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Quantum Mechanics of Electrons in Crystals

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Abstract

The electronic band structure of semiconductors reveals most of their intrinsic properties. It consists of the dispersion relation E n (k) for the various bands and is obtained from solving the Schrödinger equation for all electrons and nuclei in the solid. A manageable solution of this many-body problem requires substantial approximations for the interaction potential of all involved particles. Both empirical and ab initio approaches were developed for a one-electron scheme with different ways to approximate the actual interaction potential. Most approaches expand the wavefunction in terms of a set of orthogonal trial functions, followed by variation of the expansion coefficients for finding a self-consistent solution. The more recent density-functional method calculates self-consistently the ground-state energy of the many-electron system from the charge-density distribution.

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Notes

  1. 1.

    The latter assumption is questionable when explaining certain dielectric properties with strong polarization (Sect. 3 in chapter “Interaction of Light with Solids”). If a core–valence electron separation is not made, all potentials in Eq. 1 are simple Coulomb potentials. Otherwise, the effective interaction potentials must be obtained.

  2. 2.

    For the formal relation between the many-body problem and the one-electron band structure, see Hedin and Lundqvist (1970).

  3. 3.

    The Bloch theorem states that nondegenerate solutions of the Schrödinger equation in a periodic lattice are also solutions after translation by a lattice vector, with the amplitude function having lattice periodicity u k (r) = u k (r+R); R is any translation vector which reproduces the Bravais lattice.

  4. 4.

    Namely, k must be real and ψ(r) periodic with lattice periodicity ψ(r) = ψ(r + n a i) (Born–van Karman boundary condition).

  5. 5.

    Some of the methods are considered in more detail in this section. The following abbreviations are commonly used:

    DFT: Density functional theory

    EPM: Empirical pseudopotential method

    EEX: Exact-exchange method

    GGA: Generalized gradient approximation

    GWA: Green’s function GW approximation

    HF(A): Hartree–Fock (approximation)

    KKR: Korringa–Kohn–Rostocker (Green’s function) method

    (L)APW: (Linear) augmented plane wave method

    LCAO: Linear combination of atomic orbitals

    LDA: Local density approximation

    LMTO: Linear muffin-tin orbitals

    LSDA: Local-spin-density approximation

    OPW: Orthogonal-plane-wave method

    PSF: Pseudofunctional method (Kasowski et al. 1986)

    PW: Plane wave

    TB: Tight binding

  6. 6.

    A symmetrical wavefunction does not change sign when the coordinates of two electrons are interchanged, in contrast to an antisymmetrical wavefunction.

  7. 7.

    This model is more commonly also referred to as sps* method or spds* method.

  8. 8.

    The components of 1 and σ are

    $$ \underline{1}=\left(\begin{array}{cc}1& 0\\ {}0& 1\end{array}\right)\, ,\, {\sigma}_x=\left(\begin{array}{cc}0& 1\\ {}1& 0\end{array}\right)\, ,\, {\sigma}_y=\left(\begin{array}{cc}0& -i\\ {}i& 0\end{array}\right)\, ,\, {\sigma}_z=\left(\begin{array}{cc}1& 0\\ {}0& -1\end{array}\right). $$
  9. 9.

    See Fig. 8 in chapter “The Structure of Semiconductors”: Γ for the center at k = (0,0,0); in a face-centered cubic lattice for the point L: \( k=\frac{2\pi }{a}\left(\frac{1}{2},\frac{1}{2},\frac{1}{2}\right) \); for the point X : \( k=\frac{2\pi }{a}\left(1,0,0\right) \).

  10. 10.

    The crossing E n (k) must belong to different symmetry states that cannot interact with each other. States of the same symmetry interact and cannot cross (noncrossing rule). Examples of crossing can be seen in Fig. 11 for Si at X 1 and for noncrossing at Δ for GaAs.

  11. 11.

    Compare with a similar calculation of the density of states for phonons given in Sect. 3.2 in chapter “Elasticity and Phonons.”

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Böer, K.W., Pohl, U.W. (2018). Quantum Mechanics of Electrons in Crystals. In: Semiconductor Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-69150-3_7

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