The transverse and longitudinal optical phonons of GaAs (zinc-blende structure

PHYSICS · Multi-part · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Dielectric and optical properties of solids · Dielectric and optical properties of solids

Solid State Physics, Ch Dielectric response, local fields and ferroelectricity

Problem

The transverse and longitudinal optical phonons of GaAs (zinc-blende structure, one polar mode at the Brillouin zone centre) are routinely used to test the Lyddane-Sachs-Teller relation. ((a)) Starting from the driven equation of motion for the polar optical mode with the local field included, derive ε(ω)=ε∞+(εs−ε∞)ωTO2ωTO2−ω2\varepsilon(\omega)=\varepsilon_\infty+(\varepsilon_s-\varepsilon_\infty)\dfrac{\omega_{TO}^2}{\omega_{TO}^2-\omega^2}, and by imposing the condition for a longitudinal mode, ε(ωLO)=0\varepsilon(\omega_{LO})=0, obtain the Lyddane-Sachs-Teller relation ωLO2/ωTO2=εs/ε∞\omega_{LO}^2/\omega_{TO}^2=\varepsilon_s/\varepsilon_\infty. ((b)) GaAs has εs=12.9\varepsilon_s=12.9 and ε∞=10.9\varepsilon_\infty=10.9 (room temperature). Raman scattering gives the longitudinal mode at ωLO=8.80 THz\omega_{LO}=8.80\,\mathrm{THz} (36.4 meV). Use the Lyddane-Sachs-Teller relation to predict ωTO\omega_{TO}, in THz and meV, and compare with the Raman-measured value ωTO=8.02 THz\omega_{TO}=8.02\,\mathrm{THz} (33.2 meV). State the harmonic, rigid-ion assumption the derivation needs.

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