Introduction to the electrical effects of dislocations and other defects in semiconductors

In this chapter the historically important influence of high densities of dislocations on the electrical properties of semiconductors and on device performance is first outlined. (In the early days of semiconductor studies, high densities were generally either grown-in or introduced into the material by plastic deformation.) The mechanisms giving rise to the electronic and luminescence properties of dislocations and other defects are next treated. The role of defects in devices is discussed. The electrical properties of grain boundaries in polycrystalline semiconductors are also treated.

Introduction

The first short paper reporting that plastic deformation of Ge and Si was possible at raised temperatures, also reported that this increased the resistivity of Ge and the lifetime of photo-injected carriers was greatly reduced (Gallagher 1952). Further early studies revealed that <1% plastic strain would eliminate all the electrons in lightly doped n-type Ge and turn it p-type. (Ge was the important semiconductor at that time.) The effects of dislocations were so important that much basic dislocation theory is due to workers in the pioneering group at Bell Laboratories, especially Shockley (who introduced ‘dangling bonds’ and Shockley partial dislocations) and Read (of the Frank-Read source and the first theory of the electrical effects of dislocations). However, the industrial laboratories lost interest once it was found possible to grow low or zero dislocation density Si and effectively avoid or passivate process-induced dislocations.

Atomic core structure of dislocations

Extended defects disturb the crystal structure over many atomic distances in one or more dimensions. Those that continue to be of practical importance are interfaces and dislocations.

Extended defect cores in semiconductors are of relatively high energy due to the directional character of tetrahedral covalent bonds. If such a bond is bent from its tetrahedral direction by the displacement of the neighbouring atom, the bond energy rises rapidly. Hence sp3 bonds resist bending as if they were elastically stiff. If the neighbour atom is too far off the tetrahedral direction or at too great a distance, the bond energy would be too high and a broken or ‘dangling’ bond occurs. Hence, extended semiconductor defects can be modelled using plastic spheres with tetrahedrally drilled holes or protrusions and wire or plastic straws to connect them. Since such connections are also stiff and brittle, ball-and-wire (or caltrop-and-spoke) models can give insight, through their ease or difficulty of construction, into the likelihood of occurrence of particular atomic core arrangements in dislocations and grain boundaries. Such modelling was introduced by Hornstra (1958, 1959, 1960).

The high energetic cost of broken and strained bonds leads to a tendency for dislocations and grain boundaries to be crystallographically aligned to minimize the number of such bonds in the core. This contrasts with the curved or arbitrarily directed defects seen in many metals. Thus, dislocations in covalently bonded semiconductors are constrained to lie in deep crystallographic Peierls troughs (see Section 2.5.2 and Fig. 2.24).

Introduction

This chapter outlines the principles, advantages and limitations of the methods in use for the characterization of extended defects and should enable the reader to appreciate the experimental results presented. For additional accounts see Brundle et al. 1992, Yacobi et al. 1994, Schroder 1998, and Runyan and Shaffner 1998.

Characterization methods can be classified as (i) either surface or bulk techniques, as (ii) either destructive or non-destructive methods, and as (iii) either requiring the application of contacts or not. We have excluded the free surface from consideration on the grounds that, like point defects, its study constitutes a large specialized field already covered by many publications. We therefore also omit surface microscopy and analysis techniques here. Generally, non-destructive techniques are preferred as are contactless ones. However, in practice neither is a very important factor as generally one or a few specimens can be sacrificed for destructive examination and contacts can usually be applied.

Characterization techniques are essential for failure analysis and quality control of semiconductor materials and devices. Often failure modes are associated with manufacturing process-induced defects or with defect-dependent device degradation in service.

Electrical measurements for the analysis of a wide range of semiconductor transport properties such as, for example, resistivity (conductivity), Hall effect and capacitance-voltage measurements are made on whole bulk specimens and devices. The net influence on these properties of all the defects present then appears in the results.

Introduction

One final category of extended defect remains, although it is not generally so described. This consists of undesired, non-uniform distributions of native point defects, impurities and alloy composition variations. These point defect maldistributions render the initial material properties non-uniform and interfere with the controlled introduction of the variations required for devices.

Semiconductor processing starts with material that is uniformly sufficiently pure and perfect to exhibit the intrinsic properties of the semiconductor. Controlled concentrations of selected impurities are then introduced into chosen volumes to achieve the desired extrinsic properties. These include, for example, p- or n-type conductivity of the necessary value or luminescent emission of a certain wavelength and efficiency. For this, the impurity must occur as a uniform, random distribution of single, isolated impurity atoms of the desired element on substitutional sites in the required concentration. This chapter is concerned with crystal growth phenomena affecting point defect distributions and so materials uniformity and, therefore, capable of leading to failure to achieve successful device fabrication.

Because of their importance and relative simplicity, point defects have been studied intensively throughout the history of semiconductor physics and chemistry. The properties of point defects are therefore well treated in many review articles (Queisser and Haller 1998) and books such as Stoneham (2000) as well as series of conferences. We shall, therefore, give only the necessary minimum background on a number of points required for the present purpose.

Analogies play a central role in physics. The CF theory was motivated by the analogy between the fractional and the integral quantum Hall effects. Other analogies were proposed prior to the CF theory. As discussed in Section 5.5, the Laughlin wave function was motivated by the Jastrow wave functions used earlier for 4He superfluids. A “composite boson” approach modeled the physics of the Laughlin wave function after Bose–Einstein condensation. A “hierarchy” approach proposed understanding the FQHE at non-Laughlin fractions by using the Laughlin wave function as the basic building block. Both the hierarchy and composite boson approaches take the Laughlin wave function as their starting point. These ideas, as seen below, are distinct from the CF theory.

This chapter also presents certain other non-CF works. The simple Jastrow form of the Laughlin wave function allows for certain technical simplifications; in particular, a mapping into a classical plasma enables alternative, but 1/m-specific, derivations of certain properties that were obtained in earlier chapters by other means. In addition, we describe two quantitative approaches for excitations without using composite fermions: Laughlin's wave functions for the quasiparticle and quasihole at v =1/m, and Girvin, MacDonald, and Platzman's “single-mode approximation” for neutral excitations.

We also briefly outline the hydrodynamic theory of Conti and Vignale, and Tokatly, which treats the correlated liquid state in the lowest Landau level as a continuous elastic medium, and formulates its collective dynamics in terms of the displacement field.

Let us consider fully spin-polarized electrons in one dimension. Because of the Pauli principle, each electron is confined inside a box marked by its two neighbors on either side. As an electron moves, it collides with the neighboring electrons and, through a two-way domino effect, the dynamics becomes collective. (This is to be contrasted with higher dimensions where electrons can budge sideways to make way for a moving electron.) Electron-like quasiparticles are no longer well defined in one dimension, resulting in a breakdown of Landau's Fermi liquid theory. An understanding of the effect of interactions requires a nonperturbative treatment. Interacting liquids in one dimension are called Tomonaga–Luttinger (TL) liquids. We see in this chapter that the FQHE edge constitutes a realization of a Tomonaga–Luttinger liquid. (Certain organic or blue bronze type conductors have stacks of weakly coupled 1D chains, which behave as independent 1D conductors at high temperatures, when the coupling is irrelevant, but become three dimensional at low temperatures.)

QHE edge = 1D system

We have learned that excitations cost a nonzero energy in a pure QHE system. That is indeed true for the spherical samples of computer experiments, but not quite true for a sample in the laboratory. Excitations with arbitrarily low energies are available at the boundary of a QHE system. The dynamics of these excitations is equivalent to that of a one-dimensional system.

Whenever new emergent particles are postulated, one must ask: “So what? What do these particles do for us?” The significance of such particles is determined by the scope and the importance of experimental phenomena they produce. For particles advertised as truly exotic to show up in but minor experimental details would be hardly satisfying.

The remainder of this book concerns experimental manifestations of composite fermions. Before proceeding to laboratory experiments, we ask, in this chapter, if composite fermions can be “seen” in computer experiments.

Computer experiments

Let us recall how quantum mechanics was verified. It began by explaining the spectral lines of the hydrogen atom. With further work, its predictions were seen to be in exquisite agreement with the excitation spectra of helium and other atoms in the periodic table, and also of molecules. This vast body of exceedingly detailed comparisons, now known as quantum chemistry, is what led to the establishment of quantum mechanics as the quantitative theory of matter. Quantum mechanics was then used with great effect in systems containing many electrons, for example metals and superconductors, and now the fractional quantum Hall effect.

This chapter is devoted to the quantum chemistry of composite fermions. Nature does not produce any real “FQHE atoms,” but, fortunately, they can be created on the computer. These atoms are characterized by two parameters: the number of electrons and the amount of magnetic flux to which they are exposed. Theorists can perform computer experiments to determine their exact spectra and wave functions.

This chapter is devoted to the properties of incompressible ground states and their excitations. Ground state properties, such as pair correlation function and static structure factor, as well as energies of several kinds of charged and neutral excitations are evaluated using the CF theory. Quantities in the thermodynamic limit are obtained by an extrapolation of finite system results. Experimental studies of the FQHE state through transport, inelastic light scattering, and scattering by ballistic phonons have identified a variety of excitations which can be compared to theory. As we have seen in Chapter 6, the CF theory predicts excitation energies with an accuracy of a few % for N < 12. A reasonable expectation is that the thermodynamic limits obtained from the CF theory are also accurate to within a small percentage. Unfortunately, a comparison with real life experiments also necessitates an inclusion of the effects of nonzero thickness of the electron wave function, Landau level mixing, and disorder, which are not as well understood as the FQHE, and the accuracy of quantitative comparisons between theory and experiment is determined largely by the accuracy with which these other effects can be incorporated into theory. At present, the quantitative agreement between theory and laboratory experiment is roughly within a factor of two, although a 10–20% agreement has been achieved in some cases. Fully spin-polarized FQHE is assumed throughout this chapter; the spin physics is the topic of Chapter 11.