Abstract
In the previous chapter, we discussed the electronic states in perfect periodic solids. In this chapter, we turn our attention to a more interesting problem: the electronic states in slightly perturbed solids. The primary tool we introduce for this is the envelope function approximation. It is one of the most important concepts in the quantum theory of solids, and we derive it in detail in Section 3.2. We show that electrons in a single, uncoupled band, such as the conduction band, respond to weak external fields as if they were “free” particles, but with an effective mass that depends on the band curvature. For this reason, the envelope function approximation is also called the effective mass approximation. If we consider electrons in strongly coupled bands, such as the valence bands, the envelope function description of these states becomes similar to the k · p theory. In this case, we must solve a coupled system of equations that depend on the zone center curvatures of the unperturbed bands.
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Notes
We have not written the exact formula for the allowed values of k since, technically, they must be chosen so the Bloch functions satisfy periodic boundary conditions over a giant unit cell with volume Ω. For details, see N. W. Ashcroft and N. D. Mermin, Solid State Physics, (Saunders College, Philadelphia, 1976).
The reciprocal lattice is a fundamental idea in solid state physics. Its only function is to generalize three-dimensional Fourier analysis for functions — such as the cell-periodic wavefunctions that are periodic over regions other than rectangular prisms. For more details, see Ashcroft and Mermin.
Indeed, this is almost the definition, of the first Brillouin zone.
Since u j and u j′ are both cell-periodic wavefunetions corresponding to k = 0, we can apply (3.11).
Technically, they represent, the matrix elements taken between states u jk expanded to second order in k, as we remarked in Section 1.2.2.
A notable exception is the case of optical absorption between valence subbands in quantum wells. See Y. C. Chang and R. B. James, “Saturation of intersubband transitions in p-type semiconductor quantum wells,” Phys. Rev. B 39, 12672 (1989).
In Chapter 2 wo assumed the zone-center wavefunctions integrated to unity over a unit cell.
For a definitive account, see M. G. Burt, “The justification for applying the effective-mass approximation in microstructures,”, J. Phys.:Condens. Matter 4, 6651 (1992).
It is regrettable that many authors fail to recognize this point.
We have made an enormous leap of faith here to even talk about the conduction and valence band edges in quantum wells. As we asserted in the previous section, these ideas do have a useful meaning even in aperiodic structures. But the “band offsets” are very difficult to calculate from first principles, and cannot be measured with great accuracy. Fundamental theoretical and experimental work continues in this area.
If the quantum-confined charge density is high, it becomes important to obtain self-consistent solutions to (3.66) by solving it concurrently with the Poisson equation where ρ(r) is the total charge density and ∈(r) is the dielectric constant.
By proceeding from (3.75) instead of (3.79), it is easy to show that (3.82) also holds for nonzero kt and U ext(z).
We choose, arbitrarily, to use (2.262) rather than (2.263).
Even though (3.123) is correct and holds for all kt, we obtained it by an invalid procedure. The problem is that we cannot sensibly replace k t = in (3.119)-(3.120) with a differential operator. But we can legitimately derive (3.123) by beginning with the full 4 × 4 description (2.255) of the HH and LH states. Since all elements of (2.253) are simple polynomials in the components of k, the operation k → —i is well denned. By assuming a separable solution to the resulting 4 × 4 partial differential equation, we obtain a 4 × 4 ordinary differential equation involving only. Then since the basis transformation (2.256) involves only k x and k y, we can we can apply it to the 4 × 4 ordinary differential equation and obtain (3.123).
Recall that the adjoint A† of an operator A is defined to be the linear operator satisfying for any states f〉 and \g). Using this definition and integrating by parts, it is easy to show (3.128).
It might seem that we have taken the long way around the barn to arrive back at the bulk dispersion relation (3.132). Indeed, much of our discussion could be circumvented by introducing the idea of complex bandstructure. But several important subtleties must be mastered in order to safely apply this concept, so we have chosen to arrive at the final expressions via the straightforward technique of solving ordinary differential equations. See the references at the end of this chapter for more thorough discussions.
If we confine our attention to bound states, and avoid pathological values of E, we will always have an imaginary part of.
This notation would be horribly inconvenient for actually carrying out hand calculations, but it is quite useful for enumerating the “degrees of freedom” of the quantum well solutions.
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© 1998 Springer Science+Business Media New York
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Loehr, J.P. (1998). Electronic States in Quantum Wells. In: Physics of Strained Quantum Well Lasers. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-5673-2_3
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DOI: https://doi.org/10.1007/978-1-4615-5673-2_3
Publisher Name: Springer, Boston, MA
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