The conduction and valence bands

The model in Fig. 3.2 is sufficient to predict the general features of N E), but much more detailed calculations are needed to obtain an accurate density of states distribution. Present theories are not yet as accurate as the corresponding results for the crystalline band structure. The lack of structural periodicity complicates the calculations, which are instead based on specific structural models containing a cluster of atoms. A small cluster gives a tractable numerical computation, but a large fraction of the atoms are at the edge of the cluster and so are not properly representative of the real structure. Large clusters reduce the problem of surface atoms, but rapidly become intractable to calculate. There are various ways to terminate a cluster which ease the problem. For example, a periodic array of clusters can be constructed or a cluster can be terminated with a Bethe lattice. Both approaches are chosen for their ease of calculation, but correspond to structures which deviate from the actual a-Si H network. 

Tight binding calculations of the valence and conduction bands of crystalline and amorphous silicon. The result for amorphous silicon is obtained from a 216 atom cluster (Biswas et al. 1990). 

The Si-H bonds add further features to the density of states distribution. The Si-H bonding electrons are more localized than the Si-Si states and lead to sharper features in the density of states. One set of calculations, shown in Fig. 3.4, finds a large hydrogen-induced peak 

Calculations of the electronic states derived from SiH and SiH, bonds (Ching et al. 1980). The dashed lines are the experimental results shifted by 2 eV (see Fig. 3.7). 

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Some of tliese problems are avoided in heterojunction bipolar transistors (HBTs) [jU, 38], tlie majority of which are based on III-V compounds such as GaAs/AlGaAs. In an HBT, tlie gap of tlie emitter is larger tlian tliat of tlie base. The conduction and valence band offsets tliat result from tlie matching up of tlie two different materials at tlie heterojunction prevent or reduce tlie injection of tlie base majority carriers into tlie emitter. This peniiits tlie use of... 

Figure 9.8(a) shows how the conduction band C and the empty valence band V are not separated in a conductor whereas Figure 9.8(c) shows that they are well separated in an insulator. The situation in a semiconductor, shown in Figure 9.8(b), is that the band gap, between the conduction and valence bands, is sufficiently small that promotion of electrons into the conduction band is possible by heating the material. For a semiconductor the Fermi energy E, such that at T= 0 K all levels with E < are filled, lies between the bands as shown. 

A semiconductor laser takes advantage of the properties of a junction between a p-type and an n-type semiconductor made from the same host material. Such an n-p combination is called a semiconductor diode. Doping concentrations are quite high and, as a result, the conduction and valence band energies of the host are shifted in the two semiconductors, as shown in Figure 9.10(a). Bands are filled up to the Fermi level with energy E. ... 

The two-dimensional carrier confinement in the wells formed by the conduction and valence band discontinuities changes many basic semiconductor parameters. The parameter important in the laser is the density of states in the conduction and valence bands. The density of states is gready reduced in quantum well lasers (11,12). This makes it easier to achieve population inversion and thus results in a corresponding reduction in the threshold carrier density. Indeed, quantum well lasers are characterized by threshold current densities as low as 100-150 A/cm, dramatically lower than for conventional lasers. In the quantum well lasers, carriers are confined to the wells which occupy only a small fraction of the active layer volume. The internal loss owing to absorption induced by the high carrier density is very low, as Httie as 2 cm . The output efficiency of such lasers shows almost no dependence on the cavity length, a feature usehil in the preparation of high power lasers. 

In an intrinsic semiconductor, charge conservation gives n = p = where is the intrinsic carrier concentration as shown in Table 1. Ai, and are the effective densities of states per unit volume for the conduction and valence bands. In terms of these densities of states, n andp are given in equations 4 and... 

Global AMI.5 sun illumination of intensity 100 mW/cm ). The DOS (or defect) is found to be low with a dangling bond (DB) density, as measured by electron spin resonance (esr) of - 10 cm . The inherent disorder possessed by these materials manifests itself as band tails which emanate from the conduction and valence bands and are characterized by exponential tails with an energy of 25 and 45 meV, respectively the broader tail from the valence band provides for dispersive transport (shallow defect controlled) for holes with alow drift mobiUty of 10 cm /(s-V), whereas electrons exhibit nondispersive transport behavior with a higher mobiUty of - 1 cm /(s-V). Hence the material exhibits poor minority (hole) carrier transport with a diffusion length <0.5 //m, which puts a design limitation on electronic devices such as solar cells. 

PL is generally most usefril in semiconductors if their band gap is direct, i.e., if the extrema of the conduction and valence bands have the same crystal momentum, and optical transitions are momentum-allowed. Especially at low temperatures. 

Band gap engineetring confined hetetrostruciutres. When the thickness of a crystalline film is comparable with the de Broglie wavelength, the conduction and valence bands will break into subbands and as the thickness increases, the Fermi energy of the electrons oscillates. This leads to the so-called quantum size effects, which had been precociously predicted in Russia by Lifshitz and Kosevich (1953). A piece of semiconductor which is very small in one, two or three dimensions - a confined structure - is called a quantum well, quantum wire or quantum dot, respectively, and much fundamental physics research has been devoted to these in the last two decades. However, the world of MSE only became involved when several quantum wells were combined into what is now termed a heterostructure. 

The operation principle of these TFTs is identical to that of the metal-oxide-semiconductor field-effect transistor (MOSFET) [617,618]. When a positive voltage Vg Is applied to the gate, electrons are accumulated in the a-Si H. At small voltages these electrons will be localized in the deep states of the a-Si H. The conduction and valence bands at the SiN.v-a-Si H interface bend down, and the Fermi level shifts upward. Above a certain threshold voltage Vth a constant proportion of the electrons will be mobile, and the conductivity is increased linearly with Vg - Vih. As a result the transistor switches on. and a current flows from source to drain. The source-drain current /so can be expressed as [619]... 

At the interface of the nitride (Ef, = 5.3 eV) and the a-Si H the conduction and valence band line up. This results in band offsets. These offsets have been determined experimentally the conduction band offset is 2.2 eV, and the valence band offset 1.2 eV [620]. At the interface a small electron accumulation layer is present under zero gate voltage, due to the presence of interface states. As a result, band bending occurs. The voltage at which the bands are flat (the flat-band voltage Vfb) is slightly negative. 

Figure 4. Band diagram showing the relative positions of the conduction and valence bands in WSe2 with respect to the reduction potentials in aqueous solutions for the redox couples shown in Figure 3.

Formation of bands in solids by assembly of isolated atoms into a lattice (modified from Bard, 1980). When the band gap Eg kT or when the conduction and valence band overlap, the material is a good conductor of electricity (metals). Under these circumstances, there exist in the solid filled and vacant electronic energy levels at virtually the same energy, so that an electron can move from one level to another with only a small energy of activation. For larger values of Eg, thermal excitation or excitation by absorption of light may transfer an electron from the valence band to the conduction band. There the electron is capable of moving freely to vacant levels. The electron in the conduction band leaves behind a hole in the valence band. 

The donor electron level, cd, which may be derived in the same way that the orbital electron level in atoms is derived, is usually located close to the conduction band edge level, ec, in the band gap (ec - Ed = 0.041 eV for P in Si). Similarly, the acceptor level, Ea, is located close to the valence band edge level, ev, in the band gap (ea - Ev = 0.057 eV for B in Si). Figure 2-15 shows the energy diagram for donor and acceptor levels in semiconductors. The localized electron levels dose to the band edge may be called shallow levels, while the localized electron levels away from the band edges, assodated for instance with lattice defects, are called deep levels. Since the donor and acceptor levels are localized at impurity atoms and lattice defects, electrons and holes captured in these levels are not allowed to move in the crystal unless they are freed from these initial levels into the conduction and valence bands. 

Fig. 2-16. Electron state density distribution and electron-hole pair formation in the conduction and valence bands of intrinsic semiconductors Cf > Fermi level of intrinsic semiconductors.

Thus, for the band model the activity coefficients, y. and Vh, of electrons and holes in the conduction and valence bands are, respectively, given in Eqn. 2-33 ... 

Figure 8-11 shows as a function of electron energy e the electron state density Dgdit) in semiconductor electrodes, and the electron state density Z e) in metal electrodes. Both Dsd.t) and AKe) are in the state of electron transfer equilibrium with the state density Z>bei)ox(c) of hydrated redox particles the Fermi level is equilibrated between the redox particles and the electrode. For metal electrodes the electron state density Ai(e) is high at the Fermi level, and most of the electron transfer current occurs at the Fermi level enio. For semiconductor electrodes the Fermi level enao is located in the band gap where no electron level is available for the electron transfer (I>sc(ef(so) = 0) and, hence, no electron transfer current can occur at the Fermi level erso. Electron transfer is allowed to occur only within the conduction and valence bands where the state density of electrons is high (Dsc(e) > 0). 

The distribution of the exchange transfer current of redox electrons o(e), which corresponds to the state density curves shown in Fig. 8-11, is illustrated for both metal and semiconductor electrodes in Fig. 8-12 (See also Fig. 8-4.). Since the state density of semiconductor electrons available for electron transfer exists only in the conduction and valence bands fairly away from the Fermi level nsc), and since the state density of redox electrons available for transfer decreases remarkably with increasing deviation of the electron level (with increasing polarization) from the Fermi level CFciiEDax) of the redox electrons, the exchange transfer current of redox electrons is fairly small at semiconductor electrodes compared with that at metal electrodes as shown in Fig. 8-12. 

In addition to redox reactions due to the direct transfer of electrons and holes via the conduction and valence bands, the transfer of redox electrons and holes via the surface states may also proceed at semiconductor electrodes on which surface states exist as shown in Fig. 8-31. Such transfer of redox electrons or holes involves the transition of electrons or holes between the conduction or valence band and the surface states, which can be either an exothermic or endothermic process occurring between two different energy levels. This transition of electrons or holes is followed by the transfer of electrons or holes across the interface of electrodes, which is an adiabatic process taking place at the same electron level between the surface states and the redox particles. 

We examine an electron transfer of hydrated redox particles (outer-sphere electron transfer) on metal electrodes covered with a thick film, as shown in Fig. 8-41, with an electron-depleted space charge layer on the film side of the film/solution interface and an ohmic contact at the metal/film interface. It appears that no electron transfer may take place at electron levels in the band gap of the film, since the film is sufficiently thick. Instead, electron transfer takes place at electron levels in the conduction and valence bands of the film. 

Under light illumination, semiconductor electrodes absorb the energy of photons to produce excited electrons and holes in the conduction and valence bands. Compared with photoelectrons in metals, photoexcited electrons and holes in semiconductors are relatively stable so that the photo-effect on electrode reactions manifests itself more distinctly with semiconductor electrodes than with metal electrodes. 

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