Abstract
The chapter outlines a number of concepts that are useful in the description of periodic structures. The first sections describe the geometrical entities (characteristic vectors, direct and reciprocal lattices, translation vectors, cells, and Brillouin zones) used for describing a lattice. The analysis focuses on the lattices of cubic type, because silicon and other semiconductor materials used in the fabrication of integrated circuits have this type of structure. The next sections introduce the mathematical apparatus necessary for solving the Schrödinger equation within a periodic structure, specifically, the translation operators, Bloch theorem, and periodic boundary conditions.
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Notes
- 1.
Some important exceptions exist. Thin-Film Transistors (TFT), commonly used in flat-screen or liquid-crystal displays, are obtained by depositing a semiconductor layer (typically, silicon) over a non-conducting substrate; due to the deposition process, the structure of the semiconductor layer is amorphous or polycrystalline. Phase-Change Memories (PCM) exploit the property of specific materials like chalcogenides (for example, \(\mathrm{Ge_2Sb_2Te_5}\)), that switch from the crystalline to the amorphous state, and vice versa, in a controlled way when subjected to a suitable electric pulse.
- 2.
The meaning of “insulator” or “semiconductor”, as well as that of “conductor”, is specified in Sect. 18.2.
- 3.
The term “III-V” derives from the fact the Ga and As belong, respectively, to the third and fifth column of the periodic table of elements.
- 4.
{Zincblende} is another name for Sphalerite, an economically important mineral whence zinc is extracted. It consists of zinc sulphide in crystalline form with some contents of iron, (Zn,Fe)S.
- 5.
The symbols indicating the crystal directions are illustrated in Sect. 17.8.1.
- 6.
This aspect is further elaborated in Sect. 17.5.3.
- 7.
The indices of (16.27) are dropped for simplicity.
- 8.
The association \(b \leftrightarrow (n_1,n_2,n_3)\) can be accomplished in a one-to-one fashion by, first, distributing the triads into groups having a common value of \(d = \vert n_1 \vert + \vert n_2 \vert + \vert n_3 \vert\), then ordering the groups in ascending order of d: for example, d = 0 corresponds to \((0,0,0)\), d = 1 to \([(0,0,1),(0,1,0),(1,0,0),(0,0,-1),(0,-1,0),(-1,0,0)]\), and so on. As each group is made by construction of a finite number of triads, the latter are numbered within each group using a finite set of values of b; in order to have b ranging from \(-\infty\) to \(+\infty\), one associates a positive (negative) value of b to the triads in which the number of negative indices is even (odd).
- 9.
Typically, a graphic representation of \(E_i ({\mathbf{k}})\) is achieved by choosing a crystal direction and drawing the one-dimensional restriction of E i along such a direction. Examples are given in Sect. 17.6.5.
- 10.
For instance, in a cube of material with an atomic density of \(6{.}4 \times 10^{27}\) m-3, the number of atoms per unit length in each direction is 4000 μm-1.
- 11.
Here the periodic part of \(w_{i\,\mathbf{k}}\) is indicated with \(\zeta_{i\,\mathbf{k}}\) to avoid confusion with the group velocity.
- 12.
In the case of a free particle (Sect. 9.6) the approximation neglects only the second order because \(\omega (\mathbf{k})\) has a quadratic dependence on the components of \(\mathbf{k}\). Here, instead, the expansion has in general all terms due to the more complicate form of \(\omega_i ({\mathbf{k}})\), so the neglected rest R i contains infinite terms.
- 13.
As mentioned in Sect. 17.6, \(E_n ({\mathbf{k}})\) is considered as a function of a continuous vector variable \(\mathbf{k}\) even when the periodic boundary conditions are assumed.
- 14.
The parabolic-band approximation is not necessarily limited to absolute minima or absolute maxima; here it is worked out with reference to such cases because they are the most interesting ones. However, it applies as well to relative minima and relative maxima. The different values of the inverse, effective-mass tensor’s entries between an absolute and a relative minimum of a branch in GaAs give rise to interesting physical effects (Sect. 17.6.6).
- 15.
The extension of the energy region where the main variation of the Fermi statistics occurs is estimated in Prob. 15.1.
- 16.
From now on the band index n introduced in (17.56) is omitted from the notation.
- 17.
In contrast, the temperature dependence of the energy gap, due to the deformation of the dispersion relation, can not be neglected because of its strong effect on the carrier concentration (Sect. 18.3).
- 18.
If the magnitudes of m t and m l are significantly different, the smaller effective mass dictates the magnitude of m n .
- 19.
The reasoning seems to contradict the fact the large-area, solid-state optical sensors used in cameras and video cameras, based on the CCD or CMOS architecture, are made of silicon. In fact, the complex structure of these several-megapixel sensors and related signal-management circuitry can be realized only with the much more advanced technology of silicon. The relative ease of fabricating complex structures largely compensates for the poorer optical properties of the material.
- 20.
As mentioned in Sect. 17.5.3, the periodic boundary conditions are actually an approximation; however, the interatomic interactions typically give rise to short-range forces, hence the above reasoning holds for all the cells that are not too close to the boundaries.
- 21.
The Bloch theorem was derived in Sect. 17.5.1 with reference to the eigenfunctions of a translation operator in the continuous case; the theorem equally holds for a translation operator in the discrete case, like the dynamic matrix considered here.
- 22.
The same type of limit is applicable to the single-barrier case, whose transmission coefficient is given in (11.22).
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Rudan, M. (2015). Periodic Structures. In: Physics of Semiconductor Devices. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1151-6_17
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DOI: https://doi.org/10.1007/978-1-4939-1151-6_17
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