Nonadiabatic Kohn Anomaly in Heavily Boron-Doped Diamond

We report evidence of a nonadiabatic Kohn anomaly in boron-doped diamond, using a joint theoretical and experimental analysis of the phonon dispersion relations. We demonstrate that standard calculations of phonons using density-functional perturbation theory are unable to reproduce the dispersion relations of the high-energy phonons measured by high-resolution inelastic x-ray scattering. On the contrary, by taking into account nonadiabatic effects within a many-body field-theoretic framework, we obtain excellent agreement with our experimental data. This result indicates a breakdown of the Born-Oppenheimer approximation in the phonon dispersion relations of boron-doped diamond.

Figure 1

(a) Density-functional theory band structure of diamond for a B concentration of $1.4\times 10^{21}{\mathrm{cm}}^{-3}$. (b) Adiabatic phonon dispersions of pristine (blue lines) and B-doped diamond (dashed black lines) for momenta along $L\text{−}\mathrm{\Gamma }\text{−}X$, as obtained from density-functional perturbation theory. (c)–(e) Measured IXS spectra of pristine and B-doped diamond. The critical momentum for the onset of the KA, $q_c=2k_F$, is indicated by vertical dashed lines; see also (a). (f)–(h) Nonadiabatic spectral function, obtained from Eqs. (1) and (2), for the LO phonon of (c) pristine and (d),(e) B-doped diamond along $\mathrm{\Gamma }\text{−}X$. The phonon branch considered here is marked by the red line in panel (b). (i)–(k) Phonon energies obtained from Eq. (3) in the adiabatic approximation (${\mathrm{\Pi }}_{\mathbf{q}\nu }^{\mathrm{NA}}=0$) and from the fully nonadiabatic theory (present theory). Nonadiabatic phonon dispersions of undoped diamond are reported for comparison. All doping concentrations are in units of ${\mathrm{cm}}^{-3}$.

Figure 2

Energy renormalization of the longitudinal optical phonons of diamond, for doping concentrations of (a) $1.4\times 10^{21}{\mathrm{cm}}^{-3}$ and (b) $3\times 10^{20}{\mathrm{cm}}^{-3}$: experiment (squares) and adiabatic (dashed red line) and nonadiabatic theory (blue line).