Abstract
Origin of electron bands from atomic proximity, or from atomic periodicity in a crystal lattice are indicated. The proximity approach retains chemical atom identity. Atomic level splitting is tabulated and tunneling for electron transfer is discussed. Electronic structure of amorphous semiconductors are identified. Computed eigenvalue spectrum; Periodicity approach; Schrödinger equation for electron wave; k-vector; electron momentum; de Broglie wavelength; value; electron eigenstates; Bloch function; plane wave; Kronig-Penney Potential; explicit mathematics of band model; E(k) diagram; reduced k-vector; Newtonian description of quasi-free electrons; Kinetic electron energy; effective mass discussion; Comparison of periodicity vs. proximity approach; Band edge fuzzing of amorphous semiconductors; Discrete levels in band gap; plausibility approach.
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Notes
- 1.
Applying ΔEΔt and relating ΔE to the time an electron resides at a sufficiently high energy level Eilr (later identified as belonging to an upper band), an uncertainty of ΔE results. The time Δt is related to scattering (see Sect. 15.6); 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≤hΔp, 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. \(V( {x}) = -V_{0} \operatorname{sech2} (\gamma'x)\) 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 discussed later.
- 4.
For E>V 0, the square root in β becomes imaginary. Introducing \(\gamma\pm\sqrt{[ 2m_{0}(E -V_{0})/h^{2}]}\), and with sinh(iγ)=isin(γ) and cosh(iλ)=icos(γ), we obtain for higher electron energies a similar equation:
$$- \frac{\gamma^{2}+\alpha ^{2}}{2\alpha\gamma}\sin(\gamma a_{2})\sin(\alpha a_{1})+\cos (\gamma a_{2})\cos(\alpha a_{1})= \cos(ka). $$ - 5.
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.
- 6.
For the electron behavior, only expectation values can be given. In order to maintain Newton’s second law, we continue to use hk [Eq. (6.15)], 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 d 2 E/dk 2 to define its effective mass.
- 7.
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 still controversial. In three-dimensional lattices, other bands overlap and transitions into these bands complicate the picture.
- 8.
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 semiconductor s near the “band edge”. However, at higher energies further inside the band, there is some evidence that the mean free path 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.
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Böer, K.W. (2013). Elements of Band Structure. In: Handbook of the Physics of Thin-Film Solar Cells. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36748-9_6
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