Non-Linear Electron-Phonon Coupling in MgB₂

Since the in-plane motions of the boron will change the boron orbital overlap, significant electron-phonon coupling can be expected for the planer sigma boron conduction band at the Fermi level and this plays an important role in the superconducting pairing.

This can be most easily seen by comparing the band structure of the undistorted lattice to that of distorted one by a zone center phonon. For the harmonic phonons we did not see any significant changes near the Fermi level. However, for E_2g modes, there is a significant splitting of the bands as well as shifts near the Fermi level as shown in Fig.3 (click here to see it above). indicating strong electron-phonon coupling.

To determine the electron-phonon (EP) coupling quantitatively, we evaluated the Fermi-surface averaged deformation potential,

D(u) = < [ dE(k) - dmu ]² >,

for each zone-center frozen phonon. Here dE(k) is the change in the one-particle energy with momentum k due to the frozen phonon, dmu the corresponding change in the chemical potential. <...> denotes an average of k over the Fermi surface, which we have carried out numerically.

For the harmonic B_1g, A_2u, and E_1u phonons we calculated an insignificant coupling, and conclude that the electron-phonon coupling is negligible for these modes.

For the E_2g modes, Fig.~4 shows that the coupling D(u) is large and has both quadratic and quartic terms in frozen-phonon amplitude, u, indicating significant non-linear electron-phonon coupling.

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In conclusion, we have demonstrated that the in-plane boron modes are very anharmonic and couples to the boron sigma bands non-linearly. The following figure shows the in-plane boron mode and the sigma bands (superimposed for visualization).