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Elasticity and Phonons

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Semiconductor Physics

Abstract

Springlike interatomic forces allow macroscopic elastic deformations of the semiconductor and coupled microscopic oscillations of each atom. The strain occurring as a response to external stress is conventionally described by elastic stiffness constants. When the strain exceeds the range in which the harmonic approximation of the interatomic potential is valid, higher-order stiffness constants are used. The symmetry of crystals strongly reduces the number of independent constants. Elastic properties can be measured by static deformations or kinetically by sound wave propagation.

Different modes of sound waves propagate with different velocities from which all stiffness constants can be determined. Each mode of such collective oscillations is equivalent to a harmonic oscillator which can be quantized as a phonon. Phonons are one of the most important quasiparticles in solids. They are responsible for all thermal properties and, when interacting with other quasiparticles, for the damping of their motion.

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Notes

  1. 1.

    The ratios σ D = F D /A and σ S = F S /A are a normal stress (uniaxial pressure) and a shear stress, respectively, and P is the hydrostatic pressure. Stresses are the origin of deformations and have the unit of a force per area.

  2. 2.

    S is a fourth-rank tensor with 34 = 81 components S ijkl . The tensor (S ijkl ) is usually reduced to a 6 × 6 matrix (S ik ) by the index substitution given in the text, thereby simplifying the notation but losing the tensor transformation properties; for details, see Dunstan (1997).

  3. 3.

    The full tensor notation reads σ ij = ∑ 3 k = 1 ∑ 3 l = 1 C ijkl  ε kl . This form yields correctly transformed stress–strain relations for arbitrary stress directions, while the reduced notation applies only for stresses along the principal axes, when the generally listed constants C mn are used.

  4. 4.

    For hexagonal crystals, the compressibility is given by \( {\kappa}_{\mathrm{hex}}=\frac{C_{11}+{C}_{12}-4{C}_{13}+2{C}_{33}}{\left({C}_{11}+{C}_{12}\right){C}_{33}-2{C_{13}}^2} \).

  5. 5.

    The factors are necessary due to a simplification in Voigt notation; the full tensor notation comprises, e.g., off-diagonal summands C ijkl ε kl + C ijlk ε lk (k ≠ l), while in the shorter matrix notation, the corresponding summands C mn ε n (n = 4, 5, 6) only appear once.

  6. 6.

    Also the following labels are used (see also Eqs. 26, 27, and 28): v L (corresponding to v 1 or v 100 s, l ), v T (v 2, v 100 s, t ), v l (v 3, v 100 s, l ), v t∥ (v 4, \( {v}_{t_1} \)), v t⊥ (v 5, \( {v}_{t_2} \)), v l’ (v 6, v 111 s, l ), v t’ (v 7, v 111 s, t ).

  7. 7.

    The constant δ can be expressed as Γ 2 C v /(3ρv s 3) with Γ the Grüneisen parameter (Eq. 26 of chapter “Phonon-Induced Thermal Properties”), C v the specific heat, ρ the density, and v s the average sound velocity.

  8. 8.

    An elastic wave can thus be regarded as a stream of phonons, in analogy to an electromagnetic wave which can be described as a stream of photons. Both quasiparticles are not conserved; phonons or photons can be created by simply increasing the temperature or the electromagnetic field.

  9. 9.

    In the harmonic approximation, there is no exchange of energy between different modes of oscillation. Such exchange, necessary to return to equilibrium after any perturbation, requires anharmonicity.

  10. 10.

    To point out the similarity to the E(k) diagram for electrons in a periodic potential (see Sect. 2 in chapter “The Origin of Band Structure”), ℏω is plotted rather than ω as a function of q throughout the book. The limit of the wavenumber between –π/a and +π/a indicates the boundaries of the first Brillouin zone (Sect. 1.3 in chapter “The Structure of Semiconductors”). Extending the diagram beyond this interval provides no new information.

  11. 11.

    Strictly speaking, there are only N − 2 different modes when requiring that the surface atoms remain at rest. By bending the (long) linear chain into a circle, one can obtain an equivalent condition by introducing cyclic boundary conditions [requiring u(x = 0) = u(x = l)] to get around this “N – 2” peculiarity (Born and von Karman 1912). For large N, however, one always has N − 2 ≅ N. Such a cyclic boundary condition is also necessary to permit propagating waves.

  12. 12.

    The group velocity can be defined when at least two waves of slightly different frequencies interact and form a wave train with an envelope forming beats (Fig. 12). Since energy cannot flow past a node, one readily sees that the velocity with which energy is transmitted must equal the velocity with which the nodes move. Adding two waves with ω, q and ω + dω, q + dq, one obtains for the superposition

    $$ {u}_1+{u}_2=\left({A}_1+{A}_2\right)\cos \left(\frac{t}{2} d\omega -\frac{x}{2} d q\right). $$
    (48)

    Thus the motion of the zero-phase point of the envelope

    $$ \frac{t}{2} d\omega -\frac{x}{2} d q=0 $$
    (49)

    yields for the velocity of the groups of waves:

    $$ {v}_g=\frac{x}{t}=\frac{d\omega}{d q} $$
    (50)

    or, more generally, Eq. 47.

  13. 13.

    The velocity in which the phase of a single wave moves; for a node, it is given by

    $$ \omega t- qx=0, $$
    (51)

    resulting in the phase velocity given by Eq. 52; this is the velocity with which energy is transported in such a wave.

  14. 14.

    For instance, in a crystal with four atoms in the basis (primitive unit cell), one has three acoustic and nine optical branches; see the phonon dispersion of wurtzite GaN shown in Fig. 15b.

  15. 15.

    Some materials undergo a phase transition between diatomic and monatomic unit cells (e.g., hcp → fcc transition), whereby the optical branch disappears during the transition. For example, Ca shows such a phase transition at 450 °C.

  16. 16.

    This surface is similar to the Fermi surface in the Brillouin zone discussed in Sect. 1.1 in chapter “Bands and Bandgaps in Solids.” For phonons, however, the energy surface lies at much lower energies.

  17. 17.

    For instance, if the mass of the defect atom is much smaller, the eigenfrequency can be higher than the highest lattice phonon frequency (local mode); for heavier atoms, its optical phonon may lie within the gap between the optical and acoustic branches (gap mode). In a resonant mode, the eigenfrequency of the foreign atom lies within the band of a phonon branch of the host lattice. This mode shows a greatly enhanced amplitude.

  18. 18.

    For simplicity, the frequently applied alloying on either the anion or the cation sublattice of a compound semiconductor is considered.

  19. 19.

    The same applies for phonons localized at an interface. Both a surface and an interface break the three-dimensional translation symmetry of a solid, giving rise to such localization of the solutions of the equation of motion.

  20. 20.

    In d10-anthracene, all 10 hydrogen atoms are substituted by deuterium atoms to increase the cross section for coherent inelastic neutron scattering.

  21. 21.

    Except for compounds in which thermal neutrons react strongly with the nucleus of one of the elements, e.g., l0B or 113Cd.

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Authors and Affiliations

  1. Naples, FL, USA

    Karl W. Böer

  2. Institut für Festkörperphysik, EW5-1, Technische Universität Berlin, Berlin, Germany

    Udo W. Pohl

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  1. Karl W. Böer
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Böer, K.W., Pohl, U.W. (2018). Elasticity and Phonons. In: Semiconductor Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-69150-3_4

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