File:Phonon polaritons.svg
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Summary
This image presents the dispersion relation of phonon polaritons according to a simple theory[1], giving the two following dispersion relations for the coupled optical phonon/electromagnetic field system :
- longitudinal optical phonons (LO phonons)
- polaritons (transverse em field + TO phonons)
The dashed red curves correspond to the uncoupled modes (photons and phonons: Ω = 0), and the black curves represent solutions of the above equations with parameters chosen to fit the dispersion relation of polaritons in GaP[2], giving
- Uncoupled optical phonon frequency
- ω0 = 45.4 meV
- Coupling parameter
- Ω = 61 meV
- Electric susceptibility
- χ∞ = 7.5
Note that k is very small compared to π/a.
- ↑ Max Born, Kun Huang, Dynamical Theory of Crystal Lattices, Oxford University Press, 1954 (ISBN 9780198503699), « Chapter 8 : Infrared Dispersion and the Retardation Effect on Lattice Vibrations »
- ↑ C. H. Henry and J. J. Hopfield, Raman Scattering by Polaritons, Phys. Rev. Lett. 15, 964 - 966 (1965).
(further minor editing and conversion to svg done with inkscape)
Summary
DescriptionPhonon polaritons.svg |
English: Dispersion relation of phonon polaritons with parameters corresponding to GaP.
Français : Relation de dispersion des polaritons avec des paramètres correspondant au GaP. |
Date | |
Source | Own work |
Author | Mathieu Perrin |
SVG development InfoField | |
Source code InfoField | MATLAB codee=1.602e-19;
hbar = 1.05457168e-34;
c=2.9978e8;
ang=1e-10;
omega0=45.4e-3*e/hbar; %conversion meV->rd/s
Om=61e-3*e/hbar;
chi=7.5;
k=linspace(0,0.0004/ang)';
y1=omega0*ones(size(k)); %uncoupled phonon dispersion
y2=c*k/sqrt(1+chi); %uncoupled photon dispersion
y3=sqrt(y1.^2+Om^2/(1+chi));%longitudinal phonons
%solve for omega in
%k^2*c^2=omega^2*(1+chi+Om^2/(omega0^2-omega^2))
a1 = (1+chi)*ones(size(k));
a2 = -((1+chi)*omega0^2+k.^2*c^2+Om^2);
a3 = k.^2*c^2*omega0^2;
D = sqrt(a2.^2-4*a1.*a3);
y4=sqrt((-a2+D)./(2*a1));%upper polaritons
y5=sqrt((-a2-D)./(2*a1));%lower polaritons
plot(k*ang, [y1 y2]*1000*hbar/e,'r--','Linewidth',1.8);
hold on
plot(k*ang, [y3 y4 y5]*1000*hbar/e,'k','Linewidth',1.8);
hold off
axis([0 0.0004 35 60]);
set(gca,'FontSize',18,'TickLength',[.02 .02],'LineWidth',1.5);
xlabel('k (A^{-1})'); %A to be replaced
ylabel('h\omega (meV)'); %h to be replaced
|
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Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License.http://www.gnu.org/copyleft/fdl.htmlGFDLGNU Free Documentation Licensetruetrue |
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15 November 2008
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Date/Time | Thumbnail | Dimensions | User | Comment | |
---|---|---|---|---|---|
current | 11:20, 13 February 2010 | ![]() | 436 × 450 (14 KB) | commons>Cepheiden | fiex rendering error of one exponent |
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Width | 435.94px |
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Height | 450.11px |