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Bands and Bandgaps in Solids

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Semiconductor Physics

Abstract

Valence and conduction bands and the bandgap in between these bands determine the electronic properties of solids. For semiconductors, the band structure of the conduction band and the valence bands near the edge to the fundamental bandgap is of particular interest. Both the band structure and bandgap are influenced by external parameters such as temperature and pressure and can also be changed by alloying and heavy doping.

In low-dimensional semiconductors like superlattices and quantum wells, quantum wires and quantum dots anisotropic carrier confinement occurs. The effective gap and energies in conduction and valence bands can be varied by changing spatial dimensions and barrier height of the low-dimensional structure. The bands in amorphous semiconductors near the band edge are ill-defined since long-range periodicity is missing. Still the density-of-state distribution shows significant similarities to that of the same material in the crystallite state.

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Notes

  1. 1.

    In a real crystal this number is reduced since electron scattering limits the coherence length of the electron wave (i.e., the length in which quantum-mechanical interaction can take place). With a mean free path λ the number of atoms responsible for the band-level splitting is of the order of (λ/a)3 for a primitive cubic lattice with lattice constant a. Each of these levels is broadened by collision broadening; therefore, even for λ approaching a, the result is still bands rather than discrete levels.

  2. 2.

    AB compounds containing these elements are referred to as pnictides or chalcogenides.

  3. 3.

    In a quantum-mechanical picture, any E(k) state represents a certain mass and velocity. In a filled band, all of them add up to zero.

  4. 4.

    A positive charge was arbitrarily related to the charge of a glass rod, rubbed with silk (by Benjamin Franklin); this charge was not caused by an added electron as it became known later, but by a missing electron on the glass rod. This electron was removed by the silk.

  5. 5.

    In this and all chapters dealing with electrical conductivity, the electric field is identified as F and the energy is identified as E.

  6. 6.

    The difference between metals, where the overlap range is allowed, and semiconductors, where the overlap range is forbidden (bandgap), depends on Wigners rules (Wigner 1959), which state that eigenstates belonging to different symmetry groups of the Hamiltonian cannot mix (metals). In semiconductors they do mix, yielding sp 3 hybridization for Si.

  7. 7.

    Each band contains a large number of energy levels (see Sect. 4.2 of chapter “Quantum Mechanics of Electrons in Crystals”). Increasing number of electrons first fills the levels at the lowest energy (for T = 0) and then successively higher and higher energies. This process is referred to as band filling.

  8. 8.

    There are also Dresselhaus parameters L, M, and N (Dresselhaus et al. 1955) to describe the valence band. They are related to the Luttinger parameters by

    $$ {\gamma}_1=-2{m}_0\left(L+2M\right)/3,\,\, {\gamma}_2=-2{m}_0\left(L-M\right)/6,\,\, {\gamma}_3=-2{m}_0N/6. $$
    (14)
  9. 9.

    Note that F represents the electric field and E an energy.

  10. 10.

    Here cyclotron resonance is discussed within the same band, and quantum effects are neglected. This can be justified when, neglecting scattering, each electron describes full circles which have to be integers of its de Broglie wavelength Eq. 6 of chapter “The Origin of Band Structure”. This integer represents the quantum number n q of the circle, and for the magnetic induction discussed here, it is a large number. Resonance means absorption (or emission) of one quantum ℏω c, hence changing n q by Δn q = ±1, which is the selection rule for cyclotron transitions. Since Δn q n q , a change in circle diameter is negligible; hence, the classical approach is justified. At higher fields the circles become smaller, and when approaching atomic size, the quantum levels (Landau levels – see Sect. 2 of chapter “Carriers in Magnetic Fields and Temperature Gradients”) become wider spaced and a quantum-mechanical approach is required. For reviews, see Lax (1963), Mavroides (1972), and McCombe and Wagner (1975).

  11. 11.

    The circle diameter (πv n /ω c) is typically of the order of 10−3 cm for a magnetic induction of 1 T; here v n is the thermal velocity of an electron. In metals, however, one also has to consider the skin penetration of the probing electromagnetic field. The skin depth of a metal is usually a very small fraction of the circle diameter, so that the probing ac field can interact only at the very top part of each electron cycle close to the surface. This enhances information about near-surface behavior in metals, while in semiconductors, probing extends throughout the bulk.

  12. 12.

    The conventional term alloy of metals also encompasses crystallite mixtures of nonintersoluble metals, such as lead and tin (solder). Here, however, only materials within their solubility ranges are discussed. The Hume-Rothery rule identifies these metals as having similar binding character, similar valency, and similar atomic radii (Hume-Rothery 1936). Corresponding guidelines apply to the intersolubility of cations or anions in compounds.

  13. 13.

    This Debye–Waller factor (W) is related to the probability of phonon emission during electron or x-ray diffraction and is given in the Debye approximation by

    $$ p=\exp \left(-2W\right)=\exp \left(-\frac{6{E}_R}{k\varTheta}\left[\frac{1}{4}+\frac{T}{\varTheta }{\int}_0^{\varTheta /T}\frac{x\,\, dx}{\exp (x)-1}\right]\right), $$
    (41)

    where E R is the recoil energy Mv 2/2.

  14. 14.

    At room temperature this is λ = h/p = h/ \( \sqrt{2{m}^{\ast }E} \) ≅ 17 nm for 300 K thermal energies and a typical effective mass of 0.2 × m 0, i.e., in the 10 nm range. For excitons the relevant Bohr radius is also in this range, e.g., 11.5 nm for GaAs and 3.2 nm for GaN.

  15. 15.

    Quantum wells are fabricated from a layer sequence BAB grown using epitaxy (Sect. 3.3 of chapter “Properties and Growth of Semiconductors”); the growth direction is usually referred to as z direction, which we designate here as the direction where the confinement occurs.

  16. 16.

    The alternating potential shown in Fig. 30a is of type I, i.e., a minimum of E c(z) coincides with a maximum of E v(z). Both minima and maxima coincide in a type-II superlattice (Fig. 30b). An example for type II is the GaAs x Sbl-x superlattice. For x below 0.25, the valence band of GaAs extends above the conduction band of GaSb, resulting in quasimetallic behavior. For a review of type-II superlattices, see Voos and Esaki (1981).

  17. 17.

    This is a typical band offset at heterointerfaces; for the important GaAs/Ga x A1-x As interface, 62% of the x-dependent bandgap discontinuity ΔE g form the discontinuity at the conduction-band edge ΔE c and 38% at the valence-band edge ΔE v (Watanabe et al. 1985).

  18. 18.

    For the concept of polaron quasiparticles, see Sect. 1.2 of chapter “Carrier-Transport Equations”.

  19. 19.

    The intermolecular interaction is expressed in terms of a transfer integral (see the chapter on transport later in the book). In the tight-binding approximation of a one-dimensional molecule chain, the total bandwidth equals four times the transfer integral between neighboring molecules. The bandwidth for any molecular packing can be expressed from the amplitude of the transfer integrals between the various interacting units (Brédas et al. 2002).

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Böer, K.W., Pohl, U.W. (2018). Bands and Bandgaps in Solids. In: Semiconductor Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-69150-3_8

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