Introduction

In this chapter, an introduction to the theory of recombination in low-dimensional semiconductor structures is given. A low-dimensional semiconductor structure is one whose dimensions (i.e. layer thicknesses) are smaller than, or comparable to, the de Broglie wavelength of the carriers. An example of such a structure is a quantum well (QW), which consists of a small band-gap semiconductor, of width L, sandwiched between two larger band-gap semiconductors. If L is less than or equal to the de Broglie wavelength of the carriers, then the carriers are confined in the smaller band-gap material, although they are still free to move in the plane of the well. Thus, a two-dimensional electron (and hole) gas can be formed, whose density of states is significantly altered from the bulk (three-dimensional) density of states. The two-dimensional density of states has potential advantages for lasers utilizing such QW structures, as will be explained in the later sections of this chapter. The aim of this chapter is to give a simple description of the physics of QW structures, and to outline the reasons why devices employing such structures have possible advantages over conventional devices. The main emphasis will be on the physics of QWs for laser applications. The problems associated with long wavelength semiconductor lasers are outlined, and the use of QW lasers as a possible solution for the reduction of threshold currents and their temperature sensitivity in such systems is discussed.

A systematic interest in recombination in semiconductors dates roughly from 1950 and gave rise for example to the Shockley–Read–Hall statistics in 1952 and the application of detailed balance to radiative processes in semiconductors in 1954. But our story really started with quantum mechanics and its application to solids. In contrast to its junior cousin, the black hole, the hole of semiconductor physics was first seen experimentally (in the anomalous Hall effect) and quantum mechanics was used to elucidate it [1]. Quantum mechanics was also used later to propose the band model of a semiconductor [2]. Actually, the copper oxide plate rectifier had been a useful solid-state device since the early days of quantum mechanics in the 1920s, but its action came to be understood only just before the war using electrons, holes and the band model [3]. This work has been reviewed for example by Mott and Gurney [4] and by Henisch [5]. Solid-state electronics was already in the air then, and rudimentary solid-state amplification had been proposed by Lilienfeld [6] in the late 1920s and established by Hilsch and Pohl [7] in 1938, using potassium bromide crystals. A useful semiconductor ‘triode’ was clearly ready to be born in 1938/9. But the war intervened. Still, solid-state detection was now important for radar, and programs to study silicon and germanium were initiated partly with government funding, notably at Purdue University under K. Lark-Horovitz.

Basic assumptions for recombination statistics

General orientation

A transition in which an electron jumps from a high energy to a low energy state which is located in a different group of levels is called recombination. One can think of the electron vacancy in the lower states as a trapped hole. The transition is then an electron–hole recombination, in which both ‘particles’ disappear and energy is released, for example in the form of photons or phonons. The different groups of levels are normally different energy bands or valleys in k-space or different groups of localized states. Transitions within a band are not normally considered as recombination events as they take place very quickly.

Figure 2.1.1 gives examples of recombination transitions. As in chemical kinetics, the rate per unit volume is regarded as proportional to the frequency of collision (which is itself proportional to the appropriate product of concentrations) and a recombination coefficient. In part (a) of the figure, for example, the frequency of collision is np and the coefficient, Bs say, has the dimension [L3T−1]. The other recombination rate expressions are obtained similarly, v0 and v1 being the concentration of empty and occupied defect centers. In the fourth process, for example, one may suppose that an electron recombines with a hole trapped on a defect. In writing down the rates one must remember that full states in the valence band and unoccupied states in the conduction band ‘do not count’. There are usually enough of them so that they do not affect the rate.

Introduction

Multiphonon processes in semiconductors belong to a family of electron transfer reactions which spreads over physics, chemistry and biology [6.1.1]. A theory of these processes describes how and in what circumstances electronic energy (ranging from few tenths of an electronvolt to an electronvolt or more) can be converted into the energy of nuclear vibrations. Since a phonon carries much less energy, many phonons must be absorbed or emitted during the transition – hence the name multiphonon transitions. In this chapter, we shall be concerned solely with multiphonon (‘MP’) transitions at defects which occur with the participation of the defect levels in the band gap, and a brief review of these defect states forms therefore an appropriate starting point for a discussion of this subject.

The best known examples of impurity levels in the band gap of a semiconductor are the discrete energy levels of charged donors or acceptors, modeled most simply as in eqs. (1.8.2)–(1.8.4). In most covalent semiconductors the resulting shallow levels lie close to the respective band edge (within 0.1 eV or so) and the wavefunction spreads over a region extending over many lattice constants. The closely spaced excited states provide a suitable medium for cascade transitions (section 2.6). In ionic solids, the Coulomb potential of charged defects is stronger and it can bind electrons in states with energy levels deeper in the band gap.

Introduction

After a brief review of the statistics of radiative recombination which introduces the transition probability per unit time per unit volume (BIJ) and stimulated and spontaneous emission in section 4.2, these concepts are derived quantum mechanically by second quantization in section 4.3. However, a reader willing to accept eq. (4.4.1) can skip that section. Next, emission rates as well as optical band-band absorption phenomena are discussed in sections 4.4 and 4.5. The simple relationships which exist by virtue of detailed balance between emission and absorption make theory and experiment on absorption relevant to emission problems. The emphasis on absorption depends on the fact that in a semiconductor under normal conditions the conduction band states are occupied with a probability less than unity, typically following a Boltzmann distribution exp (— E/kT). There are therefore comparatively few initial states for radiative emission from a band, and their number drops rapidly as E increases. In absorption these states are all available and one can also sample higher lying states where band structure effects, e.g. nonparabolicity, may play a part. The importance of absorption is due to the fact that it is often experimentally more accessible than emission studies.

Both emission and absorption are affected by the crucial difference between direct and indirect semiconductors.

Introduction

Auger transitions have already been encountered (for example in section 2.1.1). They have been of interest in solids since 1940 when soft X-ray emission was still the main method of probing the density of electron states in solids and the probability of electronic transitions back into an atomic inner shell. Skinner [3.1.1], working at the University of Bristol, thought the low-energy tail in the soft X-ray spectrum of Na might be due to lifetime broadening of the empty conduction band state left behind by an electron which had made the transition back into the atomic rump. This short lifetime might be due to electron collisions of the type shown later (Fig. 5.2.9). By the uncertainty principle ΔEΔt ≳ ħ this would then lead to the broadening effect. For an almost ideal metal like Na the second electron would have to make a transition across the Fermi level in order to find an empty state with a good probability. A straightforward application of theory in the years 1947–9 to obtain the considerable (1 eV) tail was plagued by a divergence: one obtained an infinitely long tail! This led to the idea that a screened electron-electron interaction has to be used in these problems to take approximate account of correlations in the motion of the electrons [3.1.2].

Introduction

The word ‘flaw’ was introduced by Shockley and Last [5.1.1] ‘for the purpose of distinguishing between imperfections with multiple possibilities for charge condition and ordinary donors and acceptors’ following the early work on the statistics of these centers described in [5.1.2].

In fact it is a convenient term which encompasses both chemical and physical defects and it is sometimes used in this generalized sense [5.1.3]. We shall here use the term defect and use an amended form of Blakemore's classification scheme (Fig. 5.1.1). This is not done to give a systematic discussion of these defect classes (this requires a major book), but to indicate the variety of possibilities. (An As atom sitting on a Ga site (AsGa) in GaAs is a typical anti-site defect.) There is a further classification [5.1.4] which depends on the value of

Δz = valency of the impurity – valency of the host

On this basis what is often called an isoelectronic defect, defined by Δz = 0, is more logically named isovalent, since the total number of electrons on the atom involved to which the term ‘isoelectronic’ refers is often of minor relevance.