Linear Compressibility and Thermal Expansion of KMn[Ag(CN)2]3

Article pubs.acs.org/JPCC

Linear Compressibility and Thermal Expansion of KMn[Ag(CN)2]3 Studied by Raman Spectroscopy and First-Principles Calculations K. Kamali, C. Ravi, T. R. Ravindran,* R. M. Sarguna, T. N. Sairam, and Gurpreet Kaur Materials Science Group, Indira Gandhi Centre for Atomic Research, Kalpakkam 603 102, India ABSTRACT: We have investigated the linear compressibility and thermal expansion properties of KMn[Ag(CN)2]3 using Raman spectroscopy and DFT calculations. Phonon frequencies and mode assignments from polarized Raman measurements and DFT calculations agree with each other satisfactorily. Computed linear compressibilities and thermal expansion coefficients corroborate the reported measured values. Pressure variation of mean partial phonon frequencies of KMn[Ag(CN)2]3 shows that large amplitude anharmonic displacements of Ag atoms can occur with minimum enthalpy cost even with increasing pressure. This means that Ag layer can be squeezed relatively easily, which manifests as PLC in the basal plane and coupled NLC along trigonal axis due to the rigid Mn-NC-Ag-CN-Mn chain along ⟨101⟩ crystal direction. Elastic constants of KMn[Ag(CN)2]3 indicate that the crystal is highly anisotropic and becomes unstable with increasing pressure. Directional Gruneisen parameters of KMn[Ag(CN)2]3 are found to be highly anisotropic. These properties show that NLC/NTE and PLC/PTE in KMn[Ag(CN)2]3 are driven by elastic and Gruneisen anisotropies combined with anharmonic lattice vibrations of Ag atoms. Partial phonon frequencies of KMn[Ag(CN)2]3 are found to be higher than those of Ag3[Co(CN)6]. The partial frequency of K atom increases rapidly with pressure and becomes comparable to that of Ag around 2 GPa. This shows that K inclusion stiffens the lattice and changes the dynamics, causing the pressure-induced phase transformation of KMn[Ag(CN)2]3 to occur at a higher pressure rather than at 0.2 GPa as in Ag3[Co(CN)6]. The phase transformation can be attributed to the softening of two low-energy optic modes (A2 and E) and the softening of C44 shear elastic constant and hence the transverse acoustic mode.

I. INTRODUCTION Solids in general expand with temperature and shrink with pressure. However, many substances are known to exhibit anisotropic thermal expansion or linear compressibility or both.1−23 Goodwin et al.14,15 have recently studied the thermal expansion and linear compressibility of Ag3[Co(CN)6] using Xray and neutron diffraction and density functional theory calculations. They have shown that this material exhibits colossal positive and negative thermal expansion in different directions and a large negative linear compressibility. The trigonal crystal structure of Ag3[Co(CN)6] consists of alternating layers of Ag+ and [Co(CN)6]3− ions, stacked parallel to the trigonal axis and strongly bonded Co−CN−Ag− NC−Co linkages running parallel to the ⟨101⟩ directions. The interatomic separations along these linkages are therefore rigid. On the other hand, due to relatively weak binding, the interatomic separations along the basal plane and the trigonal axis can be altered easily. To preserve bond lengths along the covalent linkages, it is geometrically necessary that any decrease in interatomic separations along the basal plane occurs with an equivalent increase along the trigonal axis and vice versa. These changes in interatomic separations manifest as changes in the lattice parameters. When they occur due to temperature, they are known as positive and negative thermal expansion (PTE and NTE). When they occur due to pressure, they are known as positive and negative linear compressibility (PLC and NLC). The large PTE/PLC and NTE/NLC of Ag3[Co(CN)6] are in © 2013 American Chemical Society

particular attributed to a shallow enthalpy valley in its lattice parameters space, with a flat minimum along the floor which allows weak Ag+···Ag+ interactions to facilitate large PTE/PLC along the basal plane coupled with an equally strong NTE/ NLC along the trigonal axis. Ag3[Co(CN)6] has a NLC of χ∥ = −75 TPa−1 until a phase transition at 0.19 GPa and −5 TPa−1 thereafter. NLC is of interest in the development of sensitive seismic pressure sensors and in smart materials for armor. Though trigonal Ag3[Co(CN)6] shows a large NLC, this structure is not stable above 0.19 GPa. On the other hand, trigonal KMn[Ag(CN)2]3 shows high NLC up to 2 GPa without any phase transformation,21 which makes it more suitable for applications. The structure of KMn[Ag(CN)2]3 is closely related to Ag3[Co(CN)6]. In particular, both share ultraflexible cyanide lattice whose dimensions are determined by Ag+···Ag+ argentophilic interactions. The main difference is the inclusion of K+ ions within the cavities of the dicyanometallate framework, lowering the crystal symmetry from P-31m to P312. The coefficients of thermal expansion of KMn[Ag(CN)2]3 (α⊥ = +61 × 10−6 K−1, α∥ = −60 × 10−6 K−1) and the linear compressibilities (χ⊥ = +33.2 TPa−1 and χ∥ = −12 TPa−1) in the pressure range 0.2− 2.2 GPa show that the presence of K+ ions does not basically Received: October 15, 2013 Revised: November 11, 2013 Published: November 12, 2013 25704

dx.doi.org/10.1021/jp410214y | J. Phys. Chem. C 2013, 117, 25704−25713

The Journal of Physical Chemistry C

Article

1 ⎡ dl ⎤ χl = − ⎢ ⎥ l ⎣ dP ⎦T

affect the framework flexibility. Rather, the presence of K+ ions was shown to influence the energy of Ag displacement modes. In the present paper, we report our study of the anisotropic linear compressibility and thermal expansion properties of KMn[Ag(CN)2]3 via pressure dependence of lattice parameters, elastic constants, and phonon spectrum. The phonon spectrum and its mode assignments were obtained using Raman and IR spectroscopy. Density functional perturbation theory (DFPT) calculations combined with the quasi-harmonic approximation was used to obtain the complementing phonon spectrum. Pressure variation of elastic constants and average partial phonon frequencies gives insight into the large persistent negative linear compressibility of KMn[Ag(CN)2]3 compared to Ag3[Co(CN)6]. Experimental and computational techniques are described in the next section followed by analysis and discussion of possible mechanism for linear compressibility, thermal expansion, and pressure-induced phase transformation. We summarize our results in the final section.

(1)

Here χl is the compressibility along axis l, P is the pressure, and T is the temperature. We have also computed the linear compressibilities alternatively from elastic constants Cij. For trigonal crystals, the compressibilities in basal plane and along the trigonal axis are given by the relations26 C33 − C13

χ⊥ =

χ =

C11C33 − 2C132 + C12C33

(2)

C11 + C12 − 2C13 C11C33 − 2C132 + C12C33

(3)

Similarly, the isotropic and linear bulk moduli are expressed in terms of elastic constants by the relations Biso =

II. EXPERIMENTAL AND COMPUTATIONAL DETAILS Single crystals of KMn[Ag(CN)2]3 were grown by a layering technique, employing the procedure reported by Geiser et al.24 We obtained crystals with typical dimensions of 0.6 mm × 0.2 mm × 0.03 mm. A part of the sample was ground and characterized by X-ray powder diffraction, and the pattern is found to match with the JCPDS pattern of trigonal crystal structure [PDF No. 04-011-0797] without any impurity peaks. Distinct colorless hexagonal plates were carefully handpicked and observed through a polarizing optical microscope under cross polarized condition. The crystals that appear dark and remain dark through 360° rotation are the ones with their crystallographic c-axis (i.e., the optic axis) perpendicular to the basal plane within the crystal. Such c-oriented single crystals were then used in the polarized Raman measurements to assign the phonon modes. Spectra were measured at T = 80 K in Z(XX)Z′ (VV) and Z(XY)Z′ (VH) scattering geometries (Porto’s notation), where Z is the microscope axis, which is the direction of incident laser, Z′ is the direction of the scattered signal, and X and Y are on the crystal plane. In-situ high-pressure Raman spectra were recorded up to 14 GPa at 0.5 GPa intervals from a symmetric diamond anvil cell with diamonds of 500 μm culet diameter, using a micro Raman spectrometer (Renishaw, UK, model inVia) with 514 nm laser excitation. The sample (∼100 μm lateral dimensions) was loaded along with a few specks of ruby chips for pressure calibration and a 4:1 mixture of methanol−ethanol as pressure transmitting medium into a 200 μm hole drilled into a preintended stainless steel gasket. Pressure is hydrostatic up to 10 GPa in ethanol + methanol medium, after which it is quasihydrostatic up to ∼20 GPa. The spectra were recorded in both increasing and decreasing pressure cycles. High-pressure Raman experiments were also carried out without pressure transmitting medium under nonhydrostatic conditions. Temperature-dependent Raman measurements were carried out from 80 to 533 K at 20 K intervals using a heating and cooling microscope stage (Linkam THMS 600). The infrared (IR) spectrum of the sample, dispersed in CsI, was recorded using a BOMEM DA-8 Fourier transform IR (FTIR) spectrometer. The isothermal linear compressibilities were computed using the principal axis strain calculator PASCal.25 PASCal determines linear compressibility by fitting the lattice parameters at different volumes and the corresponding pressures to the equation23

B Λ Λ , B⊥ = , B = ⊥ 2 2+τ τ (2 + τ )

(4)

where Λ = C11 + 2C12 + C22 + 2C13τ + C33τ 2 + 2C23τ

and τ=

C11 + C12 − 2C13 C33 − C13

The linear thermal expansion coefficients are computed within the theoretical framework given in Hermet et al.26 For trigonal crystals, the different thermal expansion coefficients are given by α⊥ =

α =

⎡1 ⎤ (s11 + s12)γ⊥ + s13γ ⎥ ⎦ 2

1 V0

∑ cv(j , q)⎣⎢

1 V0

∑ cv(j , q)[s13γ⊥ + s33γ ]

j,q

(5)

(6)

j,q

Here the sij are elastic compliances, obtained by inverting the elastic constant matrix Cij, and the mode specific heat cv(j,q) is calculated using Einstein’s model.19,27 The directional Gruneisen parameters (γ⊥ and γ∥) are defined by the weighted sums over the mode Gruneisen parameters: ⎛ ∂ ln[ω(j , q , {ak })] ⎞ γk(j , q) = −⎜ ⎟ ∂ ln ak ⎝ ⎠0

(7)

where ω(j,q) is phonon frequency of mode j at wave vector q and ak are lattice parameters. First-principles calculations of the pressure-dependent lattice parameters, phonon spectrum, and elastic constants were carried out using Vienna Ab initio Simulation Package (VASP).28−30 The equilibrium geometry of KMn[Ag(CN)2]3 was computed with the crystallographic data from Geiser et al.24 A plane-wave basis set with 550 eV cutoff was used to expand the electronic wave function. Projected augmented wave pseudopotentials31,32 were used to represent the interaction of valence electrons with the ion core. Brillouin zone integrations were performed on a 7 × 7 × 7 Γ-centered Monkhorst−Pack33 k-grid, with a Gaussian smearing width of σ = 0.05 eV. Ionic relaxation was stopped when all forces were converged to smaller than 1 × 10−6 eV/Å. With the computed equilibrium geometry, the zone-center phonon spectrum was 25705



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The bands in the range 2100−2170 cm−1 arise from the internal vibrations of cyanide ions, whereas those below 500 cm−1 are due to librations of CN and translations of Mn and Ag ions. B. Results of Raman Spectroscopy as a Function of Temperature and Pressure. Figure 3 shows the Raman spectra of KMn[Ag(CN)2]3 from 80 to 533 K in 20 K intervals. Frequencies of all modes are found to decrease monotonically as temperature is increased up to 500 K, pointing to the absence of phase transition up to this temperature. The spectrum changes significantly at 503 K, indicating a possible decomposition, and visual inspection of the sample corroborates this. The mode frequency versus temperature plot, given in Figure 4, brings out these features more clearly. The spectrum at 533 K exhibits Raman bands characteristic of Mn5O8.40 The XRD pattern of the decomposed product shows cubic metallic silver as dominant phase similar to that reported in isostructural Ag3[Co(CN)6].14 Raman spectra of the sample with pressure transmitting medium in increasing pressure cycles are shown in Figure 5. Around 2.8 GPa the intensities of high-frequency symmetric stretch mode at 2164 cm−1 and some low-energy modes around 260 cm−1 fall rapidly. The bands near 160 cm−1 merge into a broad feature, and several distinct new low-energy bands appear around 53, 74, and 88 cm−1. At 6 GPa, new Raman bands appear around 2117 cm−1, but there is no further change in the low-frequency region. The intensities of low-energy bands gradually decrease and disappear around 13 GPa, indicating possible pressure-induced amorphization. The mode frequency versus pressure plot in Figure 6 clearly shows the phase transition at 2.8 GPa and amorphization around 13.5 GPa. The compound remains amorphous even in the pressure reducing cycle, unlike in the isostructural Ag3[Co(CN)6] that shows Raman bands corresponding to amorphous carbon nitride and crystalline Co3O4 in pressure reducing cycles.18 The spectra with and without medium are found to be similar except for a slight increase in the onset of pressure-induced amorphization, indicating no chemical reaction between KMn[Ag(CN)2]3 and MeOH/EtOH in contrast to Zn(CN)2.41 Pressure dependence of Raman bands below 2.8 GPa was used to obtain their corresponding mode Gruneisen parameters. Table 1 lists the mode frequencies (ωi) and the corresponding Gruneisen parameters (γi) obtained from the equation γi = (B0/ ωi)(∂ωi/∂P). Here P is the pressure and B0 is the bulk modulus obtained from high-pressure XRD lattice parameters provided by Cairns et al.21 We find that all the γi of the Raman modes are positive. Table 2 lists the implicit and explicit contribution19 of anharmonicity of each Raman modes. It is evident that total anharmonic and true anharmonic parts have the same trend, and the quasi-harmonic part is less significant. C. Results of Computational Studies. Equilibrium geometry of KMn[Ag(CN)2]3 obtained with conventional DFT calculation gives unit cell volume in agreement with measurement (within 4%), whereas the c/a ratio deviates by about 13%. A similar discrepancy has been observed in the DFT computed ground state structure of Ag3[Co(CN)6] and has been attributed to nonlocal electron correlation effects due to Ag···Ag dispersion interactions.42 Moreover, phonon dispersion of KMn[Ag(CN)2]3 computed with the above DFT equilibrium geometry is found to exhibit significant discrepancy compared to Raman spectroscopy results. Therefore, we recomputed the ground state structure of KMn[Ag(CN)2]3 using VASP implementation43 of the vDW-

calculated using the VASP implementation of DFPT. Details of DFPT calculations can be found elsewhere.19,34−37 The elastic constants were calculated using the symmetry-general leastsquares method as implemented in VASP38 and were derived from the strain−stress relationships obtained from six finite distortions of the lattice. The calculated elastic moduli include contributions of distortions with rigid ions and ionic relaxations.

III. RESULTS AND DISCUSSION A. Polarized Raman and IR Spectroscopy Results. KMn[Ag(CN)2]3 crystallizes in a trigonal structure with space group P312 (D31) with one formula unit in the crystallographic unit cell. The K, Mn, and Ag atoms occupy the Wyckoff sites e, a, and k, respectively; C and N occupy the l sites.24 Factor group analysis was carried out using the correlation method39 to obtain the total irreducible representation, Γtot = 7A1 + 10A2 + 17E, which comprises of internal modes of cyanide ion Γint = 6A1 + 6A2 + 12E, lattice modes Γext = A1 + 3A2 + 4E, and acoustic modes Γacoustic = A2 + E. Therefore, 23 Raman-active (7A1 + 16E) and 25 IR-active modes (9A2 + 16E) are expected from the sample. Figure 1 shows the polarized Raman spectra measured at 80 K. The allowed symmetry species for c-oriented hexagonal

Figure 1. Polarized Raman spectra of KMn[Ag(CN)2]3 in Z(XX)Z′ (VV) and Z(XY)Z′ (VH) scattering geometries at 80 K. Both A1 and E modes appear in VV geometry. The labeled Raman modes that disappeared in the VH geometry are assigned as A1.

single crystals in Z(XX)Z′ (VV) scattering geometry are A1 and E; in Z(XY)Z′ (VH) scattering geometry, only E species is allowed. The bands at 63, 140, 189, 298, 332, 429, and 2168 cm−1 are found to appear in the VV but not in the VH geometry, and hence they are assigned as A1 symmetry species. The remaining modes that appear in both the VV and VH geometry are labeled as E modes. The frequencies and their mode assignments are given in Table 1. We have obtained all expected 7 A1 modes and only 13 out of 16 E modes. We are thus able to obtain 20 out of the 23 bands in the Raman spectrum. The three bands that could not be observed may be because they are accidentally degenerate with the observed bands, or their Raman scattering cross section could be low. The IR spectrum from 100 to 2200 cm−1, in Figure 2, shows bands at 164, 197, 236, 391, 421, 452, 477, and 2154 cm−1, several of them being broad features. 25706



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Table 1. Comparison of Raman and IR Frequencies of KMn[Ag(CN)2]3 and Their Mode Assignments Obtained Using Polarized Raman Measurements with the Computed Phonon Frequencies and Mode Assignments; Mode Gruneisen Parameters (γi) Are Also Listed experimental Raman modes at 80 K (cm−1)

mode label

60 76 107 126

A1 E E E

140 168 180 189 203

A1 E E A1 E

A2

IR modes at 300 K (cm−1)

164

DFPT with vDW-DF2 and rPW86 mode Gruneisen parameter (γi)

zone center modes (cm−1)

mode Gruneisen parameter (γi)

mode label

mode description

0.31 0.55 0.17 0.67

38 49 51 64 76 95 125

−0.65 −0.05 2.95 2.94 1.65 1.12 1.38

A2 E E A1 A2 E E

144 154

1.28 1.21

A2 E

Ag translation Ag translation Ag translation Ag translation Mn translation K translation K + CN translation CN translation CN translation

195 201 228 233 234 253 267

0.56 1.37 0.53 0.57 0.87 0.44 0.24

A1 E A1 E A2 A2 E

293 298 371 372 380 381 427 430 442 460 468 482 2118 2121 2119 2144

0.06 0.23 0.01 0.01 0.23 0.23 0.13 0.27 0.37 0.38 0.44 0.41 0.05 0.05 0.05 0.05

E A2 A1 E A1 E E A2 E E A1 A2 E E A2 A1

0.29 0.28 0.50

197

236

265

E

0.12

290 298 304 332 357 377

E A1 E A1 E E

0.04 0.23

391

A2

421

0.21 0.19 0.18

452 428

A1

0.12 477

2119 2133 2168

E E A2 A1

0.02 0.02 2154 0.03

CN translation CN translation CN translation CN translation CN translation CN translation Mn−CN bending CN libration CN libration CN libration CN libration CN libration CN libration CN libration CN libration asym str of CN asym str of CN asym str of CN asym str of CN sym str of CN sym str of CN sym str of CN sym str of CN

Figure 3. Raman spectra of KMn[Ag(CN)2]3 at different temperatures. Distinct change in the spectrum due to the decomposition of the compound is clearly seen at 503 K.

Figure 2. IR spectrum of KMn[Ag(CN)2]3 at ambient conditions.

DF2 nonlocal correlation functional44 with rPW86 exchange functionals.45 Table 3 lists the computed lattice parameters 25707



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Table 2. Total Anharmonicities, Quasi-Harmonic (Implicit), and True Anharmonic (Explicit) Contribution of Different Phonons of KMn[Ag(CN)2]3

Figure 4. Temperature variation of phonon frequencies of KMn[Ag(CN)2]3 showing monotonic decrease of all mode frequencies with increase in temperature.

mode freq (cm−1)

total anharmonic (10−5) K−1

quasi-harmonic αγj (10−5) K−1

true anharmonic (10−5) K−1

60 76 107 126 168 180 189 265 290 298 332 357 377 428 2119 2133 2168

−50.0 −9.9 −10.0 −7.0 −21.7 −22.7 −17.9 −6.0 −7.0 −14.7 −24.5 −12.2 −7.4 −7.6 −1.02 −1.1 −1.3

0.6 1.1 0.3 1.3 0.6 0.5 1.0 0.2 0.08 0.4 0.4 0.4 0.3 0.2 0.04 0.04 0.06

−49.4 −8.8 −9.7 −5.7 −14.1 −22.2 −16.9 −5.8 −6.92 −14.3 −9.1 −11.8 −7.1 −7.4 −0.09 −1.06 −1.24

Table 3. Comparison of Computed Lattice Parameters, Unit Cell Volume, and Bulk Modulus of KMn[Ag(CN)2]3 with Experimental Valuesa property

LDA

vDW+DF2+rPW86

expt (300 K)

a (Å) c (Å) c/a V (Å3) B (GPa) B′ (GPa)

7.150 7.3216 1.024 324.28 17.09 3.26

6.887 7.7685 1.128 319.00 12.24 6.84

6.922 8.1465 1.177 338.03 12.86 5.01

a Bulk modulus (B) and its pressure derivative (B′) were obtained by fitting volume-energy data to the universal equation of state.50 Experimental bulk modulus was obtained using high-pressure XRD lattice parameters of Cairns et al.21 in PASCal.

Figure 5. Raman spectra of KMn[Ag(CN)2]3 at different pressures under hydrostatic conditions. Dramatic change at 2.8 GPa and amorphization around 13.5 GPa are evident.

They are not found to improve the predicted lattice constants. Furthermore, we also considered the LDA+U approach in view of the elements Mn and Ag comprising the substance,46 and we find that while it helps obtain lattice constants matching with experimental values, subsequent DFPT calculations produce phonon dispersion which significantly disagrees with Raman spectroscopy. Therefore, we carried out all our calculations using the vDWDF2 and rPW86 functional. To compute the mode Gruneisen parameters of KMn[Ag(CN)2]3, the phonon spectrum was calculated at three different volumes corresponding to pressures 0.0, 1.0, and 2.0 GPa. Phonon dispersion computed by the interpolation of zone-center modes shows that one of the acoustic bands becomes imaginary over some segments of Brillouin zone. This could be due to unconverged long-range interatomic forces. In the VASP-DFPT lattice dynamics calculations of a crystal made of l × m × n periodic cells, the q-point grid to compute the interatomic force constants is also l × m × n.34 This means that VASP-DFPT requires supercells for accurate calculation of interatomic force constants. We considered 2 × 1 × 1, 2 × 2 × 1, and 2 × 2 × 2 supercells. These calculations however did not improve the acoustic band appreciably. Moreover, DFPT calculations with supercells and vDW-DF2 functionals become computationally more demand-

Figure 6. Pressure variation of phonon frequencies of KMn[Ag(CN)2]3 under hydrostatic conditions. All modes, except the CN stretch band, disappear above 13 GPa.

along with experimental values. It is evident that the vDW-DF2 dispersion interaction correction predicts the c/a ratio and unit cell volume in agreement with experiments within 4% and 5%, respectively. We have also considered other options for exchange functional, such as revPBE, optPBE, or optB86b. 25708



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ing. Therefore, we carried out our calculations with the 17-atom 1 × 1 × 1 unit cell of the crystal and proceeded with the zonecenter phonons as they are known to represent the thermal expansion coefficients fairly well.13,47−49 The computed phonon frequencies and their mode assignments are compared with the corresponding Raman and IR spectroscopy results in Table 1. For most of the Raman and IR modes, we find close matching of frequencies and symmetry labels between experiment and calculation. Table 1 also shows the calculated mode Gruneisen parameters. For the low-energy A2 and E optic modes, calculation gives a negative Gruneisen parameter. For modes with frequencies above 150 cm−1, we find good matching of computed and measured values. The discrepancy at low frequencies is likely due to the fact that the computed values are for T = 0 K while the measured values correspond to finite temperatures. Moreover, it is likely that the low-frequency modes are more anharmonic which our current quasi-harmonic approximation could not capture accurately. Figure 7 shows our computed linear compressibility of KMn[Ag(CN)2]3 obtained using eq 1 with PASCal principal

Table 4. DFT Computed Elastic Constants, Bulk Modulus and Mean Linear Compressibilities of KMn[Ag(CN)2]3a elastic constants (GPa) at different pressures P0 (0.0 GPa)

elastic constants C11 C33 C44 C12 C13 C14

P1 (1.0 GPa)

P2 (2.0 GPa)

34.52 37.31 134.77 153.72 30.05 29.24 19.62 22.68 50.93 54.36 −12.32 −11.19 bulk modulus (GPa)

40.06 171.24 28.22 25.97 57.49 −10.17

Biso

B⊥

BPASCal

17.57

25.15

B∥

−44.18 10.48 compressibility (TPa−1)

method

χ⊥

χ∥

elastic constants PASCal expt21

39.7 23.9 (51) 33.2

−22.6 −14.3 (−30) −12.0

a

Linear compressibilities from elastic constants correspond to zero pressure elastic constants. PASCal computed compressibilities at P = 0 are given in parentheses.

Comparison of magnitudes of the linear compressibilities shows that NLC (−14.1 TPa−1) obtained from PASCal matches with the experimental value (−12.0 TPa−1) while the PLC (+24.0 TPa−1) is somewhat lower than the experimental value (+33.2 TPa−1). On the other hand, the bulk modulus obtained from equation of state (12.24 GPa) and PASCal fitting (10.48 GPa) are in good agreement with the experimental value (12.86 GPa). The PV equation of state and the PASCal calculations are done with reference to DFT equilibrium geometry at 0 K. We know, however, that DFT equilibrium lattice parameters are about 4% smaller than experimental values. Our calculation thus indicates that the difference in the equilibrium geometry between DFT and experiment influences the linear compressibilities more than the bulk modulus. The bulk modulus and linear compressibilities obtained from elastic constants have enhanced values (Table 4) compared to the respective experimental values. What is important however is that the elastic constants predict positive and negative linear compressibilities in accordance with experiments. No experimental values of elastic constants are available to compare with our computed values. Linear bulk moduli reported in Table 4 are calculated from eq 4. The signs of the linear bulk moduli and linear compressibilities suggest a PTE of KMn[Ag(CN)2]3 in the basal plane of trigonal unit cell and a NTE along the c-crystal direction in agreement with experimental evidence. The linear thermal expansion coefficients are calculated using eqs 5 and 6. To compute the required directional Gruneisen parameters, we first computed the phonon spectrum of KMn[Ag(CN)2]3 at three different values of the lattice parameter a while the c lattice parameter is held fixed. The three values of a are chosen such that the first value corresponds to the computed equilibrium geometry and the other two values are 1% and 2% smaller than the first. Similar calculations were carried out for three values of c. With these phonon spectra, the directional Gruneisen parameters are calculated (using eq 7) to be γ⊥ = 0.66 and γ∥ = −0.28. The total specific heat (cV) of KMn[Ag(CN)2]3 is computed to be 406.3 J mol−1 K−1. The computed equilibrium molar volume

Figure 7. Linear compressibilities of KMn[Ag(CN)2]3 calculated by fitting lattice parameters and their corresponding pressures using PASCal. Pressures are obtained from the equation-of-state calculation.

axis strain calculator. PASCal requires data of lattice constants as a function of pressure. We first computed total energy of KMn[Ag(CN)2]3 at 19 volumes ranging from 100% to 90% of the computed equilibrium volume, allowing relaxation of unit cell shape and ionic positions at each volume. Volume-energy data collected from these calculations is then used to compute the pressure−volume equation of state.50 The computed lattice constants at different volumes and the respective pressures from the equation of state are then used in PASCal calculation of linear compressibility. We find good agreement between computed and the reported linear compressibility obtained from neutron diffraction.21 Table 4 lists the elastic constants of KMn[Ag(CN)2]3 together with the linear compressibilities obtained from PASCal using eq 1 and elastic constants using eqs 2 and 3. Experimental linear compressibilities from literature are also shown for comparison. The PASCal21 computed linear compressibilities shown in Table 4 are mean values from Figure 7. We find that our linear compressibilities, from both elastic constants and PASCal fitting, show reasonable agreement with experimental values. 25709



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(Vm) is 1.92 × 10−4 m3 mol−1. The appropriate elastic compliances are calculated to be s11 = 0.118 GPa−1, s12 = −0.058 GPa−1, s13 = −0.023 GPa−1, and s33 = 0.025 GPa−1. Using these, the linear thermal expansion coefficients are computed to be α⊥ = 56 × 10−6 K−1 and α∥ = −47 × 10−6 K−1. These thermal expansion coefficients are in good agreement with measured values of α⊥ = 61 × 10−6 K−1 and α∥ = −60 × 10−6 K−1. Bulk thermal expansion coefficient (αV = 27 × 10−6 K−1) obtained from α⊥ and α∥ is found to be in agreement measured value (20 × 10−6 K−1). Moreover, αV (33 × 10−6 K−1) calculated from average bulk Gruneisen parameter (0.59), obtained from mode Gruneisen parameter listed in Table 1, is also in reasonable agreement with measured value. In Ag3[Co(CN)6], NTE behavior is partly mediated by local transverse vibrations.51 Therefore, mode Gruneisen parameters for transverse vibrations of CN are observed to be negative.26 On the other hand, in KMn[Ag(CN)2]3 all Gruneisen parameters for the transverse CN vibrations (CN librations) are positive (Table 1). This could be due to the hindrance of these vibrations by K+ cations.51 Thus, the overall negative contribution to the thermal expansion coefficient is reduced in KMn[Ag(CN)2]3 compared to Ag3[Co(CN)6]. D. Mechanism of Linear Compressibility, Thermal Expansion, and High-Pressure Phase Transformation. We have modeled the linear compressibility by computing the equilibrium lattice parameters at different volumes, allowing the shape of the unit cell and the ionic positions to relax to their equilibrium state. With increasing pressure, we find that the structure of KMn[Ag(CN)2]3 expands along its trigonal axis and shrinks along the basal plane as the overall volume decreases. Figure 8 shows the pressure dependence of

the trigonal axis, giving rise to the negative linear compressibility. Increase in the Ag−K distance with pressure is a consequence of the perpendicular displacements of Ag and N atoms. The pressure dependence of elastic constants of KMn[Ag(CN)2]3 is shown in Figure 9.

Figure 9. Computed pressure-dependent elastic constants of KMn[Ag(CN)2]3. C44 shows negative pressure derivatives, indicating pressure-induced structural instability.

It is evident that all the elastic constants increase with pressure except C44, which decreases with increasing pressure. Negative pressure derivative of the shear elastic constant C44 indicates instability of the lattice via the Born stability criterion.52 Moreover, elastic constant for KMn[Ag(CN)2]3 meets the criterion for negative linear compressibility.53 In the literature, NLC, NTE, and pressure-induced phase transformations are associated with structural instability determined by negative pressure derivative of elastic constant.53−56 Several flexible solids are also known to show pressure-induced softening of elastic constants.57,58 Moreover, elastic constants of KMn[Ag(CN)2]3 do not meet any of the conditions of isotropy for trigonal crystals.27 This shows that NLC of KMn[Ag(CN)2]3 is driven by lattice instability and anisotropy of the crystal. Next we consider the thermal expansion of KMn[Ag(CN)2]3. In view of Munn’s59 analysis of role of elastic constants in NTE, it is evident that our computed PTE and NTE coefficients are determined by significantly anisotropic Gruneisen functions (γ⊥ = 0.663 and γ∥ = −0.288) and negative elastic compliance s13. Moreover, the calculated elastic constants show that KMn[Ag(CN)2]3 is highly anisotropic.27 These things show that NTE and PTE of KMn[Ag(CN)2]3 are due to the combined Gruneisen and elastic anisotropy of the crystal.59 To gain insight into the origin of anharmonic lattice vibrations and hence linear compressibility and thermal expansion of KMn[Ag(CN)2]3, we computed the pressure variation of the mean partial phonon frequencies of the various atoms, defined by the equation14,60

Figure 8. Computed pressure dependence of interatomic separations of KMn[Ag(CN)2]3.

interatomic separations. It is evident that the C−N, Ag−C, and Mn−N bond lengths, which constitute the Mn−NC−Ag− CN−Mn covalent linkages, do not change appreciably. The separation between Mn···Ag···Mn atoms, along the linkages, also remains essentially unchanged over the pressure range of 0−2.0 GPa. The Ag−Ag bonds along the basal plane shrink with increasing pressure. Along with the Ag−Ag bonds, pressure also reduces the Mn(CN)6···Mn(CN)6 distances. Contraction of these lengths manifests as a reduction of a lattice parameter. On the other hand, the same Ag, C, and N atoms also constitute the covalent linkage along the ⟨101⟩ lattice direction. This linkage resists changes to its length by flexing at Ag and N atoms, which manifests as expansion along

⎛ 1 ⎞ ⎟⎟ ∑ ω(j , q)e(j , q , i) ωi = ⎜⎜ ⎝ 3NqNi ⎠ q , j

(8)

where Nq is the number of q-points, Ni the number of atoms of type i in the unit cell, ω(j,q) the angular frequency of phonon 25710



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mode j at wavevector q, and e(j,q,i) the magnitude of the component of the eigenvector corresponding to atom type i. The eigenvalues and eigenvectors are computed on a 9 × 9 × 9 grid of k-points using Phonopy61 with VASP computed zonecenter eigenvalues and eigenvectors. The results are shown in Figure 10. We also depict the partial frequencies of Ag3[Co-

frequencies of KMn[Ag(CN)2]3 are higher than those of Ag3[Co(CN)6], which indicates lattice stiffening in the former. Moreover, partial frequency of K, which is significantly lower than that of Ag at zero pressure, increases rapidly and becomes comparable to that of Ag with increasing pressure. This indicates that inclusion of K stiffens the lattice and changes the dynamics of KMn[Ag(CN)2]3 as inferred in Cairns et al.21 Consequently, NTE/PTE and NLC/PLC of KMn[Ag(CN)2]3 are lower than the respective values of Ag3[Co(CN)6]. Our calculation further shows that the lattice stiffening combined with modified dynamics due to K inclusion causes the phase transition of KMn[Ag(CN)2]3 to occur above 2 GPa rather than at 0.2 GPa as in Ag3[Co(CN)6]. In order to understand the pressure-induced phase transformation, we analyzed the atomic displacements of all phonon modes of KMn[Ag(CN)2]3. Figure 11 shows the atomic displacements of the A2 mode at 38 cm−1 and the E mode at 49 cm−1. These two low-energy optic modes, which show prominent displacement of Ag atoms, are predicted to have negative Gruneisen parameters. Moreover, negative pressure derivative of C44 is associated with softening of transverse acoustic modes.62 Thus, pressure-induced phase transformation is likely to be caused by softening of the transverse acoustic mode and the two low-energy soft optic modes.

Figure 10. Pressure variation of mean partial phonon frequencies of the various atoms in KMn[Ag(CN)2]3 and Ag3[Co(CN)6]. K, Mn, Ag, C, and N partial frequencies from KMn[Ag(CN)2]3 are shown by symbols ○, △, ◊, ◁, and ▷, respectively. Partial frequencies of Co, Ag, C, and N from Ag3[Co(CN)6] are shown by symbols ▽, □, × , and +, respectively. Ag frequencies shows lower slope relative to other atoms. Between Ag3[Co(CN)6] and KMn[Ag(CN)2]3, the slope of Ag frequency in the latter is higher. Average partial frequency of KMn[Ag(CN)2]3 is higher than that of Ag3[Co(CN)6] (C and N frequencies are averaged).

IV. SUMMARY AND CONCLUSION Linear compressibility and thermal expansion of KMn[Ag(CN)2]3 are investigated using pressure- and temperaturedependent Raman spectroscopy and DFT calculations. Phonon frequencies and mode assignments obtained from polarized Raman spectroscopy are found to be in good agreement with computational results. Linear compressibilities obtained from calculated pressure-dependent lattice parameters and from elastic constants corroborate the reported neutron diffraction results. Linear thermal expansion coefficients (α⊥ = 56 × 10−6 K−1 and α∥ = −47 × 10−6 K−1) obtained from calculated directional Gruneisen parameters (γ⊥ = 0.663 and γ∥ = −0.288) are in good agreement with reported measured values (α⊥ = 61 × 10−6 K−1 and α∥ = −60 × 10−6 K−1). Elastic constants of KMn[Ag(CN)2]3 show that the crystal is highly anisotropic and fulfills the criterion for NLC. Negative pressure derivative of C44 further indicates instability of the lattice. These properties show that NLC of KMn[Ag(CN)2]3 is driven by elastic anisotropy and lattice instability. Pressure variation of mean partial phonon frequencies of KMn[Ag(CN)2]3 shows that large-amplitude anharmonic displacements of Ag atoms can occur with minimum enthalpy cost. This means that Ag layer

(CN)6] in Figure 10 for comparison. The partial frequencies were calculated from all modes of the given species. We see that the partial frequencies of all atoms increase with increasing pressure. However, the rate of increase is lowest for Ag, in both KMn[Ag(CN)2]3 and Ag3[Co(CN)6]. This indicates that large amplitude displacements of Ag atoms can occur with minimum enthalpy cost even with increasing pressure. This means that the Ag layers can be squeezed relatively easily. On the other hand, the Ag atoms are also part of the rigid chain of atoms along the ⟨101⟩ crystal directions. That is when Ag layers are squeezed, the interatomic distances along the trigonal axis increase in order to maintain the length of the Mn−NC−Ag−CN−Mn covalent chain unchanged. This manifests as expansion along the c-axis. For the same reason, the Ag layer expands when KMn[Ag(CN)2]3 is heated. This manifests as PTE along the basal plane and a coupled NTE along the trigonal axis. Figure 10 also shows that the partial

Figure 11. Computed atomic displacements of the low-frequency optic modes: (a) the A2 mode at 38 cm−1; (b) the E mode at 49 cm−1. 25711



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(8) Ravindran, T. R.; Arora, A. K. Anharmonicity and Negative Thermal Expansion in Zirconium Tungstate. Phys. Rev. B 2003, 67, 064301. (9) Chapman, K. W.; Chupas, P. J.; Kepert, C. J. Direct Observation of a Transverse Vibrational Mechanism for Negative Thermal Expansion in Zn(CN)2: An Atomic Pair Distribution Function Analysis. J. Am. Chem. Soc. 2005, 127, 15630−15636. (10) Chapman, K. W.; Chupas, P. J.; Kepert, C. J. Compositional Dependence of Negative Thermal Expansion in the Prussian Blue Analogues M(II)Pt(IV)(CN)6 (M = Mn, Fe, Co, Ni, Cu, Zn, Cd). J. Am. Chem. Soc. 2006, 128, 7009−7014. (11) Ravindran, T. R.; Arora, A. K.; Chandra, Sharat; Valsakumar, M. C.; Chandra Shekar, N. V. Soft Modes and Negative Thermal Expansion in Zn(CN)2 from Raman Spectroscopy and First-Principles Calculations. Phys. Rev. B 2007, 76, 054302. (12) Ravindran, T. R.; Arora, A. K.; Sairam, T. N. A Spectroscopic Resolution of the Structure of Zn(CN)2. J. Raman Spectrosc. 2007, 38, 283−287. (13) Ding, P.; Liang, E. J.; Jia, Y.; Du, Z. Y. Electronic Structure, Bonding and Phonon Modes in the Negative Thermal Expansion Materials of Cd(CN)2 and Zn(CN)2. J. Phys.: Condens. Matter 2008, 20, 275224. (14) Goodwin, A. L.; Calleja, M.; Conterio, M. J.; Dove, M. T.; Evans, J. S. O.; Keen, D. A.; Peters, L.; Tucker, M. G. Colossal Positive and Negative Thermal Expansion in the Framework Material Ag3[Co(CN)6]. Science 2008, 319, 794−797. (15) Goodwin, A. L.; Keen, D. A.; Tucker, M. G. Large Negative Linear Compressibility of Ag3[Co(CN)6]. Proc. Natl. Acad. Sci. U. S. A. 2008, 105, 18708−18713. (16) Goodwin, A. L. The Crystallography of Flexibility: Local Structure and Dynamics in Framework Materials. Z. Kristallogr. Suppl. 2009, 30, 1−11. (17) Keen, D. A.; Dove, M. T.; Evans, J. S. O.; Goodwin, A. L.; Peters, L.; Tucker, M. G. The Hydrogen-bonding Transition and Isotope-Dependent Negative Thermal Expansion in H3Co(CN)6. J. Phys.: Condens. Matter 2010, 22, 404202. (18) Rao, R.; Achary, S. N.; Tyagi, A. K.; Sakuntala, T. Raman Spectroscopic Study of High-pressure Behavior of Ag3[Co(CN)6]. Phys. Rev. B 2011, 84, 054107. (19) Kamali, K.; Ravindran, T. R.; Ravi, C.; Sorb, Y.; Subramanian, N.; Arora, A. K. Anharmonic Phonons of NaZr2(PO4)3 Studied by Raman Spectroscopy, First-Principles Calculations, and X-ray Diffraction. Phys. Rev. B 2012, 86, 144301. (20) Mittal, R.; Zbiri, M.; Schober, H.; Achary, S. N.; Tyagi, A. K.; Chaplot, S. L. Phonons and Colossal Thermal Expansion Behavior of Ag3Co(CN)6 and Ag3Fe(CN)6. J. Phys.: Condens. Matter 2012, 24, 505404. (21) Cairns, A. B.; Thompson, A. L.; Tucker, M. G.; Haines, J.; Goodwin, A. L. Rational Design of Materials with Extreme Negative Compressibility: Selective Soft-Mode Frustration in KMn[Ag(CN)2]3. J. Am. Chem. Soc. 2012, 134, 4454−4456. (22) Li, W.; Probert, M. R.; Kosa, M.; Bennett, T. D.; Thirumurugan, A.; Burwood, R. P.; Parinello, M.; Howard, J. A. K.; Cheetham, A. K. Negative Linear Compressibility of a Metal−Organic Framework. J. Am. Chem. Soc. 2012, 134, 11940−11943. (23) Cairns, A. B.; Catafesta, J.; Levelut, C.; Rouquette, J.; Lee, A.; Peters, L.; Thompson, A. L.; Dmitriev, V.; Haines, J.; Goodwin, A. L. Giant Negative Linear Compressibility in Zinc Dicyanoaurate. Nat. Mater. 2013, 12, 212−216. (24) Geisers, U.; Schlueter, J. A. KMnAg3(CN)6, A New Triply Interpenetrating Network Solid. Acta Crystallogr., C 2003, 59, i21−i23. (25) Cliffe, M. J.; Goodwin, A. L. PASCal: A Principal Axis Strain Calculator for Thermal Expansion and Compressibility Determination. J. Appl. Crystallogr. 2012, 45, 1321−1329. (26) Hermet, P.; Catafesta, J.; Bantignies, J. −L.; Levelut, C.; Maurin, D.; Cairns, A. B.; Goodwin, A. L.; Haines, J. Vibrational and Thermal Properties of Ag3[Co(CN)6] from First-Principles Calculations and Infrared Spectroscopy. J. Phys. Chem. C 2013, 117, 12848−12857.

can be squeezed relatively easily. This manifests as PLC along the basal plane and coupled NLC along trigonal axis due to the rigid Mn−NC−Ag−CN−Mn chain along the ⟨101⟩ crystal direction. For the same reason, it is reasonable to consider that Ag layer can expand easily when KMn[Ag(CN)2]3 is heated. This manifests as PTE along the basal plane and a coupled NTE along the trigonal axis. Moreover, the directional Gruneisen parameters are found to be highly anisotropic. These properties show that NTE/NLC and PTE/PLC in KMn[Ag(CN)2]3 are driven by elastic and Gruneisen anisotropies combined with anharmonic lattice vibrations of Ag atoms. Comparison of mean partial phonon frequencies of KMn[Ag(CN)2]3 and Ag3[Co(CN)6] shows that Mn and Ag atoms from the former have higher frequencies than Co and Ag from the latter. Average CN partial frequencies are almost the same for both. K partial frequency is lower than that of Ag at zero pressure and increases rapidly with pressure, becoming comparable to that of Ag around 2 GPa. This shows that K inclusion stiffens the crystal and changes the dynamics, causing the pressure-induced phase tranformation of KMn[Ag(CN)2]3 to occur at a higher pressure than found in Ag3[Co(CN)6]. The phase transformation can be associated with the softening of two low-energy optic modes (A2 and E) and the softening of the C44 elastic constant.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (T.R.R.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Atsushi Togo for help with calculation of irreducible representation using Phonopy, A. K. Arora, Arindham Das, and K. Sivasubramanian for fruitful discussions, B. V. R. Tata, B. K. Panigrahi, and P. Ch. Sahu for interest in the work, C. S. Sundar for encouragement, and P. R. Vasudeva Rao for support.

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ABBREVIATIONS NLC, negative linear compressibility; PLC, positive linear compressibility; NTE, negative thermal expansion; PTE, positive thermal expansion.

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REFERENCES

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Linear Compressibility and Thermal Expansion of KMn[Ag(CN)2]3

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