Anharmonic Lattice Dynamics in MgB₂

To investigate the anharmonicity of phonons in MgB₂, we plot in Fig.~2a the energy as the atoms are displaced by an amount u according to each of the four eigenmodes.

For the B_1g, A_1u, and E_1u modes the distortion energy can be very well approximated by the harmonic expression
E(u)= A₂ u² up to u/a = 0.065. It is unusual that the phonons remain harmonic at these large distortions, but this is consistent with the fact that we did not observe any substantial changes in the measured GPDOS with temperature.

The most interesting observation, though, is the totally opposite behavior of the E_2g in-plane boron mode, as shown in Fig.~2b. The potential well is very shallow at small displacements and increases rapidly at large displacements, and can be fit very well to
E(u)= A₂ u² + A₄ u⁴
with a large ratio of A₄/A₂² = 8. This indicates that this mode is unusually anharmonic and that we are in a non-perturbative regime.

Hence the estimated harmonic energy, w_H (E_2g) = 60.3 meV will be lower than the actual energy.
Self Consistent Harmonic (SCH) approximation yields a better estimate of w_SCH (E_2g) = 70 meV, a 17% enhancement of the harmonic phonon energy w_H (E_2g).
We can calculate the exact energy levels of the potential by numerically solving the Schrodinger equation and obtain w(E_2g) =74.5 meV, a 25% enhancement of the harmonic value and also the Tc.

WE NOTE THAT

For the energy of E_2g mode, there are large discrepancies between the results reported for various calculations. This discrepancies are resolved when the strong anharmonicity associated with the in-plane displacements of the boron atoms is taken into account.
Due to strong anharmonic effects, the harmonic theories such as lattice dynamics calculations from linear-response theory for MgB₂ have to be modified!