Relationship between phonons and thermal expansion in Zn(CN)${}_2$ and Ni(CN)${}_2$ from inelastic neutron scattering and ab initio calculations

Zn (CN)${}_2$ and Ni(CN)${}_2$ are known for exhibiting anomalous thermal expansion over a wide temperature range. The volume thermal expansion coefficient for the cubic, three-dimensionally connected material, Zn(CN)${}_2$, is negative (${\alpha }_V=-51\times {10}^{-6}\hspace{0.5em}{\mathrm{K}}^{-1}$) while for Ni(CN)${}_2$, a tetragonal material, the thermal expansion coefficient is negative in the two-dimensionally connected sheets (${\alpha }_a=-7\times {10}^{-6}\hspace{0.5em}{\mathrm{K}}^{-1}$), but the overall thermal expansion coefficient is positive (${\alpha }_V=48\times {10}^{-6}\hspace{0.5em}{\mathrm{K}}^{-1}$). We have measured the temperature dependence of phonon spectra in these compounds and analyzed them using ab initio calculations. The spectra of the two compounds show large differences that cannot be explained by simple mass renormalization of the modes involving Zn (65.38 amu) and Ni (58.69 amu) atoms. This reflects the fact that the structure and bonding are quite different in the two compounds. The calculated pressure dependence of the phonon modes and of the thermal expansion coefficient, $\alpha$${}_V$, are used to understand the anomalous behavior in these compounds. Our ab initio calculations indicate that phonon modes of energy ∼ meV are major contributors to negative thermal expansion (NTE) in both compounds. The low-energy modes of ∼8 and 13 meV also contribute significantly to the NTE in Zn(CN)${}_2$ and Ni(CN)${}_2$, respectively. The measured temperature dependence of the phonon spectra has been used to estimate the total anharmonicity of both compounds. For Zn(CN)${}_2$, the temperature-dependent measurements (total anharmonicity), along with our previously reported pressure dependence of the phonon spectra (quasiharmonic), is used to separate the explicit temperature effect at constant volume (intrinsic anharmonicity).

Figure 1

The structure of Zn(CN)${}_2$ in $P$43$m$. Key: Zn, gray spheres; C, green spheres; N, blue spheres.

Figure 2

The structure of part of one layer of Ni(CN)${}_2$ with $D_{4h}$ symmetry. Key: Ni, gray spheres; C, green spheres; N, blue spheres.

Figure 3

The temperature dependence of the phonon spectra for Zn(CN)${}_2$. The phonon spectra are measured with an incident neutron wavelength of 5.12 Å using the IN6 spectrometer at ILL. The calculated phonon spectra from ab initio calculations are also shown. The calculated spectra have been convoluted with a Gaussian of FWHM of 10% of the energy transfer in order to describe the effect of energy resolution in the experiment carried out using the IN6 spectrometer.

Figure 4

The temperature dependence of the phonon spectra for Ni(CN)${}_2$. The phonon spectra are measured with an incident neutron wavelength of 4.14 Å using the IN6 spectrometer at ILL. The calculated phonon spectra from ab initio calculations are also shown. The calculated spectra have been convoluted with a Gaussian of FWHM of 10% of the energy transfer in order to describe the effect of energy resolution in the experiment carried out using the IN6 spectrometer.

Figure 5

Comparison of the experimental phonon spectra for Zn(CN)${}_2$ and Ni(CN)${}_2$.

Figure 6

The calculated partial density of states for the various atoms in Zn(CN)${}_2$ and Ni(CN)${}_2$.

Figure 7

Raman spectrum of Ni(CN)${}_2$. Inset shows enlargement of the low-wave number region.

Figure 8

The ν${}_{\mathrm{C}\equiv \mathrm{N}}$ region of the Raman (black) and infrared (gray) spectra for Ni(CN)${}_2$.

Figure 9

The calculated phonon dispersion curves for Zn(CN)${}_2$ and Ni(CN)${}_2$. The Bradley-Cracknell notation is used for the high-symmetry points along which the dispersion relations are obtained. Zn(CN)${}_2$: Γ$=$(0,0,0), $X$$=$(1/2,0,0), $M$$=$(1/2,1/2,0), and $R$$=$(1/2,1/2,1/2). Ni(CN)${}_2$: Γ$=$(0,0,0), $X$(1/2,0,0), and $M$(1/2,1/2,0). In order to expand the $y$ scale, the sets of four and three dispersionless modes, respectively, in Zn(CN)${}_2$ and Ni(CN)${}_2$ owing to the cyanide stretch at ∼280 meV are not shown.

Figure 10

The experimental Bose-factor corrected $S$($Q$,$E$) plots for Zn(CN)${}_2$ and Ni(CN)${}_2$ at 180 and 160 K, respectively. The values of $S$($Q$,$E$) are normalized to the mass of the sample in the beam. For clarity, a logarithmic representation is used for the intensities. The measurements for Zn(CN)${}_2$ and Ni(CN)${}_2$ were performed with an incident neutron wavelength of 5.12 Å (3.12 meV) and 4.14 Å (4.77 meV), respectively.

Figure 11

The comparison between the experimental and calculated $\frac{{\Gamma }_i}{B}$ plotted as a function of phonon energy $E$. The $\frac{{\Gamma }_i}{B}$ values derived from ab initio calculations from Ref. [33] are shown by closed circles. The $\frac{{\Gamma }_i}{B}$ values represent the average over the whole Brillouin zone.

Figure 12

(a) The calculated $\frac{{\Gamma }_i}{B}$ and (b) volume thermal expansion coefficient (α${}_V$) derived for Ni(CN)${}_2$ from ab initio calculations.

Figure 13

(a) The comparison between the volume thermal expansion coefficient (α${}_V$) derived from the ab initio calculations and experimental $\frac{{\Gamma }_i}{B}$ values (Ref. [17]) at 165 K. (b) The comparison between the volume thermal expansion derived from the present ab initio calculations (solid line), high-pressure inelastic neutron scattering experiment (Ref. [17]) (dashed line), and that obtained using x-ray diffraction (Ref. [5]) (open circles).

Figure 14

The contribution of phonons of energy $E$ to the volume thermal expansion coefficient (α${}_V$) as a function of $E$ at 165 K in Zn(CN)${}_2$ and Ni(CN)${}_2$. The contribution from phonons for Zn(CN)${}_2$ is obtained from experimental $\frac{{\Gamma }_i}{B}$ as indicated in Ref. [17].

Figure 15

The calculated contribution to the mean squared amplitude of the various atoms arising from phonons of energy $E$ at $T$$=$300 K in Zn(CN)${}_2$ and Ni(CN)${}_2$.

Figure 16

Polarization vectors of the zone-center and zone-boundary modes in Zn(CN)${}_2$. For each mode the energy and Grüneisen parameter is indicated. The lengths of arrows are related to the displacements of the atoms. The absence of an arrow on an atom indicates that the atom is at rest. At the zone center, where the phase factor is zero, the eigenvectors have only real components. The numbers after the mode assignments give the phonon energies and Grüneisen parameters, respectively. Zone-boundary eigenvectors have both real and imaginary components. The actual displacements of atoms are a combination of both these components. Key: Zn, gray spheres; C, red spheres; N, blue spheres.

Figure 17

Polarization vector of the zone-center and zone-boundary modes in Ni(CN)${}_2$. For each mode the energy and Grüneisen parameter is indicated. The lengths of arrows are related to the displacement of the atoms. The absence of an arrow on an atom indicates that the atom is at rest. At the zone center, where the phase factor is zero, the eigenvectors have only real components. The numbers after the mode assignments give the phonon energies and Grüneisen parameters, respectively. Zone-boundary eigenvectors have both the real and imaginary components. The actual displacements of atoms are a combination of both these components. Key: Ni, gray spheres; C, red spheres; N, blue spheres.

Figure 18

The total anharmonicities $(\frac{1}{E_i}\frac{{\mathit{dE}}_i}{\mathit{dT}}|{}_P)$ of different phonons of energy $E$ in Zn(CN)${}_2$ and Ni(CN)${}_2$. The $\frac{1}{E_i}\frac{{\mathit{dE}}_i}{\mathit{dT}}|{}_P$ has been obtained using the cumulative distributions of the experimental data of DOS of Zn(CN)${}_2$ at 180 and 240 K, while for Ni(CN)${}_2$ the experimental data at 160 and 220 K have been used. The inelastic neutron scattering measurements are carried out in the energy-gain mode. The data for Ni(CN)${}_2$ at 160 K were obtained only up to 50 meV.

Figure 19

Total anharmonicity, and implicit (quasiharmonic) and explicit (true anharmonic) contributions of different phonons in Zn(CN)${}_2$. The total anharmonicity $(\frac{1}{E_i}\frac{{\mathit{dE}}_i}{\mathit{dT}}|{}_P)$ has been obtained using the cumulative distributions of the experimental data of DOS of Zn(CN)${}_2$ at 180 and 240 K. The quasiharmonic contribution has been obtained (Ref. [17]) using the pressure dependence of DOS at 165 K. The bulk modulus $B$ value of 34.19 GPa (Ref. [39]) has been used for estimating ${\Gamma }_i$ values from our measured (Ref. [17]) $\frac{{\Gamma }_i}{B}$ values. The large contributions to the measured spectra from the high-pressure cell did not allow the experimental quasiharmonic contribution for Zn(CN)${}_2$ to be obtained beyond 30 meV (Ref. [17]). The data have therefore only been shown up to 30 meV.