Direct and indirect semiconductors

To make a diagonal indirect transition, both an appropriate energy photon and an appropriate momentum phonon must be present together with the electron. This is referred to as a three-body interaction because there are three particles (electron, phonon, and photon) participating. Such collisions are over 1000 times less likely than a simple electron-photon interaction at common temperatures. This means that electrons and holes of different momenta do not recombine rapidly. Typically, electrons and holes in pure direct-gap semiconductors last no more than 10 s. 

Because in indirect-gap semiconductors generation and recombination of carriers are equally difficult, even in indirect-gap materials the distribution of electrons and holes is governed by the Fermi function. Because the Fermi energy can never be farther than half the energy gap from one band edge or the other, the density of carriers in an indirect-gap material is still determined by the minimum energy gap in spite of the difference in momenta of the band minima in indirect-gap materials. 

The consequences of the need for a phonon to permit an electron and a hole to interact in indirect-gap materials include  

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In many semiconductors employed for LEDs, and especially in mixed alloys, the direct and indirect minima ate separated by smaller energies than those of the purely direct and indirect semiconductors. As a result, finite electron concentrations exist within both minima. The total electron concentration, n, is given by equation 5 ... 

This chapter reviews important aspects of inorganic LED structures. Section 1.2 introduces the basic concepts of optical emission. Band diagrams of direct and indirect semiconductors and the spectral shape of spontaneous emission will be discussed along with radiative and nonradiative recombination processes. Spontaneous emission can be controlled by placing the active region in an optical... 

Direct and Indirect Energy Gap. The radiative recombination rate is dramatically affected by the nature of the energy gap, E, of the semiconductor. The energy gap is defined as the difference in energy between the minimum of the conduction band and the maximum of the valence band in momentum, k, space. Eor almost all semiconductors, the maximum of the valence band occurs where holes have zero momentum, k = 0. Direct semiconductors possess a conduction band minimum at the same location, k = O T point, where electrons also have zero momentum as shown in Eigure la. Thus radiative transitions that occur in direct semiconductors satisfy the law of conservation of momentum. 

Semiconductors can be divided into two groups direct and indirect band gap materials. In direct semiconductors the minimum energy in the conduction band and the maximum in the valence band occur for the same value of the electron momentum. This is not the case in indirect materials. The difference has profound consequences for the transitions of electrons across the band gap in which light is emitted, the radiative transitions, of interest here. 

In discussing deep levels in wide band-gap semiconductors, the first requirement is to define deep and wide. The latter can be done relatively easily, although arbitrarily. We list in Table I the 4°K band gaps of the various III-V semiconductors, based on the tabulation by Strehlow and Cook (1973). We shall call those with Eg> 1.5 eV the wide band-gap ones. In practice, our review will present data only on GaAs and on GaP as prototypes of direct and indirect gap materials of this class. These are also the only two materials of this class that have been extensively studied and that are in common use. Discussion of deep levels in ternary and quarternary alloys of III-V semiconductors are omitted since treating these in detail might well have doubled the size of this chapter. 

The above-mentioned theoretical background shows that, irrespective of the chemical nature of the photosensitizer and its binding mode to the semiconductor surface, one should consider two main ways of the semiconductor CB populating direct and indirect. Direct processes include VB -> CB excitations, photosensitization via bulk doping (TTRS-driven processes) and photophysical processes involving the TTRMs term. Indirect processes, in turn, involve excitation of the surface and a subsequent electron transfer reaction (WRV1 + TTet). 

The optical band-gap of the semiconductor (Section 1.2) is an important parameter in defining its light absorption behavior. In this quantized process, an electron-hole pair is generated in the semiconductor when a photon of energy hv (v = frequency and hv > Ef) is absorbed. Optical excitation thus results in a delocalized electron in the CB, leaving behind a delocalized hole in the VB this is the band-to-band transition. Such transitions are of two types direct and indirect. In the former, momentum is conserved and the top of VB and the bottom of CB are both located at /c = 0 (A is the electron wavevector). The absorption coefficient (a) for such transitions is given by [202]... 

Saalfrank, P., Boendgen, G., Corriol, C. and Nakajima, T. (2000) Direct and indirect, DIET and DIMET from semiconductor and metal surfaces What can we learn from toy models Faraday Discuss., 117, 65-83. 

Harrison D., Abram R. A. and Brand S. (1999), Characteristics of impact ionization rates in direct and indirect gap semiconductors , J. Appl. Phys. 85, 8186-8192. 

An electron hole pair is created in a semiconductor when a photon of sufficient energy is absorbed, resulting in excitation of an electron from the valence band to the conduction band [115]. In the context of semiconductor photoelectrochemistry, it is useful to distinguish between direct and indirect optical transitions. If the top of the valence band and the bottom of the conduction band are both situated at = 0 (A being the electron wave vector), one-step optical processes between delocalised states in the valence and conduction band can occur. The absorption coefficient for direct absorption of photons of energy hv, in a semiconductor with bandgap Eg is given by... 

A detailed band gap analysis involves plotting and fitting the absorption data to the expected trendlines for direct and indirect band gap semiconductors. Ideally, the absorbance A is first normalized to the path length I of the light through the material to produce the absorption coefficient a as per Eq. (5.3). Values of a > 10" cm often obey the following relation presented by Tauc and supported by Davis and Mott [4, 5] ... 

The exponent is = 0.5 for semiconductors with direct transition, and = 2 for semiconductors with indirect transition. Direct and indirect transitions are explained in Figure 9.11. 

These are two groupings of semiconductors based on their general band structure, namely, direct and indirect [5.42], In direct materials, a high, sharply rising absorption constant is found immediately above the bandgap... 

The luminescence mainly originates from the inter-band transitions, which are divided into direct and indirect transitions according to the transition modes. If the electrons jump at the same point between the VBM (valence band maximum) and the CBM (conduction band minimum), this transition is direct. In contrast, there is indirect transition. The semiconductors silicon (Si) and gallium arsenide (GaAs) are typical examples as shown in Fig. 6.6. They have an indirect and direct band gap with the values 1.95 and 0.17 eV, respectively. When the crystal size becomes smaller, e.g., forming quantum dots, the Si becomes a better self-activated luminescence material. 

The energy of the electron varies with the wave vector k. However, a more simplified band structure can be used for semiconductors without referring to the value of k. This approximation is acceptable for limits comparison of semiconductors of different structures and distinguishes between the direct and indirect band gap semiconductor. This is illustrated in Fig. 18.4. 

Figure 24.11 Direct and indirect band gaps for semiconductors.

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