Soft modes and NTE in Zn(CN)2 from Raman spectroscopy and first
  principles calculations
Microsoft Word - word_Zn_CN__high_pressure
Soft modes and NTE in Zn(CN)2 from  
Raman spectroscopy and first principles calculations 
T. R. Ravindran*, A. K. Arora, Sharat Chandra, M. C. Valsakumar and N. V. Chandra 
Shekar 
Materials Science Division, Indira Gandhi Centre for Atomic Research, Kalpakkam 603 
102, India 
We have studied Zn(CN)2 at high pressure using Raman spectroscopy, and report 
Gruneisen parameters of the soft phonons.  The phonon frequencies and eigen vectors 
obtained from ab-initio calculations are used for the assignment of the observed phonon 
spectra.   Out of the eleven zone-centre optical modes, six modes exhibit negative 
Gruneisen parameter.  The calculations suggest that the soft phonons correspond to the 
librational and translational modes of C≡N rigid unit, with librational modes contributing 
more to thermal expansion.  A rapid disordering of the lattice is found above 1.6 GPa 
from X-ray diffraction. 
PACS numbers: 62.50.+p, 63.20.Dj, 78.30.–j, 78.20.Bh 
*Corresponding author: Email: trr@igcar.gov.in 
Interest in materials that exhibit negative thermal expansion (NTE) was renewed 
after the report [1] of high and isotropic NTE in Zr(WO4)2 over a wide temperature range, 
leading to extensive work and several reviews on the subject [2-5].  The structure of 
Zr(WO4)2 and several other NTE materials consist of corner sharing tetrahedral and 
octahedral units.  From Raman spectroscopic investigations on Zr(WO4)2 as a function of 
pressure and temperature, the phonons responsible for NTE have been identified, and it 
has been shown that in addition to the librational (rigid-unit) mode at 5 meV, several 
other phonons of much higher energy also contribute significantly to the NTE in this 
material [6-9].  Based on structural analysis transverse displacements of the shared 
oxygen atoms and consequent rotation of polyhedra [1] was suggested as the cause of 
NTE in Zr(WO4)2.  In the context of corner linked structures, Zn(CN)2 is remarkable, as it 
has C≡N as the linking species between tetrahedral units instead of a single atom and 
exhibits twice as much coefficient of NTE (-17x10-6 K-1) [10] as that of Zr(WO4)2.  The 
structure of Zn(CN)2 consists of three-dimensional, inter-penetrating, tetrahedral 
frameworks of Zn-CN-Zn chains  [11].  Two different cubic structures, mP 34  (CN-
ordered) [11] and mPn3 (CN-disordered) [10] have been reported to fit well to the 
diffraction patterns.  In the ‘ordered structure’ the CN ions lying along the body diagonal 
are orientationally ordered such that they form ZnC4 and ZnN4 coordination tetrahedra 
around alternate cations.  On the other hand, in the ‘disordered structure’, C and N atoms 
are randomly flipped so as to occupy the sites with equal probability.  It was shown 
recently from factor group analysis in conjunction with Raman and IR spectroscopic 
measurements that the structure is indeed disordered [12]. 
From a topological model treating ZnN4/ZnC4 as rigid units the structure was 
argued to support a large number of low frequency rigid unit phonon modes (ωph< 2 THz, 
≈70 cm-1) that contribute to NTE [13].  On the other hand, it was shown recently by 
spectroscopic measurements [12] that the lowest energy optical mode in Zn(CN)2 is an IR 
mode at 178 cm-1.  It has been shown by atomic pair distribution function analysis of X-
ray diffraction data and suitable modelling of the displacements of C/N away from the 
body-diagonal that this displacement increases as a function of temperature [14].  This is 
in any case expected from the increased amplitude of atomic vibrations as temperature is 
increased.  However, there is no report of the role of different phonons to thermal 
expansion.  Since phonon modes and their Gruneisen parameters are directly responsible 
for thermal expansion in a material, it becomes vitally important to study them. 
Here we report the first study of phonons in Zn(CN)2 at high pressure using 
Raman spectroscopy and ab-initio calculations. High pressure X-ray diffraction 
measurements are also carried out for obtaining the bulk modulus to calculate Gruneisen 
parameters.  From high pressure Raman measurements soft phonons are identified.  In 
addition, first-principles ab-initio density functional calculations are performed at 
different volumes and phonon dispersion curves obtained using frozen phonon 
approximation with SIESTA code [15].  The phonon eigen vectors are used for the 
assignment of phonon modes.  The thermal expansion coefficient is calculated from 
Gruneisen parameters of all the phonons and compared with the reported value. 
Zn(CN)2 (>99.5%) was obtained from Alfa Aesar.  X-ray diffraction pattern of 
this powder sample showed no observable impurity phases.  A small piece of sample of 
lateral dimensions ~100 μm was loaded into a gasketed, Mao-Bell type diamond anvil 
cell.  Raman spectra were recorded at different pressures in the backscattering geometry 
using the 488-nm line of an argon ion laser.   Methanol-ethanol (4:1) mixture was used as 
pressure transmitting medium.  Ruby fluorescence was used to measure pressure.  
Scattered light from the sample was analyzed by a SPEX double monochromator, and 
detected with a cooled photomultiplier tube operated in the photon counting mode. 
Scanning of the spectra and data acquisition were carried out using a PSoC 
(Programmable System on Chip) hardware controlled by LabVIEW® 7.1 program [16].  
The spectral range covered was 10-2400 cm-1 that also includes the C≡N stretch mode 
around 2220 cm-1.  High pressure X-ray diffraction (HPXRD) was carried out in an angle 
dispersive mode using Guinier diffractometer [17].  The incident Mo Kα1 radiation is 
obtained from a Rigaku 18 kW rotating anode x-ray generator. 
Ab-initio calculations were carried out in the framework of the density functional 
theory using the Perdew–Burke–Ernzerhof generalized gradient approximation for 
exchange and correlation [18].  A 3×3×3 supercell of Zn(CN)2 unit cell was used for 
determining the relaxed atomic configuration, and phonon frequencies calculated using 
the SIESTA code.  The calculations were performed using a Monkhorst-Pack grid of 
8×8×8 k-points with a shift of 0.5.  The energy cut off was 350 Rydbergs and a double 
zeta plus polarization (DZP) basis set was used.  Standard norm-conserving, fully 
relativistic Troullier-Martins TM2 pseudopotentials were used.  The computations were 
performed in a 16-node linux cluster. 
The 30 degrees of freedom arising from the 10 atoms in the cubic unit cell of 
Zn(CN)2 result in 3 acoustic and 27 optical branches.  Out of the three structural units 
viz., C≡N ion, ZnC4 tetrahedron and ZnN4 tetrahedron, C≡N is the most strongly bound 
unit and hence taken as a ‘rigid molecular unit’.  The 6 degrees of freedom corresponding 
to the linear molecular ion C≡N can be divided into 1-internal (stretching vibration), 3 
rigid-translations and 2 rigid rotational degrees of freedom.  The ‘disordered’ structure of 
zinc cyanide has the following irreducible representations of optical phonons [12]: 
.OptΓ = A1g + Eg + F1g + 3F2g + A2u + Eu + 2F1u + F2u 
Out of these, the A1g, Eg and F2g modes are Raman active and F1u mode is IR active.  The 
remaining four modes are optically inactive. 
  Figure 1 shows the Raman spectra of Zn(CN)2 at several pressures including 
ambient.  There are three Raman modes clearly seen at 2221, 342, and 200 cm-1.  Out of 
these, the asymmetric peak about 342 cm-1 is actually a doublet that can be resolved into 
339 and 343 cm-1 [12].  The linewidth of all three modes increase and their intensities 
reduce at high pressures.  While the C≡N stretch mode about 2220 cm-1 hardens as 
pressure is increased, the other two modes are seen to soften.  The modes at 342 cm-1 and 
200 cm-1 are too weak to follow above 1 GPa.  Figure 2 depicts the phonon frequency (ω) 
vs. pressure (P) for the three modes observed by Raman spectroscopy.  Most 
measurements were carried out under hydrostatic conditions using (methanol + ethanol) 
as pressure transmitting medium.  One set of measurements in which no medium was 
used (open circles in Fig. 2) resulted in a weaker P dependence for the 2220 cm-1 mode 
up to a pressure of 1.5 GPa and a negative coefficient above 2 GPa.  The reason for this 
change of slope from positive to negative could be a structural transition occurring under 
non-hydrostatic pressure.  However, high pressure X-ray diffraction measurements 
(discussed later) under hydrostatic pressure have not indicated any structural phase 
transition between 1.5 and 2 GPa or at any pressure up to 5.2 GPa, the highest pressure 
up to which measurements were made. 
X-ray diffraction patterns were recorded at several pressures up to 5.2 GPa.  Only 
three reflections, viz., (110), (211) and (321) could be observed.   As pressure is 
increased, the intensity of all lines reduces drastically.  At a pressure as low as 0.2 GPa 
the intensities of the peaks reduce by about 50%.  Above 0.6 GPa the (321) line 
disappears and above 1.6 GPa, only the (110) line is present, which continues up to the 
highest pressure of 5.2 GPa, indicating possible disordering of C≡N has taken place 
above 1.6 GPa.  Such a partial/sublattice amorphization has been reported earlier also in 
other compounds [19, 20].  Lattice parameters at several pressures were obtained from 
the three lines using a disordered cubic space group ( mPn3 ) structure.  The unit cell 
volume obtained as a function of pressure was fitted to Murnaghan equation of state and 
resulted in a bulk modulus B0=25±11 GPa.  The large error in B0 is due to the scatter in 
the XRD data and also the small number of reflections that were used to calculate the 
lattice parameters.  With this input of B0 the mode Gruneisen parameters (γi  = B0ωi−1 
∂ωi/∂P) of the three Raman modes could be calculated (Table 1, last column).  In the 
absence of γi values of other vibrational modes, the thermal expansion coefficient has 
been calculated using simulation data as detailed in the next paragraph. 
It is not straight forward to incorporate random disorder in ab-initio calculations.  
When such a disorder is introduced by randomly flipping half the C≡N species in the 
supercell, it is found that for this disordered structure of cubic Zn(CN)2 - when the system 
is allowed to relax - the ground state energy does not converge to a stable configuration 
but evolves into a tetragonal structure (space group nmP 24 ) with c-parameter ~0.5% 
larger than the a- and b-parameters.  Upon further relaxation, the structure slowly 
becomes triclinic.  Additionally, the inter-atomic forces are large and do not converge to 
small values.  On the other hand, for the ‘ordered structure’ the forces converged to 
values less than 10-6 eV/Å due to geometrical considerations.  Hence the ordered 
structure of Zn(CN)2 is used for computational purposes.  It should be pointed out that the 
values of the vibrational frequencies obtained from either of the space groups are not 
expected to be different from each other, since the same kind of atomic motions are 
involved in the vibrational modes.  The number of zone-centre optical phonon modes is 
also the same in either space group.  The total energy of the system was computed in the 
relaxed configuration for different volumes of the cell up to V/V0=0.844.  The energy vs. 
volume data was fitted to Murnaghan equation and the bulk modulus obtained is 88 GPa.  
A similar result (90 GPa) is obtained when WIEN2K is used to calculate the bulk 
modulus.  Phonon dispersion curves at different volumes were calculated using the 
frozen-phonon method using the VIBRA module in the SIESTA package.  Eigen 
frequencies for the various modes were obtained by diagonalizing the dynamical matrix.  
The phonon dispersion curves obtained at ambient volume from simulations are shown in 
Figure 3.  Eigenvectors were viewed using the Visual Molecular Dynamics (VMD) 
package [21].  The highest compression corresponds to a pressure of 8.3 GPa.  From the 
pressure dependence of the various zone centre optical phonons (inset in Fig. 2) the mode 
Gruneisen parameters were obtained [Table I].  Using Einstein’s specific heat Ci = R 
[xi2exp(xi)]/[exp(xi)-1]2, where xi=ħωi/kBT, for the various modes the total specific heat 
CV was obtained.  Here R is the universal gas constant.  Thermal expansion coefficient 
α=(γavCV)/(3VmB0), (where γav=½∑piCiγi)/CV, pi are the degeneracies of the respective ωi 
phonon branches at the Brillouin zone centre, Vm is the molar volume and B0 taken as 88 
GPa) is calculated to be -22×10-6 K-1, in good agreement with the reported value.   
In view of the non-availability of polarized Raman measurements on oriented 
single crystals of Zn(CN)2, the observed modes were assigned (Table I) based on eigen 
vectors of calculated phonons.  The CN stretching mode at 2200 cm-1 can be assigned to 
A1g.  In the internal mode region A1g and F2g modes arise due to correlation splitting [12] 
and are often degenerate.  It is noteworthy that six out of the eleven optical modes exhibit 
negative Gruneisen parameters.  Furthermore, all the modes of energy lower than 360 cm-
1 have negative γi.  Figure 4 shows the displacement vectors of different atoms for the 
phonons that exhibit large negative γi.  The 143 cm-1 mode corresponds to translational 
motion of CN ions whereas the other three modes involve librations of CN ions about the 
axis joining Zn-Zn’ atoms.  The difference in the values of the calculated and the 
observed values of γi could partly arise from the different values of B0 used.  However, 
this does not affect the calculation of α, since B0 gets cancelled in the definition of α.  
Further, the total Gruneisen parameter (∑piγi) for the C≡N librational modes is -57 
whereas for the translational modes this value is -41. 
As mentioned earlier, using a topological model the network structure of Zn(CN)2 
has been argued to have a large number of low-frequency rigid units modes of 
ZnC4/ZnN4 in analogy with Zr(WO4)2.  On the other hand, in the present lattice 
dynamical calculations the lowest frequency mode turns out to be a CN-translational 
mode.  This is because the topological model treated ZnC4/ZnN4 as rigid units, whereas 
actually only the strongly bound CN ions should be considered as rigid units.  Recent 
atomic pair distribution function analysis shows that the displacements of C/N away from 
the line joining Zn…Zn’ increases as a function of temperature.  Though this appears 
physically reasonable, this displacement, when extrapolated to 0 K, remains as large as 
0.42 Å (Fig.9 of Ref.[14]) suggesting inconsistency between the Rietweld refined 
structure and that obtained from PDF analysis.  Furthermore, the reason for Zn…Zn’ 
distance (which is directly related to the lattice parameters) estimated from PDF analysis 
being different from that obtained from XRD analysis [14] remains unclear.  On the other 
hand, the present phonon calculations and Raman measurements at high pressure provide 
the first insight into the relative role of the different phonons in causing negative thermal 
expansion in Zn(CN)2. 
In conclusion, we have identified the optical phonons responsible for NTE in 
Zn(CN)2 from high pressure Raman spectroscopic studies and from first principles 
density functional simulation studies at different volumes.  Gruneisen parameters of all 
the vibrational modes were obtained from simulations.  A large number of phonon modes 
in Zn(CN)2 are soft, and all contribution to NTE arises from C≡N librational and 
translational modes.  The value of thermal expansion coefficient α calculated from the 
Gruneisen parameters is in good agreement with experimental value.  X-ray diffraction 
investigations suggest growth of  disorder at high pressure.   
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Table I 
Modes 
(deg. of 
freedom) 
metry 
Calc. 
freq. 
(cm-1) 
Obs. 
freq. 
(cm-1) 
Calc. 
Obs. γi 
Zn-trans. 
F2g 388 343 (R) 0.45 - 
F1u 143 178 (IR) -14.3 - 
Eu 255 Inactive -1.5 - 
F2g  352 216 (R) -0.13 -0.50 
(15) 
A2u 564 Inactive 1.1 - 
CN-trans. 
(12) 
F1u 596 461 (IR) 1.4 - 
F1g 288 Inactive -8.0 - 
Eg 357 339 (R) -6.2 -0.54 
CN-libr. 
F2u 326 Inactive -7 - 
F2g 2232 2218(IR) 1.5 0.14(1) CN-int. 
(4) A1g 2245 2221 (R) 1.5 - 
Figure and table captions: 
Table 1. Calculated and observed phonon frequencies in Zn(CN)2, their classification, 
mode assignments and Gruneisen parameters.  Observed IR frequencies are from [12]. 
Figure 1.  Raman spectra of Zn(CN)2 at several pressures.  Spectra are scaled and shifted 
for clarity.  The modes at 342 and 200 cm-1 could not be followed above 1 GPa due to 
weak intensities. 
Figure 2.  Mode frequency vs. Pressure for the observed Raman modes in Zn(CN)2.  
Open symbols: results without pressure medium.  The inset shows ω vs. P for all the 
eleven modes obtained from the phonon calculations.  Though data were generated up to 
8.3 GPa, the trend after the first three pressures (shown here) is non-linear, and hence not 
considered for obtaining γi. 
Figure 3.  Phonon dispersion curves obtained from First Principles density functional 
simulations on a 3×3×3 supercell of Zn(CN)2.  Note that the acoustic phonon branch 
interacts with the lowest energy optical phonon branch at 143 cm-1.  Both the branches 
change character due to the non-crossing rule. 
Figure 4. Atomic displacements of  vibrational modes corresponding to (a) 143 cm-1, (b) 
288 cm-1, (c) 326 cm-1 and (d) 357 cm-1.  The arrows fixed to the atoms are proportional 
to the amplitude of atomic motion.  In the 326 cm-1 mode neighbouring Zn atoms also 
move (opposite direction) 
200 250 300 350 2200 2250
CN-translation
CN stretch
CN libration
x0.6x1.6
0.8 GPa
2.4 GPa
1.2 GPa
0.1 GPa
Raman shift (cm-1)  
Figure 1. Ravindran et al 
Figure 2. Ravindran et al. 
1 0 0
2 0 0
3 0 0
4 0 0
5 0 0
6 0 0
2 2 0 0
2 3 0 0
Figure 3. Ravindran et al. 
Figure 4. Ravindran et al.
