Abstract
Characteristic for much of the electronic behavior in solids is the existence of energy bands, separated by bandgaps. The bands are permitted for occupation with carriers, and their origin can be described by two complementary models. The proximity approach considers the effect of the neighborhood in a solid on the energy levels of an isolated atom; this model is particularly suited for organic semiconductors, amorphous semiconductors, and clusters of atoms. The periodicity approach emphasizes the long-range periodicity of the potential in a crystal. Electrons near the lower edge of a band in a crystal behave akin to electrons in vacuum; the influence of the crystal potential is expressed by an effective electron mass which increases with increasing distance from the band edge. This chapter describes the basic elements of the electronic band structure in solids.
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Notes
- 1.
Applying ΔEΔt ≅ ħ and relating Δt to the time an electron resides at a sufficiently high energy level E ik (later identified as belonging to an upper band), an uncertainty of ΔE results. The time Δt is related to scattering (see Sect. 2 in chapter “Carrier-Transport Equations”); the electron is removed from this level after λ/v rms ≈ 10−12 s, yielding an uncertainty of ~1 meV, which is on the same order as the splitting provided by only 104 atoms (assuming a band width of ~1 eV and an equidistant splitting of 1 level per added atom – that is, within a crystallite of <100 Å diameter. With larger crystallites the splitting is even closer and results in a level continuum.
- 2.
The de Broglie wavelength is on the same order of magnitude as the uncertainty distance obtained from Heisenberg’s uncertainty principle Δx ≥ ħ/Δp x , which has the same form as λ DB. This yields uncertainty distances of 10 Å for thermal (free) electrons at room temperature.
- 3.
In one dimension, there are other periodic potentials for which the Schrödinger equation can be integrated explicitly. V(x) = −V 0 sech2(γx) is one such potential, which yields solutions in terms of hypergeometric functions (see Mills and Montroll 1970). The results are quite similar to the Kronig-Penney potential.
- 4.
For E > V 0, the square root in β becomes imaginary. Introducing \( \gamma =i\sqrt{2{m}_0\left(e-{V}_0\right)/{\hbar}^2} \) and with sinh(iγ) = i sinγ and cosh(iγ) = i cosγ, we obtain for higher electron energies a similar equation:
$$ -\frac{\gamma^2+{\alpha}^2}{2\alpha \gamma}\sin \left(\gamma {a}_2\right)\sin \left(\alpha {a}_1\right)+\cos \left(\gamma {a}_2\right)\cos \left(\alpha {a}_1\right)=\cos (ka). $$ - 5.
This concept must be used with caution, since k is a good quantum number only when electrons can move without scattering over at least several lattice distances. That is certainly not the case in most amorphous semiconductors near the “band edge” (see Sect. 4 in chapter “Carrier Transport Induced and Controlled by Defects”). However, at higher energies further inside the band, there is some evidence that the mean free path (Sect. 2 in chapter “Carrier-Transport Equations”) is much larger than the interatomic distance even in amorphous semiconductors. In bringing the two approaches together, the argument presented here lacks rigor and has plausibility only in terms of correspondence.
- 6.
In an infinite crystal, the electron (when not interacting with a localized defect) is not localized and is described by a simple wavefunction (i.e., having one wavelength and the same amplitude throughout the crystal). The probability of finding it is the same throughout the crystal (∝ψ 2). When localized, the electron is represented by a superposition of several wavefunctions of slightly different wavelengths. The superposition of these wavefunctions is referred to as a wave packet. A moving electron is represented by a moving wave packet \( \psi =\frac{1}{2\delta k}{\int}_{k-\delta k}^{k+\delta k}u\left(x,k\right)\exp \left(i,\left( kx-\omega t\right), \right) dk \) which quickly spreads out over time. It has its maximum at a position \( \overline{x}=\frac{1}{\hbar}\frac{\partial E}{\partial k}t \), yielding for the group velocity, i.e., the velocity of the maximum of the wave packet, \( {v}_g=\frac{\partial \overline{x}}{\partial t}=\frac{1}{\hbar}\frac{\partial E}{\partial k} \). With E = ħω, we obtain \( {v}_g=\frac{\partial \omega }{\partial k} \).
- 7.
For the electron behavior, only expectation values can be given. In order to maintain Newton’s second law, we continue to use ℏk (Eq. 15), which is no longer an electron momentum. It is well defined within the crystal and is referred to as crystal momentum. We then separate the electron properties from those of the crystal by using ∂ 2 E/∂k 2 to define its effective mass.
- 8.
In theory, the electron will continue to accelerate in the opposite direction to the field and lose energy, thereby descending in the band, and the above-described process will proceed in the reverse direction until the electron has reached the lower band edge, where the entire process repeats itself. This oscillating behavior is called the Bloch oscillation . Long before the oscillation can be completed, however, scattering interrupts the process. Whether in rare cases (e.g., in narrow mini-bands of superlattices or ultrapure semiconductors at low temperatures) such Bloch oscillations are observable, and whether they are theoretically justifiable in more advanced models (Krieger and Iafrate 1986), is controversial. In three-dimensional lattices, other bands overlap and transitions into these bands complicate the picture.
- 9.
This energy difference represents the binding energy of the exciton; its value is much larger than values found in inorganic semiconductors. A large binding energy corresponds to a strong spatial localization, a typical feature of excitons in organic crystals.
- 10.
The polaron character of mobile carriers in organic crystal is often not explicitly considered; in analogy to the quasiparticles of inorganic semiconductors, the carriers are simply termed electrons and holes.
- 11.
The mobility of electrons is defined in Sect. 2.2 in chapter “Carrier-Transport Equations” by μ = (q/m*) × τ, with effective mass m*, charge q, and a mean time τ between scattering events; in organic crystals μ 300Κ is usually below 1 cm2/(Vs), often orders of magnitude smaller, compared to values of 103 cm2/(Vs) for inorganic semiconductors.
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Böer, K.W., Pohl, U.W. (2018). The Origin of Band Structure. In: Semiconductor Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-69150-3_6
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