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Lattice Dynamics and Thermal Expansion Behavior in Metal Cyanides, MCN (M=Cu, Ag, Au): Neutron Inelastic Scattering and First Principles Calculations

We report measurement of temperature dependence of phonon spectra in quasi one-dimensional metal cyanides MCN (M=Cu, Ag and Au). Ab initio lattice dynamics calculations have been performed to interpret the phonon-spectra as well as to understand the anamolous thermal expansion behavior in these compounds. We bring out the differences in the phonon mode behavior to explain the differences in the thermal expansion behavior among the three compounds. The chain-sliding modes are found to contribute maximum to the negative thermal expansion along c axis in the Cu and Ag compounds, while the same modes contribute to positive thermal expansion in the Au compound. Several low energy transverse modes lead to positive thermal expansion along a and b axis in all the compounds. The calculated elastic constants and Born effective charges are correlated with the difference in nature of bonding among these metal cyanides.

The discovery of colossal thermal expansion behavior in metal cyanides compounds 15,18,19 has further accelerated the research in this field. The discovery of NTE has lead to industrial applications of these compounds in various areas like fiber optics, coatings, electronics and mirror substrates to tooth fillings Recently metal cyanides have gained attention. Zn(CN) 2 was the first compound in this category 27 . The word "colossal" is used to emphasis the huge thermal expansion behavior of the compound. The structure consists of tetrahedral framework of Zn connected with four cyanide units.
The tetrahedral units are connected through CN, which provide the flexibility to bend the tetrahedra and results in NTE behaviour. The isostrucral cyanide Cd(CN) 2 also exhibit NTE behavior. The magnitude of negative thermal expansion coefficient is found to be larger 17 in comparison to Zn(CN) 2 . The first principles calculations have been done to understand the difference in magnitude of negative thermal expansion coefficient in these two compounds. The calculations shows that low energy phonons in Cd(CN) 2 are more sensitive to pressure than in Zn(CN) 2 . NTE has been explained in terms of rotations, translations and deformations of M(C/N) 4 coordinated tetrahedra 28,29 .
Recently a number of prussian blue analogue were also found to be exhibit NTE, where the feature is attributed from metollopholic interactions 19,30,31   Raman and infrared measurements have also been done 43 on MCN compounds. These measurements are limited to zone centre, hence they do not provide complete information about dynamics of the compounds. Here we present the temperature dependent inelastic scattering measurements on these cyanides. The measured spectra have contributions from all phonon modes from entire Brillouin zone. Our studies provide vibrational properties of these cyanides and the analysis of vibrational spectra using ab-initio phonon calculations is useful to understand the thermal expansion behavior of these cyanides.

II. EXPERIMENTAL DETAILS
The  [45][46][47] to the phonon density of states g (n)( E) as follows: where the + orsigns correspond to energy loss or gain of the neutrons respectively and where 1 B n(E,T) exp(E/k T) 1 . A and B are normalization constants and b k , m k , and g k (E) are, respectively, the neutron scattering length, mass, and partial density of states of the k th atom in the unit cell. The quantity between < > represents suitable average over all Q values at a given energy. 2W(Q) is the Debye-Waller factor averaged over all the atoms. The weighting factors

III. COMPUTATIONAL DETAILS
The ab-initio phonon spectra can be calculated using either from supercell method or density functional perturbation theory (DFPT) 51 method. We have used the former approach of supercell method to calculate the phonon frequencies. The PHONON software 52 is used to generate supercell with various (±x, ±y, ±z) atomic displacement patterns. The Hellman-Feynman forces on the atoms in the super cell have been calculated using density functional theory as implemented in the VASP software 53 .
The phonon calculations for HT-CuCN, AgCN and AuCN are performed considering the periodic lattice model using the experimental structure parameters as given in Table I However the peak at about 55 meV becomes more diffusive as temperature increases from 150 to 310 K. The intensity of these peaks decrease as the temperature rises. The increase in temperature will enhance the vibrational mean square amplitude of atoms; hence the Debye Waller factor would in turn reduces the intensity of the peaks. Also the C/N disorder is known to increase 39 with temperature which will further reduce the sharp features in the phonon spectra. In Fig. 3(b) we have shown the neutron inelastic scattering spectra measured at 150 K for high temperature and low temperature phases of CuCN. We find that the peak at around 20 meV in LT-CuCN seems to be broader in comparison to that in HT-CuCN, while at 300 K ( Fig. 3(a)) the width of peaks in both the compounds seems to be same. The larger width at low temperature in LT-CuCN may be due to the fact that low temperature phase is a modulated structure of the high temperature phase. It seems at higher temperature the effect due to the CN disorder and anharmonicity dominate and the inelastic spectra as measured in both the phases appear similar.

B. Calculated Phonon Spectra and Elastic Constants
The crystal structure of all three metal cyanides is known to show C/N disorder 39  We have also calculated the phonon dispersion (Fig 5)  The comparison between the experimental and calculated phonon spectra is shown in Fig. 4 The calculated elastic constants as given in however, significant distortion is observed in HT-CuCN and AgCN.

C. Partial Phonon Density of States
The partial density of states provides the contributions of the individual atoms to the total phonon spectra. We have calculated (Fig. 6) (Table IV)

D. Born Effective Charges
The computed Born effective charges in all three compounds are listed in Table III Further for AuCN (Table II)

E. Thermal Expansion Behavior
The lattice parameter as a function of temperature has been reported from neutron diffraction measurements at temperature ranging from 90 K to 450K 39 . The measurements show that the c lattice parameter decreases with increase in temperature, however, lattice parameter a (=b) shows positive expansion behaviour. The overall volume thermal expansion is found to be positive in all three cyanides and has similar magnitude. The negative thermal expansion behavior along c axis is largest in CuCN and least in AuCN. So also, the positive expansion along a and b-axis is largest in CuCN and least in AuCN. We have computed the thermal expansion behavior using the quasiharmonic approximation. Each phonon mode of energy E i contributes to the volume thermal expansion , where V is the unit cell volume, B is the bulk modulus, i ( = -lnE i / lnV) are the mode Grüneisen parameters and C Vi the specific-heat contributions of the phonons in state i (= qj) of energy E i . The volume dependence of phonon frequency is used to calculate the thermal expansion behavior. The phonon spectra in the entire Brillouin zone have been calculated at two volume corresponding to the experimental structures at 10 K and 310 K. The calculated Grüneisen parameters are shown in Fig.8(a). It can be seen that low energy modes below 4 meV have large positive Grüneisen parameters. The calculated partial density of states shows that the contribution at such low energies is mainly from the M( =Cu, Ag and Au) atoms. As shown in Fig. 5 the calculated transverse acoustic modes in HT-CuCN and AgCN and AuCN are unstable. For the thermal expansion calculation the phonon energies have been calculated at 8000 q points (72000 phonon modes) in the entire Brillouin zone. We find that among these 72000 modes, the number of unstable modes in HT-CuCN and AgCN and AuCN are 558, 453 and 75 respectively, which is less than 1%. The temperature dependence of the unit cell volume is calculated without including the unstable modes. The calculations are qualitatively in good agreement with the observed thermal expansion behavior in AuCN, but underestimated the experimental magnitude in other two compounds (Fig 9). The underestimate might be related to the C/N disorder as discussed below.
The experimental 39,44 value of the coefficient of negative thermal expansion (NTE) along the chain direction (α c ) for HT-CuCN, AgCN and AuCN is -27.9×10 -6 K -1 , -14.8×10 -6 K -1 and -6.9×10 -6 K -1 respectively, while positive thermal expansion (PTE) in the a-b plane (α a ) is 74.8×10 -6 K -1 , 65.7×10 -6 K -1 and 57.4 ×10 -6 K -1 respectively. As noted above, among the three compounds HT-CuCN has the highest C/N disorder and it has also the highest positive as well as negative thermal expansion coefficients. AuCN has the least C/N disorder and has the smallest values of NTE and PTE coefficients. As mentioned above, the ab-initio calculations performed with the ordered structures exhibit the highest number of unstable modes for HT-CuCN, while AuCN show the least number of unstable modes. It seems C/N disorder stabilizes the structure. Among cyanides, nickel cyanide Ni(CN) 2 has a long-range ordered structure in two dimensions (a-b plane) but a high degree of stacking disorder in the third dimension. The compound exhibits 59 NTE in two dimensions (α a = −7 10 -6 K -1 ) along with a very large PTE coefficient (α c = 61.8 10 -6 K -1 ) perpendicular to the layers. Here again it can be seen that disorder along c-axis results in large overall volume thermal expansion (α V = 48.5 10 -6 K -1 ). It appears that the C/N disorder contributes towards positive thermal expansion behavior. The order-disorder transitions at around 400 K in ZrW 2 O 8 reduces 21 the overall NTE coefficient.
As noted above, the linear thermal expansion coefficient along "a-" and "c-" axis are found to be positive and negative respectively. We are interested to find the modes which have large negative and positive Grüneisen parameters and contribute towards thermal expansion behavior. The estimated Grüneisen parameters ( i ) and the specific-heat contribution of modes (C Vi ) from the ab-initio calculations have been used to estimate the contribution of the various phonons to the thermal expansion ( Fig. 8(b)) as a function of phonon energy at 300 K. The maximum contribution to α V seems to be from the low-energy modes below 10 meV.   42,56 (T=10 K) as used in the ab-initio calculations of phonon spectra. The "a' and "c' lattice constants and atom coordinates in the hexagonal unit cell are given.