The previous chapters described the unusual electronic phenomena brought about by disorder and interaction and how they can be understood at least qualitatively. To conclude this book, a number of open questions and possible future research directions are addressed. These demonstrate that the field of electron glass is far from being closed and that some of its major issues are not understood or are under debate. First and foremost, there exists no statistical or thermodynamic theory of nonergodic systems, and for that reason there is no full understanding of any glasses – less so of quantum glasses such as is the electron glass. Indeed, the purpose of this book is to present a comprehensive review of the current state of the field, with the anticipation that further experimental, theoretical, and numerical work will follow and help elucidating the issues that remain unclear today.

• Processes responsible for glassy properties At the present, there is no consensus as to the electronic processes that lead to long relaxation and memory phenomena. Does the glassy behavior require many-electron transitions or are single hops enough? Is it only very low energy transitions that linger at very long times? The proposed suggestions for the slow relaxation of conductivity are slow formation of the Coulomb gap, slow formation of quasi-particles, slow manyelectron transitions off the percolation path affecting the paths, many-electron transitions traversing the current carrying path, exchange of particles between the percolation path and clusters off the path, all as discussed in Chapter 7.

Chapter 3 serves as an introduction to glasses and in doing so focuses on what is characteristic of glasses in general. To briefly summarize, the phenomenological common features are an extremely long or perpetual state of nonequilibrium, disobeying time homogeneity (aging), and memory effects. Structurally, the common features are interaction and disorder, the combination of which leads to frustration. As far as experiments are concerned, one can excite the glass with some generalized force – thermodynamically an intensive quantity – and measure a responding dynamical variable (an extensive quantity) or a generalized susceptibility. For example, in structural glasses, one measures deformation or viscosity responding to stress, in spin glasses one measures magnetization or magnetic susceptibility responding to applied magnetic fields, and in electron glasses one measures conductivity or electrical susceptibility in response to applied electric fields. Alternatively, one can measure slow relaxation of this variable after cooling from high temperature.

In relation to Chapter 3, this chapter is primarily intended to point out the more subtle differences between the electron glass and other glasses, with some particular attention to the spin glass, which is a close relative of the electron glass and is the most widely studied glass.

Structural glasses

Before all else, it is useful to start with structural glasses, which, in more than one sense, is the mother of other glasses. The silica glass (“window glass”) is by far the oldest glass that humans took interest in, mainly as an object of fashion.

The evolution of the field of electron glasses has a long glassy history of over half a century. As in experiments on glasses where it is hard to determine the beginning of slow relaxation, so it is also difficult to pinpoint the beginning of the field. Perhaps one should assign the beginning to the first experiments on impurity conduction on SiC by Busch and Labhart in 1946. Experiments on impurity conduction were greatly extended by Fan and Fritzcsche and later by Fritzsche and collaborators at Purdue and Chicago, by Zabrodskii and collaborators in Leningrad, and by Davis in Cambridge in the 1950s. Those experiments dealt with doped germanium and silicon. The surprising result of the experiments was a transition from an Arrhenius behavior due to thermal excitation of carriers into the conduction band to a much smaller activation energy as the temperature was lowered. The effect remained unexplained until Mott (1956) and Conwell (1956) independently attributed the lower activation energy to transitions between impurity states, a process that became known as “hopping.” A more quantitative transport theory based on these explanations is due to Miller and Abrahams (1960). These authors also showed that the problem of hopping conduction can be mapped on a random network of resistors, each resistor connecting a pair of impurities with a resistance in accordance with the hopping rate between them.

A parallel very important development came in 1958 with Anderson's paper “Absence of diffusion in certain random lattices,” which studied systems of spins where the disorder prevents spins from diffusing over long distances, a phenomenon that soon after became known as Anderson localization.

This chapter presents a survey of the dc conductivity in electron glasses within the linear response regime. A deep understanding of the processes responsible for conductivity in the steady state is necessary for attempting to treat the out-of equilibrium phenomena of the conductivity. Special emphasis is given to the effect of electron-electron interactions, which lead to the Coulomb gap and to correlated electronic transitions. These are essential for understanding the glassy properties in these materials.

Section 5.1 is devoted to review the main experimental results on dc conductivity, with emphasis on materials where glassy effects are observed. The different elements needed for conductivity theory in an electron glass are discussed in Section 5.2. The different types of hopping transport are described in Section 5.3, with emphasis on variable range hopping (VRH). The most frequently used approach to solve the previous model is percolation theory, which will be elaborated upon in Section 5.4. Scaling theory is the other approach employed to understand these problems and is reviewed in Section 5.5. The algorithms employed in numerical simulations are detailed in Section 5.6, together with their main results on conductivity. Finally, concluding remarks are summarized.

dc Conductivity: experimental

Impurity conduction in doped semiconductors

If one could pinpoint when serious interest began in understanding the microscopic physics of disordered systems, many would probably agree that observation of impurity conduction at low temperatures provided the driving incentive. These date back to the late 1940s in SiC (Busch and Labhart, 1946).

A glass is a system that does not reach thermodynamic equilibrium during any reasonable experimental time. Therefore, the traditional well-established statistical mechanics methods cannot be applied to such systems. In particular, two concepts that are fundamental to statistical mechanics, namely, ergodicity and the fluctuation-dissipation theorem, cannot be utilized in discussing properties of glasses. This chapter will discuss in more detail the problems inherent in applying these two concepts to glasses and will outline some more of the basic properties of glassy systems. Among glasses, spin glasses are the most similar to electron glasses so their main properties and proposed models will be reviewed. The chapter ends with an introduction to two-level systems, which is suggested as the possible origin of some of the properties of glasses, in particular, those of electron glasses.

The modern concept of glass

Fundamental scientific interest in glasses started a few decades ago with the pioneering works of Anderson and Mott in disordered (i.e., noncrystalline) solids. For fairness' sake, it should be mentioned that much earlier Schrödinger tried to raise attention to such systems when he conceived of the “aperiodic solid” as a probable candidate for carrying the genetic code. He also emphasized the long-term stability of such structures. The advent of DNA proved him of course right.

The first known glass is the common window glass.Awell-known feature typical of this glass has been that it reaches a crystalline state extremely slowly, if ever.

The subject of the electron glass has been evolving from a number of different directions so the terminology in the literature is not uniform. Fleishman and Anderson (1980) first referred to a system of disorder-localized electrons with Coulomb interactions as Fermi glass. Subsequently the term was used for noninteracting Anderson-localized electrons while in the presence of interactions such systems became known in a broad sense as the electron glass (or sometimes Coulomb glass), and it is used in this way here. Sometimes the term is used more narrowly referring to materials with localized interacting electrons exhibiting the glassy properties described in Chapter 7.

The complications resulting from the need to include disorder, interactions, and in many cases quantum effects necessitated a number of approximations to facilitate a solution and an understanding of the properties of the electron glass, whether analytically or by computer simulations. Both methods have been amply employed. In the final count, the success or failure of the theory must be evaluated according to how well predictions agree with experiment.

A proper description of the electron glass must contain the disorder that is localizing the states and the poorly screened Coulomb interactions. The combined effect of disorder and interactions is bound to produce frustration (see Chapter 3), which is often considered to be the key ingredient to glassiness.

The dynamics of disordered systems with localized wave functions are usually slow, since the motion of their particles is by hopping rather than by diffusion, characteristic of systems with extended states.

The understanding of the linear response dc conductivity, described in length in Chapter 5, is fundamental for the treatment of the glassy properties of electron glasses. This chapter presents a few other transport properties, which are more loosely connected to the heart of this book. There is a great volume of literature on much of the material – experimental, theoretical, and computational. Such material is presented here rather briefly, focusing on properties relevant to the glassy properties and including references to more detailed discussions for the interested reader.

High field conductivity

Nonlinear effects in the conductivity are especially important in electron glasses. Interactions usually increase nonlinearities and can also establish an effective temperature for the electronic system higher than the phonon bath temperature. At the low temperatures where hopping systems are studied, the thermal coupling between the electrons and the phonons is not large enough to dissipate all the electrical power provided to the system even for relatively small values of the applied electric field.

Large electric fields – the “activationless” regime

There are many experimental studies of nonlinear effects on systems showingVRH, but there is no proper theory addressing the problem, except for extremely large values of the electric field. For electric fields larger than FT = kT/eξ, the socalled ‘activationless’ regime, the electric field plays a somewhat similar role to that played by the temperature in the linear regime. Shklovskii (1973) derived the field dependence of the conductivity in this regime.