Band structure of silicon

Piezoresistive sensors. To measure the pressure, the resistance change to stress (the piezoresistance effect) may be employed. When silicon is stressed, the resulting strain breaks the cubic symmetry of the underlying crystal structure. The band structure of silicon is very sensitive to its crystal structure and, as a result, the consequent modification causes changes in the resistivity of the material (holes in the case of p " material). This change is... 

The role of crystal symmetry properties in determining the shape of the bands has been emphasized, but the few examples reported have also shown that the existence of a gap and the energy range of bands depend on the mutual interactions of all particles, electrons, and nuclei, in the lattice. Therefore, the correctness of a calculation is largely dependent on the kind of approximation used in the evaluation of such interactions. In fact, different approximations of the Hamiltonian can produce a variety of results and, in particular, band structures that are not only quantitatively but also qualitatively different in some cases. In Figure 18, the HF band structure of silicon is compared with that obtained with DFT methods, both in the LDA, in the form of Slater-Vosko-Wilk-Nusair ° functional, and with the Becke 3 (B3) parameter-Lee-Yang-Parr (LYP) approximation, which incorporates a part of the exact exchange... 

Figure 18 LDA, B3LYP and HF upper valence and lower conduction band structure of silicon.

Figures 14.4 and 14.5 show calculated band structures of silicon and germanium, respectively. At the typical temperatures that will concern us (0-300 K), the only external action (that is, thermal excitation of electrons and holes, optical absorption edges) will occur dose to the highest point in the valence band and the lowest points in the conduction band.

Fig. 2.2 Left Formation of the valence and conduction bands in covalent semiconductors liom bonding and antibonding sp orbitals, respectively. Right Calculated electronic band structure of silicon [1]. The gray area indicates occupied states in the valence band of the material...

In most conventional semiconductors, such as Si and Ge, covalent bonding dominates. In silicon, for example, the outer 3s and 3p orbitals combine to form hybrid sp orbitals. Neighboring sp orbitals interact to form bonding and antibonding combinations that form the valence and conduction bands of the material, respectively. This is schematically illustrated in Fig. 2.2, which also shows the electronic band structure of silicon. 

The band-structure of silicon obtained in this calculation is shown in Fig. 6.10. It was calculated at the LDA equilibrium lattice constant, even in the GGA case. These band-structures exhibit the well-known band-gap problem of DFT the predicted band-gap is too small roughly by a factor of two. This is true for the LDA and the GGA. In fact, the GGA does not show a great improvement, even when the band-structure is calculated at its predicted equilibrium lattice constant (Table 6.4). The failure of these two DFT schemes in predicting the band-gap of silicon is not a surprise. Even if the true xc potential was known, the difference between the conduction and valence bands in a KS calculation would differ from the true band-gap (Eg). The true band-gap may be defined as the ground-state energy difference between the N and N l systems... 

E. O. Kane, Band structure of silicon from an adjusted Heine Abarenkov calculation, Phys. Rev. 146, 558-67 (1966). 

The band structure of bulk silicon, with possible optical transitions for (c) absorption and (d) emission of a photon, together with (e) the dispersion curves of phonon branches, is shown on the right. After [Kol5],... 

The presence of an impurity such as an As or a Ga atom in silicon leads to an occupied level in the band gap just below the conduction band or a vacant level just above the valence band, respectively. Such materials are described as extrinsic semiconductors. The n-type semiconductors have extra electrons provided by donor levels, and the p-type semiconductors have extra holes originating from the acceptor levels. Band structures of the different types of semiconductors are shown in Fig. 4.3.4. 

Fig. 2.11 Effect of n- and p-type doping on the band structure of a semiconductor (e.g. silicon).

Compounds containing silicon bonded to only one other atom are unstable and are usually only generated and observed as reactive intermediates of short half-life. Silicon compounds subjected to flash photolysis or electrical discharges in the gas phase produce short-lived species SiX (X = H, F, Cl, Br, I, C, Si, etc.), the band structure of which have been studied in detail. The structures, electronic configurations, and so on of Six (X = H, F, Cl, Br, I, N, O, etc.) have also been the subject of MNDO (modified neglect of diatomic overlap) and other calculations. ... 

Eckelt, P. (1967). Energy band structures of cubic ZnS, ZnSe, ZnTe and CdTe (Korringa-Kohn-Rostoker method). Phys. Status Solidi 23, 307-12. Edwards, A. H., and W. B. Fowler (1985). Semiempirical molecular orbital techniques applied to silicon dioxide MIND03. J. Phys. Chem. Solids 46, 841-57. 

Local density functional (LDF) quantum mechanical calculations for materials science. deMon for density functional calculations. Turbomole for Hartree-Fock and MP2 ab initio calculations. ZINDO for extended Fliickel, PPP, CNDO, and INDO semiempirical molecular orbital calculations and prediction of electronic spectra. Plane Wave for band structures of semiconductors. ESOCS for electronic structure of solids. Silicon Graphics and IBM workstation versions. 

An example of a one-dimensional superlattice structure is structure 1, which is an ordered copolymer. The skeleton is formed by silicon and germanium atoms. A unit cell is three times larger than that of a homopolymer. The band structure of this ordered copolymer changes to the zone-folded profile, which may result in a characteristic absorption spectrum. 

An LCAO (linear combination of atomic orbitals) local-density functional approach was used to calculate the band structures of a series of polymer chain conformations unsubstituted polysilane in the all-trans conformation and in a 411 helical conformation, and all-trans poly(dimethylsilane). Calculated absorption spectra predict a highly anisotropic absorption for the all-trans conformation of polysilane, with the threshold absorption peak arising strictly from polarizations parallel to the chain axis. The absorption spectrum for the helical conformation is much more isotropic. Results for the dimethyl-substituted polysilane chain suggest that the states immediately surrounding the Fermi level retain their silicon-backbone a character upon alkyl-group substitution, although the band gap decreases by I eV because of contributions from alkyl substituent states both below the valence band and above the conduction band to the frontier states. 

The local-density functional approach was used to compare the band structures of the sW-trans conformation of unsubstituted polysilane with a 4/1 helical conformation and with an dll-trans conformation of dimethyl-substituted poly silane. In line with previous theoretical studies, the electronic wave functions in the vicinity of the Fermi level are primarily silicon-back-bone states, with the major effect of methyl substitution being a decrease in the gap. The predicted absorption spectra for the dll-trans conformations of unsubstituted and dimethyl-substituted polysilane are similar for nearthreshold absorption. Given this similarity, we believe that the shift in energy and strong anisotropy of threshold absorption that we predict for the two extremes of the dll-trans conformation and the dll-gauche model will also occur in alkyl-substituted systems, which are currently under investigation. 

We have performed ab initio calculations of electronic band structures of nonhydrogenated silicon nanowires in the <001>, <011>. <111> and <112> orientations. Our results clearly indicate that silicon nanowires with the <001>, <111> and <112> axes have turned out to be metallic, while the one with the <011> axis displays the semiconducting behavior. 

Figure 5.9 Effect on the band structure of doping silicon (i) with phosphorus (ii) and aluminium (Hi)...

The band structure of solids has been studied theoretically by various research groups. In most cases it is rather complex as shown for Si and GaAs in Fig. 1.5. The band structure, E(kf is a function of the three-dimensional wave vector within the Brillouin zone. The latter depends on the crystal structure and corresponds to the unit cell of the reciprocal lattice. One example is the Brillouin zone of a diamond type of crystal structure (C, Si, Ge), as shown in Fig. 1.6. The diamond lattice can also be considered as two penetrating face-centered cubic (f.c.c.) lattices. In the case of silicon, all cell atoms are Si. The main crystal directions, F —> L ([111]), F X ([100]) and F K ([110]), where Tis the center, are indicated in the Brillouin zone by the dashed lines in Fig. 1.6. Crystals of zincblende structure, such as GaAs, can be described in the same way. Here one sublattice consists of Ga atoms and the other of As atoms. The band structure, E(k), is usually plotted along particular directions within the Brillouin zone, for instance from the center Falong the [Hl] and the [HX)] directions as given in Fig. 1.5. 

The FEs produced at low temperature by illumination with photons in the vicinity or above Eg have finite lifetimes that depend on temperature (see [34] for silicon), their binding energies, and on the band structure of the semiconductor (the lifetime is larger in semiconductors with indirect gap than direct gap). During their lifetime, they can diffuse in the crystal and be trapped by impurities and defect to become bound excitons (BEs) with energies slightly different from that of the FE. 

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