Phonon properties of copper oxide phases from first principles

We present density functional theory (DFT) calculations on phonon dispersions, phonon density of states, and thermodynamic quantities for the three copper oxide phases ${\mathrm{Cu}}_2\mathrm{O},{\mathrm{Cu}}_4{\mathrm{O}}_3$, and CuO. For monoclinic CuO we consider the correct antiferromagnetic ground state. Sound velocities for the acoustic phonon branches and Debye temperatures are calculated and are found to be in good agreement with experiment. We further show how the method for the treatment of dipole-dipole interactions in dynamical matrices of Gonze and Lee [Phys. Rev. B 55, 10355 (1997)] may be incorporated in the real-space (direct) method for interatomic force constants (FCs). The role of the long-ranged dipole-dipole interactions in the phonon dispersion is discussed. Based on this method, we outline a perturbationlike scheme to compute first-order derivatives of the phonon mode frequencies with respect to the wave vector which can be used to compute velocities of sound.

Figure 1

Phonon band structures (upper panel) and phonon density of states (lower panel) for (a) ${\mathrm{Cu}}_2\mathrm{O}$, (b) ${\mathrm{Cu}}_4{\mathrm{O}}_3$, and (c) CuO. Experimental data for ${\mathrm{Cu}}_2\mathrm{O}$ (red-bounded stars) is extracted from figures in Ref. [66], and—apart from the latter reference—covers Refs. [60, 126]. For simplicity, different references are not distinguished through use of different colors or symbols in the plot.

Figure 2

Phonon band structure along selected directions in reciprocal space for (a) ${\mathrm{Cu}}_2\mathrm{O}$, (b) ${\mathrm{Cu}}_4{\mathrm{O}}_3$, and (c) CuO. Dispersions are shown without (red; left to $\mathrm{\Gamma })$ and with dipole-dipole corrections (dark blue; right to $\mathrm{\Gamma })$. Shaded areas mark the frequency range bounded by the LO and TO components of the respective mode at the $\mathrm{\Gamma }$ point (see indication of the irreducible representation on the right). Note that only modes subject to clearly visible changes due to the dipole-dipole corrections have been indicated.

Figure 3

Phonon band structure of ${\mathrm{Cu}}_2\mathrm{O}$ (blue solid line) along the $\mathrm{\Gamma }\to \mathrm{X}$ direction together with phonon frequencies at ${\mathbf{q}}_{\mathrm{c}}$ vectors commensurable with a $13\times 1\times 1$ supercell (red-bounded stars). Note that, among the frequencies at the commensurable wave vectors, the topmost frequency at the $\mathrm{\Gamma }$ point $(\sim 630$ ${\mathrm{cm}}^{-1})$ is not missing but coincides with the frequency directly below $(\sim 610$ ${\mathrm{cm}}^{-1})$. The reason is that the highest-frequency $\mathbf{q}=\mathbf{0}$ phonon mode is accompanied by a macroscopic electric field which is incompliant with periodic boundary conditions imposed in DFT calculations.

Figure 4

Constant-volume lattice specific heat (a)–(c) and lattice contribution to entropy (d)–(f) for all three copper oxide phases. Experimental values in (a) for the constant-pressure heat capacity $\left(C_p\right)$, and in (d) for the entropy of ${\mathrm{Cu}}_2\mathrm{O}$ have been taken from Hu and Johnston [127] (blue stars). Constant-volume heat capacities $\left(C_v\right)$ are derived from the relation $C_p-C_v=VT{\alpha }^2/{\beta }_T$ $(V$: sample volume; $\alpha$: coefficient of thermal expansion; ${\beta }_T$: isothermal compressibility) using the data of Refs. [128, 129, 130]. In (c) and (f), in order to get $C_v$, experimental $C_p$ vs $T$ data for CuO from Refs. [70, 131] have been modified following the procedure of Loram et al. [132]. The boundaries of the gray-shaded regions in (c) and (f) mark the antiferromagnetic phase transitions characterized by temperatures $T_{\mathrm{N}}^{(1,2)}$.

Figure 5

Debye temperature ${\mathrm{\Theta }}_{\mathrm{D}}$ versus temperature $T$ extracted from calculated (measured) heat capacities for (a) ${\mathrm{Cu}}_2\mathrm{O}$, (b) ${\mathrm{Cu}}_4{\mathrm{O}}_3$, and (c) CuO. Calculated results are shown as solid black lines where Eq. (28) has been used. Experimental values in (a) for ${\mathrm{Cu}}_2\mathrm{O}$ are taken from Ref. [127] (blue stars). In (c) for CuO experimental data from Loram et al. [132] (light-gray circles), and Gmelin et al. [134] (dark-gray triangles) is shown.