This last chapter is devoted to the study of resonant tunnelling through laterally confined, ultra-small, double-barrier heterostructures. Recent rapid advances in nanofabrication techniques have naturally led to the idea of resonant tunnelling through three-dimensionally confined ‘quantum dot’ structures. Since electrons are confined laterally as well as vertically in these structures, the devices are often called zerodimensional (0D) RTDs and have become of great interest both from the standpoint of the physics of quantum transport through 0D electronic states and also for device miniaturisation towards highly integrated functional resonant tunnelling devices. The 0D RTD is a virtually isolated quantum dot only weakly coupled to its reservoirs and thus is well suited to investigating electron-wave transport properties through 3D quantised energy levels. By designing the structural parameters such as the barrier thickness, the quantum well width and dimensionality of lateral confinement, it is possible to realise a ‘quantum dot’ in which the number of electrons is nearly quantised so the effect of single-chargeassisted transport, or the so-called Coulomb blockade (CB), becomes significant. After Reed et al. reported their pioneering work in 1988 on resonant tunnelling through a quantum pillar which was fabricated by electron beam lithography and dry etching, several theoretical and experimental studies have been reported which investigate the mechanism of the observed fine structures. Transport in the 0D RTD is generally much more complicated than that in the conventional large-area RTDs which we have studied so far in this book: problems such as lateral-mode mixing due to a non-uniform confinement potential, charge quantisation in a quantum well and the interplay between resonant tunnelling and Coulomb blockade single-electron tunnelling have recently been invoked for the 0D RTDs. Such difficulties are still far from being fully resolved.

The tremendous progress of crystal growth and microfabrication technologies over the last two decades has allowed us to explore a new field of semiconductor device research. The quantum mechanical wavenature of electrons, expected to appear in nanometre-scale semiconductor structures, has been used to create novel semiconductor devices. The Resonant Tunnelling Diode (RTD), which utilises the electron-wave resonance occurring in double potential barriers, emerged as a pioneering device in this field in the middle of the 1970s. The idea of resonant tunnelling (RT) was first proposed by Tsu and Esaki in 1973, shortly after Molecular Beam Epitaxy (MBE) appeared in the research field of compound semiconductor crystal growth. Since then, RT has become of great interest and has been investigated both from the standpoint of quantum transport physics and also its application in functional quantum devices. Despite its simple structure, the RTD is indeed a good laboratory for electron-wave experiments, which can investigate various manifestations of quantum transport in semiconductor nanostructures. It has played a significant role in disclosing the fundamental physics of electron-waves in semiconductors, and enables us to proceed to study more complex and advanced quantum mechanical systems.

This book is designed to describe both the theoretical and experimental aspects of this active and growing area of interest in a systematic manner, and so is suitable for postgraduate students beginning their studies or research in the fields of quantum transport physics and device engineering.

We start this chapter with a brief review of some basic concepts. First in Section 1.1 we introduce the gallium arsenide (GaAs)/aluminum gallium arsenide (AlGaAs) material system which provides a very high quality two-dimensional conduction channel and has been widely used in meso-scopic experiments. Section 1.2 summarizes the free electron model that is commonly used to describe conduction electrons in metals and semiconductors. Next we discuss different characteristic lengths like the de Broglie wavelength, mean free path and the phase-relaxation length which determine the length scale at which mesoscopic effects appear (Section 1.3). The variation of resistance in the presence of a magnetic field is widely used to characterize conducting films. Both the low-field properties (Section 1.4) and the high-field properties (Section 1.5) yield valuable information regarding the electron density and mobility.

In Section 1.6 we introduce the concept of transverse modes which plays a prominent role in the theory of mesoscopic conductors and will appear repeatedly in this book. Finally in Section 1.7 we address an important conceptual issue that arises in the description of degenerate conductors, that is, conductors with a Fermi energy that is much greater than kBT. Normally we view the current as being carried by all the conduction electrons which drift along slowly. However, in degenerate conductors it is more appropriate to view the current as being carried by a few electrons close to the Fermi energy which move much faster. One consequence of this is that the conductance of degenerate conductors is determined by the properties of electrons near the Fermi energy rather than the entire sea of electrons.

Following the study in Chapter 3 of the effects of elastic and inelastic scattering on the transmission probability function, this chapter investigates non-equilibrium electron distribution in RTDs. Electron distribution in the triangular potential well in the emitter is studied first (Section 4.1). Then dissipative quantum transport theory is presented based on the Liouville–von-Neumann equation for the statistical density matrix (Section 4.2.1). Numerical calculations are carried out in order to analyse the femtosecond dynamics of the electrons (Section 4.2.2) and the dynamical space charge build-up in the double-barrier structure which gives rise to the intrinsic current bistability in the NDC region (Section 4.3.1). Next experimental studies of the charge build-up phenomenon are presented using magnetoconductance measurements (Section 4.3.2) and photoluminescence measurements (Section 4.3.3). Finally, the effects of magnetic fields on intrinsic current bistability are studied (Section 4.4).

Non-equilibrium electron distribution in RTDs

Let us start with a discussion on electron distribution in the emitter. We have seen in Section 2.4 that the electronic states in the emitter become 2D in the pseudo-triangular potential well formed between the thick spacer layer and the tunnelling barrier (see Fig. 2.16). Sharper current peaks observed for Materials 2 and 3 (Fig. 2.18) have been attributed to the 2D–2D nature of resonant tunnelling. This interpretation is based upon an assumption that the electrons in the triangular well are well thermalised, and that local equilibrium is achieved.

The 1980s were a very exciting time for mesoscopic physics characterized by a fruitful interplay between theory and experiment. What emerged in the process is a conceptual framework for describing current flow on length scales shorter than a mean free path. This conceptual framework is what we have tried to convey in this book. The activity in this field has expanded so much over the last few years that we have inevitably missed many interesting topics, such as persistent currents in normal metal rings, quantum chaos in microstructures, etc.

The development of the field is far from complete. So far both the theoretical and the experimental work has been almost entirely in the area of steady-state transport and many basic concepts remain to be clarified in the area of time-varying current flow as well as current fluctuations. Another emerging direction seems to be the study of mesoscopic conductors involving superconducting components. Finally, as we study current flow in smaller and smaller structures it seems clear that electron–electron interactions will play an increasingly significant role. As a result it will be necessary to go beyond the one-particle picture that is generally used in mesoscopic physics. Single-electron tunneling is a good example of this and it is likely that there will be many more developments involving current flow in strongly correlated systems.

It is well-known that the conductance (G) of a rectangular two-dimensional conductor is directly proportional to its width (W) and inversely proportional to its length (L); that is,

G = σW/L

where the conductivity a is a material property of the sample independent of its dimensions. How small can we make the dimensions (W and/or L) before this ohmic behavior breaks down? This question has intrigued scientists for a long time. During the 1980s it became possible to fabricate small conductors and explore this question experimentally, leading to significant progress in our understanding of the meaning of resistance at the microscopic level. What emerged in the process is a conceptual framework for describing current flow on length scales shorter than a mean free path. We believe that these concepts should be useful to a broad spectrum of scientists and engineers. This book represents an attempt to present these developments in a form accessible to graduate students and to non-specialists.

Small conductors whose dimensions are intermediate between the microscopic and the macroscopic are called mesoscopic. They are much larger than microscopic objects like atoms, but not large enough to be ‘ohmic’. A conductor usually shows ohmic behavior if its dimensions are much larger than each of three characteristic length scales: (1) the de Broglie wavelength, which is related to the kinetic energy of the electrons, (2) the mean free path, which is the distance that an electron travels before its initial momentum is destroyed and (3) the phase-relaxation length, which is the distance that an electron travels before its initial phase is destroyed.

Our purpose in this chapter is to describe an approach (often referred to as the Landauer approach) that has proved to be very useful in describing mesoscopic transport. In this approach, the current through a conductor is expressed in terms of the probability that an electron can transmit through it. The earliest application of current formulas of this type was in the calculation of the current-voltage characteristics of tunneling junctions where the transmission probability is usually much less than unity (see J. Frenkel (1930), Phys. Rev., 36, 1604 or W. Ehrenberg and H. Honl (1931), Z. Phys., 68, 289). Landauer [2.1] related the linear response conductance to the transmission probability and drew attention to the subtle questions that arise when we apply this relation to conductors having transmission probabilities close to unity. For example, if we impress a voltage across two contacts to a ballistic conductor (that is, one having a transmission probability of unity) the current is finite indicating that the resistance is not zero. But can a ballistic conductor have any resistance? If not, where does this resistance come from? These questions were clarified by Imry [2.2], enlarging upon earlier notions due to Engquist and Anderson [2.3]. Büttiker extended the approach to describe multi-terminal measurements in magnetic fields and this formulation (generally referred to as the Landauer–Büttiker formalism) has been widely used in the interpretation of mesoscopic experiments.

In Section 2.6 we briefly studied the effects of electron scattering on resonant tunnelling which are inevitable in a real system operating at room temperature. The phenomenological Breit–Wigner formula was introduced to describe the incoherent aspect of the electron tunnelling which in general results in a broadening of the transmission peak and thus degraded current P/V ratios in RTDs. In this chapter we look in more detail at various scattering processes, both elastic and inelastic, which have been of great interest not only from a quantum transport physics point of view but also because of the possibility of controlling and even engineering these interactions in semiconductor microstructures. The inelastic longitudinal–optical (LO) phonon scattering, introduced in the preceding chapter, is the most influential process, with Г–X-intervalley scattering and impurity scattering also affecting the resonant tunnelling electrons. Section 3.1 describes the dominant electron–LO-phonon interactions. Both theoretical and experimental studies of a postresonant current peak are presented, which provide much information about the electron–phonon interactions in the quantum well. Section 3.2 then discusses the effects of the upper X-valley which become more significant in AlxGa1−xAs/GaAs systems with an Al mole fraction, x, higher than 0.45 since the energy of the X-valley then becomes lower than that of the Г-valley. Finally, in Section 3.3, we study elastic impurity scattering, which may be caused by residual background impurities or those diffused from the heavily doped contact regions.

One of the most significant discoveries of the 1980s is the quantum Hall effect (see K. von Klitzing, G. Dorda and M. Pepper (1980), Phys. Rev. Lett., 45, 494). Normally in solid state experiments, scattering processes introduce enough uncertainty that most results have an ‘error bar’ of plus or minus several per cent. For example, the conductance of a ballistic conductor has been shown (see Fig. 2.1.2) to be quantized in units of (h/2e2). But this is true as long as we are not bothered by deviations of a few per cent, since real conductors are usually not precisely ballistic. On the other hand, at high magnetic fields the Hall resistance has been observed to be quantized in units of (h/2e2) with an accuracy that is specified in parts per million. Indeed the accuracy of the quantum Hall effect is so impressive that the National Institute of Standards and Technology is interested in utilizing it as a resistance standard.

This impressive accuracy arises from the near complete suppression of momentum relaxation processes in the quantum Hall regime resulting in a truly ballistic conductor of incredibly high quality. Mean free paths of several millimeters have been observed. These unusually long mean free paths do not arise from any unusual purity of the samples. They arise because, at high magnetic fields, the electronic states carrying current in one direction are localized on one side of the sample while those carrying current in the other direction are localized on the other side of the sample. Due to the formation of this ‘divided highway’ there is hardly any overlap between the two groups of states and backscattering cannot take place even though impurities are present.

Abstract

If lattice polarons exist in high-temperature superconducting oxides then there must be evidence of local lattice distortion associated with polarons. While the distortions are dynamic and subtle, making direct observation difficult, there are numerous indications that some anomalous local deviations from the crystallographic lattice structure exist in superconducting oxides. Based largely upon the results of pulsed neutron scattering measurements, we present an argument in favor of the presence of local lattice distortions consistent with lattice polarons. A few implications of the observation in relation to other physical properties are discussed.

Introduction

Even though polarons have been known for a long time, direct experimental observation of lattice distortions associated with them is surprisingly scarce, largely because the density of polarons is usually low and consequently the lattice distortion is small on average, making observation very difficult. While some observations of lattice distortion associated with polarons have been made for low-dimensional organic conductors in which the periodic lattice distortion (Peierls distortion) can be regarded as an array of localized polarons [1], there are very few such reports for oxides [2]. Moreover, most known cases of polarons are heavy, small polarons, while in high-temperature superconducting (HTSC) oxides the presence of mobile large polarons is suspected. For those reasons, local lattice distortion has been observed so far mostly by nontraditional methods of structural study, while the crystallographic community has largely been skeptical. In this paper we discuss why observation is difficult, whether there is sufficient experimental evidence to support the presence of polarons in high-temperature superconducting oxides or not, and the implications of these observations.

Abstract

Motivated by aspects of layered high-temperature superconductors and related quasi-one-dimensional materials, we consider polaron structure, dynamics, and coherence in certain extended Peierls–Hubbard models. We emphasize the qualitative importance of electron–lattice interactions even in the presence of dominant electron–electron correlations, the signatures of polaron structure and dynamics in energy-resolved pair-distribution structure functions, and the effect of disorder on polaron propagation and stability.

Introduction

The formation and dynamics of polarons (and bipolarons), despite a halfcentury of theoretical and experimental study, remain fascinating topics in many-body physics, combining as they do (often competing) aspects of coupled fields with distinct natural time scales (e.g., electron–phonon, spin–phonon, exciton–phonon), electron–electron interactions, lattice discreteness, nonadiabaticity, collective quantum tunneling, thermal fluctuations, competitions between disorder and polaronic localization, etc. Direct observation of polarons through real-space imaging of any of the coupled fields is rare even with the advent of STEM, AFM techniques, etc. Likewise, global measurements such as that of electronic band-structure are insensitive to polaron features. Thus, experiments and theoretical techniques have had to focus on indirect effects on microscopic probes such as transport coefficients, electronic absorption, vibrational spectroscopy, and so forth.

Our purpose here is to briefly review three qualitative effects in polaron physics that have arisen in modeling two components of the lattice structure of the layered cuprate superconductors [1] – namely, extended multi-band Peierls–Hubbard models of (a) the active CuO2 planes and (b) polarizable Cu–O clusters in the axial direction (polarizable interplanar medium).

Abstract

The spin-polaron concept is introduced in analogy to ionic and electronic polarons and the assumptions underlying the author's approach to spinpolaron-mediated high-Tc superconductivity are discussed. Elementary considerations about the spin-polaron formation energy are reviewed and the possible origin of the pairing mechanism illustrated schematically. The electronic structure of the CuO2 planes is treated from the standpoint of antiferromagnetic band calculations that lead directly to the picture of holes predominantly on the oxygen sublattice in a Mott–Hubbard/charge transfer insulator. Assuming the holes to be described in a Bloch representation but with the effective mass renormalized by spin-polaron formation, equations for the superconducting gap, Δ, and transition temperature, Tc, are developed and the symmetry of Δ discussed. After further simplifications, Tc is calculated as a function of the carrier concentration, x. It is shown that the calculated behavior of Tc(x) follows the experimental results closely and leads to a natural explanation of the effects of under- and over-doping. The paper concludes with a few remarks about the evidence for the carriers being fermions (polarons) or bosons (bipolarons).

Introduction

A carrier (electron or hole) moving through an ionic lattice will induce displacements of the ions and, under certain conditions, the carrier plus ionic displacements may form a good quasi-particle, i.e., the ionic polaron [1]. Similarly, electronic polarons may form when the carriers induce polarization of localized or quasi-localized electronic distributions. In an analagous manner, a spin polaron is a spin ½ carrier moving in a magnetic medium accompanied by deviations of localized ionic spins.