Ab initio lattice dynamics and thermal expansion of ${\text{Cu}}_2\text{O}$

Using ab initio density-functional perturbation theory the lattice dynamics of ${\text{Cu}}_2\text{O}$ has been extensively studied. Discrepancies with older neutron-scattering results triggered a neutron-scattering investigation of our own which confirmed the theoretical prediction. Based on the accurate description of the phonon dispersion the vibrational part of the free energy was calculated as a function of lattice constant and temperature. The thermal expansion could be obtained in good agreement with experiments. For $T<300\text{K}$ negative thermal expansion (NTE) was observed while for higher temperatures normal thermal expansion was obtained. The origin of the NTE is due to anomalous behavior of phonon modes with energies $<20\text{meV}$ which is highly mode and wave-vector sensitive. Our calculations also showed that free energies based on the exact phonon dispersion and those based on an Einstein-model description (widely used) differed by more than 20%, which clearly indicates the importance of proper phonon-dispersion treatment for obtaining reliable thermodynamic data.

Figure 1

Phonon dispersion of ${\text{Cu}}_2\text{O}$ along the [001] direction. The modes have been classified according to symmetry. Lines represent theoretical results while crosses represent old neutron data (Ref. 26) and open circles mark our neutron measurements. Squares are data taken from Ref. 31.

Figure 2

Phonon dispersion along the [011] direction. For explanation of the symbols see Fig. 1.

Figure 3

Phonon dispersion along the [111] direction. For explanation of the symbols see Fig. 1.

Figure 4

Generalized phonon density of states and partial contributions. A broadening of 1 meV has been used.

Figure 5

Phonon dispersion for different lattice constants. Full lines $a_0=4.30\text{Å}$ and dotted lines $a=4.24\text{Å}$ (compressed lattice).

Figure 6

Phonon dispersion for different lattice constants. Full lines $a_0=4.30\text{Å}$ and dotted lines $a=4.36\text{Å}$ (expanded lattice).

Figure 7

Theoretical Grüneisen parameter $\gamma \left(E\right)$ averaged over phonons of energy $E$.

Figure 8

Free energy as a function of temperature between 0 and 1000 K for $a_0=4.30\text{Å}$. Full line represents the exact calculation while the dashed line gives the Einstein-model result.

Figure 9

(Color online) Normalized percent variation in the lattice constant of cuprous oxide, ${\text{Cu}}_2\text{O}$. Theoretical results (open squares) are compared with experimental data (filled circles) taken from Refs. 32, 33, 34. Lines are only added as guides to the eye.