Introduction

The ever-growing demand for fast optical data transmission calls for lasers offering high modulation rates and low energy consumption at the same time. Advances in growth and processing methods make quantum dot (QD) based lasers better candidates for this challenge than ever before. Placed in microresonators able to confine light in regions roughly the size of their wavelength, QDs pave the way to ultra-low threshold lasing. The most common resonator geometries aimed at three-dimensional light confinement are microdisks, photonic crystal membrane cavities and micropillars. The latter are especially good candidates for realizing microlasers suitable for applications as they offer directed emission and allow for parallel device processing. However, this increased efficiency also results in modified emission properties of QD lasers [8]. Semiconductor-specific processes like Pauli-blocking of states, the composite nature of the carriers involved and Coulomb interactions between carriers cause deviations from the standard atomistic laser picture. The main aim of our studies is to characterize microlaser emission in terms of photon statistics and coherence properties. Following Glauber, the most detailed description of a light field is given in a series of correlation functions describing coherence in different orders [10].

This chapter is organized as follows. Section 10.2 contains a brief review on the characteristic properties of micropillar lasers and discusses the emission properties of microlasers operated below and above threshold. Section 10.2.1 focuses on photon statistics and the classification of light fields.

Introduction

Quantum dots are often referred to as artificial atoms, since they trap carriers in discrete energy-levels due to the nanoscale three-dimensional finite potential energy well they provide. As such, dots exhibit a coherent light–matter interaction that is similar to an atom. This is evidenced by observations of atom–optics phenomena such as Rabi oscillations [43, 26], power broadening [27], Autler–Townes doublet [14, 42], Mollow triplet [42, 8], and coherent population trapping [6]. In this chapter, Rabi rotation measurements are used to examine how an exciton transition deviates from an ideal two-level atom due to its interaction with a reservoir of phonons.

The neutral exciton transition may be regarded as a two-level system, or qubit, composed of the crystal ground-state ∣0〉 and a single electron-hole pair ∣X〉. The state-vector of a qubit can be described as a pseudo spin-half. When an oscillating electro-magnetic field resonantly excites the two-level transition it drives an oscillation in the population inversion known as a Rabi oscillation. This results from the oscillations of the driving field and the dipole of the two-level system being synchronous, such that in its rotating frame, the driving field acts as a static magnetic field that causes the pseudo-spin to rotate. Coherent control of the pseudo-spin can be achieved by applying well-defined driving fields, enabling the preparation, and manipulation of superposition states. Such coherent control concepts have found widespread use in electron spin, and nuclear magnetic resonance spectroscopy.

Scalability requirements in future device application of self-assembled quantum dots for non-classical light generation necessitate control of the quantum dot nucleation site. In this chapter we discuss a site-control technique based on directed self-assembly of InAs/InP quantum dots emitting at telecommunication wavelengths. The site-control method preserves the high optical quality inherent in self-assembled quantum dots and the characteristic signatures of a strongly confined system are observed in the emission spectra. The efficacy of site-control manifests in the coupling of single quantum dots to microcavities required for the fabrication of efficient devices. The a priori knowledge of the quantum dot position is used to deterministically couple single dots to high-finesse microcavities with the assurance that one and only one quantum dot is coupled to each cavity. Such devices form the basis of efficient sources of single photons and entangled photon pairs for telecommunications applications that can be manufactured in a scalable manner using conventional semiconductor processing.

Introduction

Self-assembled quantum dots possess the two-level emitter characteristics required for non-classical light generation [37] in quantum information processing and quantum key distribution. The performance of a quantum dot-based single photon source or entangled photon pair source will depend on how well the dot can be coupled to a high-quality factor Q, small volume Veff microcavity [44]. The cavity is required to channel photons from the exciton decay into an optical mode that can be collected by an external optical system.

Introduction

Quantum dots have proven to be exciting systems to study light-matter interaction [32, 9]. Self-assembled quantum dots obtained by the Stranski–Krastanow growth mode have been the main system to date [32, 9]. Here we introduce a new type of quantum dot embedded in a one-dimensional nanowire. Quantum dots in nanowires offer a range of advantages over strain-driven Stranski–Krastanov quantum dots. In the case of quantum dots in nanowires, the light extraction efficiency can be very high for the quantum dot emission due to a waveguide effect in the nanowire [14, 45], theoretically approaching 100% according to simulations [14]. Since strain is not the driving mechanism during growth, unprecedented material freedom is available to the quantum engineer in the choice of materials for the quantum dot and the barrier material. At the scale of nanowires, both zincblende and wurtzite crystal structures can coexist, opening the door to a new type of confinement based not only on the material composition, but also on the phase of the crystal lattice [1]. The ability to electrically contact a single nanowire implies that all the current injected in a nanowire will flow through a single quantum dot, enabling an efficient interface between single electrons and single photons [33, 44]. In addition, electrostatic gating is highly versatile, allowing for coherent spin manipulation [36], charge state control [54], and the ability to control the exciton–biexciton splitting by an in-plane electric field [43].

Introduction

In 1998, Daniel Loss and David DiVincenzo published a seminal paper describing how semiconductor quantum dots could be used to create spin qubits for quantum information processing [28]. They recognized that a single spin in a magnetic field forms a natural two-level system that can serve as a quantum bit. Moreover, owing to the weak magnetic moment of the electron, the spin is relatively well isolated from the environment leading to long coherence times. To confine single spins, Loss and DiVincenzo envisioned the quantum dot architecture shown in Fig. 15.1. A GaAs/AlGaAs heterostructure confines electrons to a two-dimensional electron gas (2DEG). Depletion gates are fabricated on top of the structure to provide a tunable confinement potential, trapping a single electron in each quantum dot. Neighboring quantum dots are tunnel coupled, with the coupling strength controlled by the electrostatic potential. The orientation of a single spin can be controlled by using electron spin resonance (ESR), while nearest-neighbor coupling is mediated by the depletion gate tunable exchange interaction.

It is fair to say that in 1998 many of the requirements of the Loss–DiVincenzo proposal had not been implemented, starting with the most basic necessity of a single electron lateral quantum dot [8]. The purpose of this chapter is to describe several experiments inspired by the Loss–DiVincenzo proposal. Many powerful experiments have been performed since 1998 and, given the space constraints here, we cannot give each experiment the attention it deserves.

Introduction

Self-assembled quantum dots (QDs) have been at the center of research on the quantum properties of zero-dimensional semiconductor nanostructures. The deep understanding of the physical properties and mechanisms that are active in QDs have allowed for their application in quantum secure single photon communication, quantum processing, etc. This would have been impossible without the progress in the growth control of self-assembled QDs. Nowadays, we can accurately control QD parameters such as height, composition, and strain which determine the optoelectronic and spintronic properties. Several double capping approaches have been developed that allow trimming of the QD height and Sb capping was shown to eliminate QD erosion during capping. Droplet epitaxy is a novel approach to obtain non-strained QDs, which is of great advantage because strain is one of the most complicating factors in understanding and utilizing self-assembled QDs.

In this chapter we will review our recent cross-sectional scanning tunneling microscopy (X-STM) analysis of self-assembled QDs in various hosts and obtained by a range of techniques to control their structural properties. The X-STM technique allows to image atomic scale details in semiconductor structures that are cleaved along a natural cleavage plane perpendicular to the growth direction. Although we have been able to analyze many intricate aspects of the studied QDs by X-STM, this technique is limited by its twodimensional (2D) nature. Recently, atom probe tomography (APT) has been able to extend its field of application to semiconductor materials.

Self-assembled quantum dots as host for spin qubits

A coherent spin in the solid-state would be very attractive for a number of applications. A single spin has an obvious application as a magnetic field sensor; entangled spin states can potentially enhance the sensitivity [54]. An optically active spin is a potential component of a quantum repeater, a technology to extend fibre-based quantum cryptography to large distances [41]. Also, a spin qubit is a potential building block of a quantum information processor [62]. But, applications aside, the targeted investigation of spin coherence in the solid-state is leading to new insights into the microscopic nature of the complex spin environment, allowing some old problems, for instance the central spin problem, to be fruitfully revisited.

The search for spin coherence in the solid-state has led most spectacularly so far to the NV− centre in diamond whose spin coherence can reach ˜1 ms even at room temperature [5]. However, diamond is difficult to process into a real device. Electron and hole spins in III–V semiconductors have yet to achieve the coherence of the NV− in diamond, but these materials have some considerable advantages. First, quantum dots can be used to confine electron spins to nanometer length scales [47, 61]. The quantum dots can either be defined electrostatically by local depletion of a two-dimensional electron gas, or they can be self-assembled during growth, for instance InAs on GaAs. Second, both a mature heterostructure technology and post-growth nanofabrication can be used to add functionality to the quantum dots.

Introduction

In the previous chapter, we saw how the long-range order of particle positions in a crystal could be described in a rather elegant manner using a space lattice that extends indefinitely. In this chapter, we examine instead disordered matter such as liquids or glasses in which a long-range repeated pattern is absent. These amorphous materials might not seem as glamorous as their crystalline counterparts, but they are increasingly prevalent in our world as they comprise the windows, computer screens and vast array of plastic components that surround us on a daily basis. In comparison with crystalline structures, these amorphous materials pose a challenge to describe, and their structure can only be defined in a statistical sense by introducing an ensemble-averaged, pair distribution function. In spite of their disordered nature, a robust pattern of particle positions emerges over short distances. This short-range order reflects the local coordination of particles and we briefly review the random close pack and the continuous random network systems as common examples of amorphous structure.

A statistical structure

Disordered or amorphous condensed matter has a clear disadvantage in that particle positions lack any long-range repeating pattern akin to that found in crystals. This is evident in Fig. 2.1, which illustrates the typical particle positions of either a glass or a liquid captured at a particular instant in time.

STRUCTURE

Picture with me an old cottage nestled in the woods. There is a small house built of clay bricks that were thoughtfully stacked and interlaced by a master bricklayer so as to produce a repeated interlocking pattern. The house has a thatched roof consisting of bundles of straw. The straws in each bundle are oriented in a common direction to direct rainwater off the roof, and are lashed together with twine. Around the house is a garden enclosed by a stone wall. Like the brick walls of the house, the stones in the wall are bonded together with mortar. But unlike the bricks, the stones lack any sense of a repeating pattern.

In this part of the textbook, we examine the basic structures that are found in condensed matter as well as the forces (the mortar and twine) that maintain these structures over long time periods. For our purposes, structures are divided into two main categories: ordered (like the bricks and the straw of the house) and disordered (like the stones in the garden wall).

Introduction

In Chapter 5 we introduced the structure factor, S(q), as the Fourier representation of the positions of a collection of fixed, elastically scattering, particles. In reality, these particles are rarely fixed. In a solid (crystal or glass), the particles are bound together by bonds and, while unable to wander about, are able to oscillate or vibrate about a fixed center of motion. In a liquid, the particles are even less constrained and are free to wander around over considerable distances. In this chapter we develop the dynamic structure factor as a straightforward extension of the static structure factor introduced previously, and apply it to examine the dynamics of liquid-like systems. In one instance, we consider the Brownian diffusion of macromolecules in a solvent, where the motion mimics that of the random walk we discussed in the previous chapter. In another instance, we show how thermodynamically driven density fluctuations present in a simple liquid are responsible for the characteristic Rayleigh–Brillouin spectrum of light scattering. We also take this opportunity to consider the special case of slow dynamics in polymer liquids and to briefly consider the nature of the liquid-to-glass transition that separates amorphous solids from their liquid counterparts.

Dynamic structure factor

We can think of a liquid as a time-dependent amorphous structure. In many ways, the structure of a liquid resembles the structure of a glass in that, at any instant in time, a “snapshot” of its S(q) resembles that of the glass. Indeed, the only real difference between a liquid and a glass is the presence or absence, respectively, of long-range translational motion. In the liquid, the translational motion results from the incessant jostling of the particles allowing them to wander about. By virtue of this motion, particles of the liquid are able to rearrange on some characteristic time scale (related to the viscosity of the liquid) into different, but thermodynamically equivalent, amorphous configurations whose instantaneous structure resembles that of a glass. For certain glass forming liquids near their glass transition point, the characteristic time scale for these rearrangements can become exceedingly long with some unusual consequences, as we will discuss later.

Introduction

In the last two chapters we explored the behavior of phonons in a crystal. There we saw how these discrete, quantized pieces of propagating energy contributed to both the specific heat and thermal conductivity of the solid. Here we turn our attention to crystalline metals whose metallic bonding results in the formation of a sea of mobile electrons present within the crystal. Like phonons, these mobile electrons carry around energy and consequently contribute to the specific heat. But they also carry around charge and so contribute also to the electrical conductivity of the metal.

In this chapter, we begin with a simplistic model of the mobile electrons as quantum mechanical waves trapped within an infinite square well potential. This model is known as the free electron model because the interaction of the electron with the ion cores of the metal lattice is disregarded. The electron is only trapped by the confines of the crystal itself. Although this simple model is unable to capture all the experimental features of conduction in metals, it readily accounts for the smallness of the electron contribution to the specific heat and does provide a simple interpretation of such electron emission phenomena as the photoelectric effect.

Introduction

In this final chapter on the subject of structures, we turn our attention to magnetic materials. Why? Because magnetic materials are illustrative of yet another level of structure that often arises in condensed matter, beyond that of particle arrangements. As we will see, magnetic particles have a property of net spin and a magnetic moment whose orientation in space is largely unrestricted. Regardless of whether a large system of magnetic particles is positioned in an ordered or disordered manner, their spins represent an additional layer of ordering. The moments could be randomly oriented or aligned in a common direction. In ferromagnetic systems, these moments interact with one another to promote a local alignment of the moments which can eventually spread over the entire system. This is reminiscent of how pairwise bonding between particles eventually leads to crystallization of a liquid, and it is the archetype for a wide variety of phase transitions in which order appears in the form of correlated regions emerging from a disordered host.

The ordering process

So far, our discussion of structure has focused entirely on particles: their relative positions and the forces that hold them together. We have seen that arrangements of particles fall into either an ordered or disordered pattern which can be characterized by the level of symmetry present. By virtue of its disorder, the liquid has rotational invariance and an infinite symmetry (on average). The crystal, however, conforms to a space lattice and possesses only a discrete set of symmetry operations. In the process of forming a crystal from the liquid, the symmetry is often said to be “broken”.

Introduction

In this chapter, we develop some fundamental understanding of the nature of phase transitions by examining two well-studied examples: the vapor-to-liquid transition of fluids and the paramagnetic-to-ferromagnetic transition in magnetic materials. Here, our focus is on the experimentally observed features of these two transitions and how to interpret and navigate the many phase diagrams that describe them. The theoretical interpretation will be tackled later in Chapter 17. We will find that, in general, a phase transition is accompanied by some change in the amount of order as when, for example, liquid water freezes into crystalline ice. Moreover, we can describe this amount of ordering quantitatively by introducing an appropriate order parameter, whose value changes significantly only during the transition. Based upon the manner in which the order parameter changes, we can distinguish two different types of phase transitions: those of first order for which the order parameter changes discontinuously and those of second order for which it changes continuously. Second-order transitions are possible for both the vapor-to-liquid transition and the paramagnetic-to-ferromagnetic transition and are of interest due to the way in which many properties diverge near the transition in a similar, power law manner.

DYNAMICS

A steaming cup of coffee is sitting on my desk. Aside from the steam, there is little else that would suggest any other activity is present. The cup and it contents appear to be “at rest”. But a little closer examination reveals a small ripple of waves on the surface of the liquid caused by a mechanical pump in the room next door. Indeed, if I rest my finger gently on the lip of the cup, I can feel the vibration. I can also feel the heat that has developed in the cup, now several minutes since I poured the coffee from the pot and added some creamer.

If I could examine this evenmore closely Iwould actually see that nothing is truly “at rest”. The particles of the liquid are jostling about incessantly. The particles of the creamer that I added have clearly taken flight and diffused rapidly out into all regions of the coffee. The cup itself is also in motion. Its particles are undergoing incessant vibrations that are ultimately responsible for the heat I feel when I hold it.

Perhaps most interesting is that all this microscopic motion appears to be driven entirely by thermodynamics. The coffee and cup are sitting in a climatecontrolled office and there is a constant flux of thermal energy (heat) entering and exiting both the coffee and the cup to keep things moving.