This spectral region, however, offers great opportunities for both science and technology. Here the wavelengths are considerably shorter than visible or ultraviolet radiation, thus permitting one to see smaller features in microscopy, and to write finer patterns in lithography. Furthermore, optical techniques such as high spatial resolution lenses and high reflectivity mirrors have been developed that enable these applications to a degree not possible at still shorter wavelengths. Photon energies in the soft x-ray/EUV spectral region are well matched to primary resonances of essentially all elements. While this leads to very short absorption lengths, typically one micrometer or less, it provides a very accurate means for elemental and chemical speciation, which is essential, for instance, in the surface and environmental sciences. Interestingly, water is relatively transparent in the spectral region below the oxygen absorption edge, providing a natural contrast mechanism for imaging carbon-containing material in the spectral window extending from 284 to 543 eV. This provides interesting new opportunities for both the life and the environmental sciences.

A lot of features connected with absorption and emission of light in nanocrystals can be understood in terms of the quantum confinement approach. In this approach, a nanocrystal is considered as a three-dimensional potential box in which photon absorption and emission result either in a creation or in an annihilation of some elementary excitations in an electron subsystem. These excitations are described in terms of quasiparticles known for bulk crystals, that is, electrons, holes, and excitons.

This chapter is meant to remind readers of some principal results from elementary quantum mechanics and to provide an elementary introduction to solid state physics, which is essential for the following chapters. We then depart from elementary “particle-in-a-box” problems and consider the properties of an electron in a periodic potential. In the next step, we introduce the concepts of effective mass and quasiparticles as elementary excitations of a many-body system. Finally, we give an idea of the low-dimensional structures that constitute, undoubtedly, one of the major fields of research in modern condensed-matter physics.

A few problems from elementary quantum mechanics

Particle in a potential well

To restate some basic properties of quantum particles that are necessary to consider electrons in a crystal, we start with a particle in a one-dimensional potential well (Fig. 1.1).