Uniaxial Negative Thermal Expansion, Negative Linear

Nov 30, 2017 - Compressibility, and Negative Poisson's Ratio Induced by Specific. Topology ... The Poisson's ratio, which is defined as the ratio of t...
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Article Cite This: Inorg. Chem. 2017, 56, 15101−15109

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Uniaxial Negative Thermal Expansion, Negative Linear Compressibility, and Negative Poisson’s Ratio Induced by Specific Topology in Zn[Au(CN)2]2 Lei Wang,† Hubin Luo,‡ Shenghua Deng,† Ying Sun,† and Cong Wang*,† †

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Center for Condensed Matter and Materials Physics, School of Physics, Beihang University, Beijing 100083, People’s Republic of China ‡ Key Laboratory of Magnetic Materials and Devices, Ningbo Institute of Materials Technology and Engineering, Chinese Academy of Sciences, Ningbo 315201, People’s Republic of China ABSTRACT: The well-known idea of “structure determines properties” can be understood profoundly in the case of hexagonal zinc dicyanometalate. Using density functional theory (DFT) calculations, we show the uniaxial negative thermal expansion (NTE) and negative linear compressibility (NLC) properties of Zn[Au(CN)2]2. The temperature dependence of phonon frequencies within the quasiharmonic approximation (QHA) is investigated. The abnormal phonon hardening (frequency increase on heating) is detected in the ranges of 0−225, 320−345, and 410−430 cm−1, which can be indicative of the unusual physical properties of Zn[Au(CN)2]2. Due to the significance of low-energy phonon frequencies in Zn[Au(CN)2]2, in this work, the corresponding vibrational mode of the lowest-frequency optical phonon at the zone center is analyzed. The specific topology of a springlike framework that will produce the effects of a compressed spring on heating and an extended spring under hydrostatic pressure is identified and leads to the coexistence of uniaxial-NTE and NLC behaviors in Zn[Au(CN)2]2. The distinguishing phonon group velocity along the a axis and c axis facilitates different responses for both the axes under temperature and hydrostatic pressure field. Through an analysis and visualization of the spatial dependence of elastic tensors, it is found that a negative Poisson’s ratio (NPR) is presented in all projection planes due to the specific topology.

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proposed.5 Owing to stretch densification along the NLC orientation, NLC materials may lay a foundation for the development of effectively incompressible materials for various practical applications, including amplification of piezoelectric response in sensors and actuators, “smart” body armor made of robust shock-absorbing materials, sensitive pressure detectors, etc.6 The NPR enables materials to function as unusual press-fit fasteners, to conform by bending to convex surfaces, and to enhance the characteristics of piezoelectric transducers.4 If these abnormal properties can coexist in materials, it will enable the materials themselves to have a extensive application facing temperature and pressure field. The investigation of materials with multi-negative properties is very significant in the development of advanced functional materials. The most significant challenges in this field have been the apparent rarity of materials showing multi-negative properties, the extreme weakness of the abnormal effects found in these materials, and scarce understanding of mechanisms governing multi-negative properties in particular materials.7,8 Presently, related studies have demonstrated that NLC can be detected in framework structures exhibiting very anisotropic NTE behaviors. For anomalous materials with

INTRODUCTION Crystalline materials in general expand on heating and shrink under hydrostatic pressure with typically the observed elongation and reduction occurring in all crystallographic cell parameters. However, some materials can exhibit an abnormal response to temperature and hydrostatic pressure field where materials shrink on heating and expand under hydrostatic pressure in a specific direction while maintaining positive thermal expansion (PTE) and positive volume compression. These behaviors of unaxial shrinkage on heating and uniaxial expansion under hydrostatic pressure are perceived as uniaxial negative thermal expansion (NTE) and negative linear compressibility (NLC), respectively.1−3 Another interesting and counterintuitive physical property is a negative Poisson’s ratio (NPR),4 which is permitted by the theory of elasticity. The Poisson’s ratio, which is defined as the ratio of transverse contraction strain to longitudinal extension strain during stretching, is usually positive, as one can see in the stretching of a rubber band. An NPR means that the material undergoes lateral expansion when stretched longitudinally. These abnormal physical properties have attracted widespread interest from scientists. The potential uses of NTE materials in controlled thermal expansion composites have been readily recognized, and possible applications ranging from fiber optics, electronics, and mirror substrates to tooth fillings are © 2017 American Chemical Society

Received: September 25, 2017 Published: November 30, 2017 15101

DOI: 10.1021/acs.inorgchem.7b02416 Inorg. Chem. 2017, 56, 15101−15109

Article

Inorganic Chemistry

forces generated by nonequivalent atomic displacement in a supercell for a crystal structure. In the present work, the dimension of the supercell is 2 × 2 × 1 unit cells of Zn[Au(CN)2]2. Fourier interpolation is used to obtain the dynamic matrices on a k-point sampling that is limited to 9 × 9 × 9. A dynamic matrix is constructed from HF forces acting on all atoms in the supercell with a displaced atom, and phonon frequencies are calculated by solving the eigenvalue problem for the dynamic matrix. Effects of Finite Temperature and Pressure. The thermal properties at constant pressure may be obtained from the Gibbs free energy defined as

simultaneous uniaxial-NTE and NLC behaviors, some characteristics are indispensible, including a flexible and low-density framework structure, extreme mechanical anisotropy, unusual phonon response in temperature or pressure field, and specific geometric topologies such as wine-rack, honeycomb, helices, and so on.2 Uniaxial-NTE and NLC behaviors have been identified in a number of metallocyanide framework materials. Goodwin et al. had reported that the trigonal Ag3[Co(CN)6] showed colossal uniaxial-NTE and NLC phenomena along the c axis.9,10 The coefficient of thermal expansion (CTE) (ac = −130 × 10−6 K−1) of Ag3[Co(CN)6] was much higher than those of other typical framework NTE materials.11,12 With a phase transition from a trigonal to a monoclinic structure (at 0.19 GPa), the NLC behavior of Ag3[Co(CN)6] was qualitatively unaffected, but the value of compressibility varied from Kc = −75 TPa−1 (0 GPa ≤ P ≤ 0.19 GPa) to Kc = −5 TPa−1 (0.19 GPa ≤ P ≤ 7.65 GPa).9 In comparison to other NLC materials such as elemental Se (Kc = −2.5 TPa−1),13 TeO2 (Kc = −5.1 TPa−1),14 CH3OH·H2O (Kc = −2.7 TPa−1),15 BiB3O6 (Kc = −6.7 TPa−1),16 and so on, the NLC of trigonal Ag3[Co(CN)6] was extremely large. To explore the underlying mechanism contributing to uniaxial-NTE and NLC behaviors of Ag3[Co(CN)6], density functional theory (DFT) had been employed to study the Grüneisen parameters, the lattice dynamics, and the Ag−Ag dispersive interactions.17−20 Moreover, the NPR behavior of Ag3[Co(CN)6] was detected by the analysis of elastic tensors.21 Recently, huge uniaxial-NTE (ac = −57.58 × 10−6 K−1) and NLC (Kc = −42 TPa−1) behaviors were observed along the c axis in zinc dicyanoaurate (Zn[Au(CN)2]2) using a combination of X-ray single-crystal and powder diffraction.22,23 At the moment, very critical and indispensable research on the correlation between uniaxial-NTE and NLC behaviors in Zn[Au(CN)2]2 is still absent. In addition, some questions need to be resolved: for example, can an NPR exist in Zn[Au(CN)2]2, and what is the origin of these abnormal physical behaviors? On this basis, within the framework of DFT combined with the quasi-harmonic approximation (QHA), we perform a study of the correlation between uniaxial-NTE and NLC behaviors in Zn[Au(CN)2]2 via the temperature and pressure dependence of lattice parameters, abnormal phonon response on heating, phonon group velocity along different axes, and analysis of spatial dependence of elastic tensors. Also of relevance to the discussion is the commonality and differences of abnormal physical properties in a series of metallocyanide framework materials.

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G(T , P) = U (V ) + Fphonon(V , T ) + PV

(1)

A part of the temperature effect can be included into total energy of electronic structure through phonon (Helmholtz) free energy at constant volume. Here the “quasi-harmonic approximation (QHA)” is adopted for an approximation that introduces the volume dependence of phonon frequencies as a part of the anharmonic effect. The Helmholtz free energy F({ai}, T) with respect to all its geometrical degrees of freedom {ai} at any temperature T can be expressed as

F({ai}, T ) = E({ai}) + Fvib(ωq , j({ai}), T )

(2)

with

Fvib(ωq , j({ai}), T ) =

∑

ℏωq , j({ai}) 2

q,j

⎡ ⎛ ℏωq , j({ai}) ⎞⎤ ⎟⎟⎥ + kBT ∑ ln⎢1 − exp⎜⎜− ⎢⎣ kBT ⎝ ⎠⎥⎦ q,j (3) where E({ai}) is the ground state energy and the sums run over all the Brillouin zone wave vectors and the band index j of the phonon dispersions. The term ∑q,j(hωq,j({ai})/2) denotes the zero-point motion. Since volume dependences of energies in electronic and phonon structures are different, the volume giving the minimum value of the energy function shifts from the value calculated only from the electronic structure even at 0 K. When the temperature is increased, the volume dependence of the phonon free energy changes, and then the equilibrium volume at temperatures changes. Minimizing the Helmholtz free energy F({ai}, T) with respect to the degrees of freedom {ai}, while holding the temperature fixed, we can obtain the equilibrium structure at different pressures. Phonon Group Velocity. The phonon group velocity can be derived analytically from the Gibbs free energy.27 Owing to phonon anharmonic interactions, an atomic displacement operator is expressed as

⎛ ℏ ⎞1/2 uα(lκ ) = ⎜ ⎟ ∑ wq−j 1/2[aq̂ j + a−̂ †qj] × eiq·r(lκ)Wα(κ , qj) ⎝ 2Nmκ ⎠ qj (4)

COMPUTATIONAL METHODOLOGY

where uα(lκ) is the atomic displacement of the κth atom in the lth unit cell. N is the number of unit cells in the crystal, and mκ is the atomic mass of type κ. ℏ is the reduced Planck constant. α, β, and γ are Cartesian indices. r(lκ) is the equilibrium atomic position. â̂−qj and âqj are the phonon creation and annihilation operators of normal mode of band index j and wave vector q. The frequency ωqj and polarization vector W(κ, qj) are obtained from the eigenvalue problem of a dynamic matrix D(q)

First-Principles Calculations. The equilibrium lattice structure of Zn[Au(CN)2]2 has been determined by ground state calculations. The total energy calculations are performed using the VASP code, which is based on DFT. The exchange and correlation functionals are given by the generalized gradient approximation (GGA) as proposed by Perdew and Wang.24,25 Electron−ion interactions are represented by the projector augmented wave (PAW) method with plane waves up to an energy of 500 eV. Lattice constants and internal positions in primitive cells at various volumes are fully optimized. The k-point meshes of Brillouin zone sampling in the primitive cell, on the basis of the Monkhorst−Pack scheme, are 5 × 5 × 1 in order to obtain an absolute energy convergence below 1 meV/atom. Lattice Dynamics Calculations. Dynamic properties were obtained from the direct method under the QHA.26 In this method, phonon frequencies are calculated from Hellmann−Feynman (HF)

∑ Dαβ(κκ′, q)Wβ(κ′, qj) = wq2jWα(κ , qj) (5)

κ′β

with

Dαβ (κκ′, q) = 15102

1 mκ mκ ′

∑ Φαβ (lκ , l′κ′)eiq·[r(l′ κ ′) − r(lκ)] l′

(6)

DOI: 10.1021/acs.inorgchem.7b02416 Inorg. Chem. 2017, 56, 15101−15109

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Inorganic Chemistry Φαβ(lκ, l′κ′) denotes the second-order force constant. l, l′, ... are the indices of unit cells, and κ, κ′, ... are the indices of atoms in a unit cell. The phonon group velocity vα(qj) can be obtained directly from the eigenvalue equation vα(qj) ≡

∂wqj ∂q α

=

1 2wqj

∑

Wβ(κ , qj)

∂Dβγ (κκ′, q) ∂q α

κκ ′ βγ

The minimal values f(θ,φ) = minχ X(θ,φ,χ) and maximal values g(θ,φ) = maxχ X(θ,φ,χ) over all possible values of χ can be obtained by plotting each spherical coordinates (θ,φ).

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RESULTS AND DISCUSSION Positive Volume Thermal Expansion and Volume Compressibility. The hexagonal Zn[Au(CN)2]2 comprises six formula units in the crystallographic unit cell corresponding to 66 atoms. The Zn and Au atoms occupy the 6f and 12k Wyckoff positions, respectively; C and N atoms occupy the 12k positions. By fitting of the total energies as a function of volume to the Murnaghan equation of state, the resulting equilibrium lattice parameters of Zn[Au(CN)2]2 are a = b = 8.58 Å and c = 20.93 Å, which are in good agreement with experimental data31 (a = b = 8.45 Å and c = 20.62 Å) within the accuracy of GGA calculations. The volume thermal expansion aV, defined as aV = (1/V) × (dV/dT)P, is expected to be positive due to the normal increase of entropy (disorder) with volume. It is only in this limited sense that NTE is anomalous. Volume compressibility KV is quantified by isothermal compressibility, measured in TPa−1 and defined as the relative rate of change in volume with respect to pressure (KV = −(1/V) × (dV/dP)T). The calculated aV and KV values of Zn[Au(CN)2]2 are shown in Figure 1. The

Wγ(κ′, qj)

(7) The average phonon group velocity vα̅ as a function of temperature can be expressed as

vα̅ =

∑qj vqjnq̅ j ∑qj nq̅ j

(8)

where nq̅ j is the Bose−Einstein distribution. Elasticity Calculations. The elastic constants are calculated for Zn[Au(CN)2]2 in an equilibrium structure by computing the second derivative of total energy as a function of strain.28−30 Zn[Au(CN)2]2 crystallizes in a hexagonal structure with space group P6222,31 hence yielding five independent elastic constants. On the basis of linear elasticity theory, the distortion energy can be defined as a function of strain as

ΔE = E − E0 =

V0 2

6

∑ Cijεiεj + O(ε3) (9)

i,j=1

where E and E0 are the total energies of the distorted and equilibrium structures, respectively. V0 is the equilibrium volume. The unit cell of Zn[Au(CN)2]2 is deformed by five different strain modes; nonzero strains are shown in Table 1. We then fit the distortion energies

Table 1. Deformation Matrices Used To Calculate the Five Independent Elastic Constants in a Hexagonal Crystal strain I

param

1 2

εi = 0 if not listed ε1 = ε2 = δ ε6= δ

3

ε3 = δ

4 5

ε4 = ε5 = δ ε1 = ε2 = ε3 = δ

relationship between k2 and elastic constants k2 = (C11 + C12)V0 1

k 2 = 4 (C11 − C12)V0 1

k 2 = 2 C33V0 k2 = C44V0

(

1

)

k 2 = C11 + C12 + 2C13 + 2 C33 V0

calculated at δ = ±0.005 and ±0.01 to the function ΔE(δ) = k2δ2 + k3δ3 and obtained the elastic constant Cij from the quadratic coefficient k2. The analysis and visualization of elastic tensors in Zn[Au(CN)2]2 were performed by ELATE.21,32 Young’s modulus E and linear compressibility β can alternatively be parametrized by two angles in spherical coordinates:

⎛ sin θ cos φ ⎞ ⎜ ⎟ a = ⎜ sin θ sin φ ⎟ ⎜ ⎟ ⎝ cos θ ⎠

Figure 1. (a) Calculated temperature dependence of aV in Zn[Au(CN)2]2. For comparison, experimental data22 are shown in the inset. (b) Calculated pressure dependence of compressibility KV. The related experimental results from ref 23 are shown in the inset.

experimental results by Goodwin et al. are presented in insets for comparison, including the temperature dependence of the unit cell volume and the pressure dependence of volume change ratio determined by both single-crystal X-ray diffraction and powder X-ray diffraction.22,23 From Figure 1, the aV value increases to approximately 18 ppm K−1 over the temperature range 0−1000 K, and the KV value increases with hydrostatic pressure and reaches about 37 TPa−1 at 8 GPa, suggesting that Zn[Au(CN)2]2 follows the normal volume thermal expansion and volume compression natures. These are consistent qualitatively with experimental measurements where the volume of Zn[Au(CN)2]2 expanded from 1277 to 1292 Å3 in the temperature range 100−773 K and the volume change ratio decreased with hydrostatic pressure.22,23 In axial-NTE and NLC materials, the values of aV and

(10)

and thus both moduli can be represented as a 3D parametric surface as well as 2D projections on planes. The shear modulus G and Poisson’s ratio v depend on two orthogonal unit vectors a and b, which are the direction of the stress applied and the direction of measurement respectively. The vector b is written as

⎛ cos θ cos φ cos χ − sin ϕ sin χ ⎞ ⎜ ⎟ b = ⎜ cos θ sin φ cos χ + cos ϕ sin χ ⎟ ⎜ ⎟ − sin θ cos χ ⎝ ⎠

(11) 15103

DOI: 10.1021/acs.inorgchem.7b02416 Inorg. Chem. 2017, 56, 15101−15109

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Inorganic Chemistry KV may be relatively low under the net effect of combined axialPTE and axial-NTE and of combined positive linear compressibility (PLC) and NLC. Additionally, in comparison to the experimental measurements, the calculated results can be relatively small on the basis of the QHA, which neglects the quartic anharmoncity. Note that hexagonal-symmetry Zn[Au(CN)2]2 is intrinsically chiral, belonging to either one of the enantiomorphic groups P6222 (I-phase) or P6422 (II-phase). At a hydrostatic pressure of about 1.8 GPa, a displacive I/II transition was evident in the emergence of a new set of superlattice Bragg reflections in both single-crystal and powder X-ray diffraction patterns associated with mode softening at the Brillouin zone boundary (L point).23 The occurrence of the high-pressure II-phase with enantiomorphic space group symmetry P6422 (a doubling of the unit cell volume) in Zn[Au(CN)2]2 is hard to observe directly by a classical DFT method. The possible phase instability will be discussed later by an unusual phonon response in the temperature field. Uniaxial-NTE and NLC Behaviors. Our results shown in Figure 2 are the temperature-dependent relative changes in

Figure 3. Calculated pressure-dependent relative change (a − a0, c − c0) of unit cell parameters in Zn[Au(CN)2]2. The experimental values23 are added for comparison. The experimentally observed I/II phase transition occurs at about 1.8 GPa.

symbols, respectively. It can be observed that the c axis in Zn[Au(CN)2]2 exhibits abnormal elongation with increasing hydrostatic pressure, while the a axis has a normal compression nature. There is a consistency between calculated and experimental values. The NLC characteristic of the c axis in Zn[Au(CN)2]2 persists throughout the phase transition under hydrostatic compression in experimental measurements. The situation is similar to that in another metallocyanide framework, Ag3[Co(CN)6]. Fang et al. reported that it was only by including the dispersive interaction in the calculations that the interdigitated structure of the high-pressure phase could be reproduced in Ag3[Co(CN)6].19 Recently, Gupta et al. performed a enthalpy calculation for both phases in Zn[Au(CN)2]2 and obtained a phase transition at about 1.5 GPa.33 In the future we will give specialized independent research on the effects of classical DFT and modern implementations of DFT, including a correction for the long-range dispersive interactions on the high-pressure phase transition in Zn[Au(CN)2]2. In this work, exploring the underlying mechanism of anomalous physical properties in Zn[Au(CN)2]2 at ambient pressure with space group P6222 is our main focus. In Figure 3, the changes in structural characteristics in conventional material and NLC material on hydrostatic compression are sketched. The variations of lattice parameters in Zn[Au(CN)2]2 under the combination of temperature and hydrostatic pressure field are displayed in Figure 4, where the a axis and c axis are calculated for five different pressures (P = 0, 2, 4, 6, 8 GPa) as a function of temperature. The opposite trend of a axis and c axis is obvious in the combination effect of temperature and hydrostatic pressure field. The uniaxial-NTE at constant compression and the NLC at constant temperature of the c axis are clearly reflected. Phonon Spectra and Abnormal Phonon Response. The unit cell of Zn[Au(CN)2]2 with 66 atoms yields 198 vibrational modes in phonon spectra, as shown in Figure 5a. It is noted that no imaginary frequency exists in the phonon spectra, indicating that this crystal structure is dynamically stable. A factor group analysis gives the following set of irreducible representations that characterize the entire phonon modes at the center of the Brillouin zone: Γtot = 16A1 + 16A2 + 17B2 + 17B1 + 64E2 + 68E1 (Γacoustic = A2 + E2). Among these modes, A1, E1, and E2 are Raman-active modes, A2 and E1 are

Figure 2. Calculated relative change (a − a0, c − c0) of the a axis and c axis in Zn[Au(CN) 2 ] 2 as a function of temperature. The experimentally observed relative changes in lattice parameters as a function of temperature22 are also presented.

lattice parameters denoted as solid symbols. The experimental results from 100 to 773 K22 are displayed as empty symbols for comparison. It is clear that the relative change of the a axis is positive with increasing temperature, namely the axial-PTE; in contrast, the c axis has a negative relative change, behaving as an axial-NTE. These calculated results agree qualitatively well with the experimental measurements. We know that the thermal expansion of solids depends on the anharmonicity of lattice vibrations. The Hamiltonian operator can be decomposed into harmonic and anharmonic parts: namely, H = H0 + H′, where H0 = −(ℏ2/2m)(∂2/∂q2) + (k/2)q2, and H′ = ξq3+ςq4 + ... Our calculations are based on the QHA method, which contains the harmonic part H0 and the ξq3 term in the anharmonic part H′, while the ςq4 + ... terms are neglected. This may lead to the quantitative discrepancy between our calculations and the experimental results. In Figure 2, a diagram of the difference between uniaxial-NTE material and the conventional material on heating is shown. The pressure dependence of relative change in lattice parameters is presented in Figure 3, where calculated results and experimental data22 are represented by solid and empty 15104

DOI: 10.1021/acs.inorgchem.7b02416 Inorg. Chem. 2017, 56, 15101−15109

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Inorganic Chemistry

values corresponding to phonon modes of five different frequency ranges are shown separately with the colors corresponding to the frequency ranges in Figure 5a. It is found that there is a coexistence of phonon softening and phonon hardening in Zn[Au(CN)2]2 under temperature field. The low-frequency (0−225 cm−1) phonons have normal phonon softening (up to 0.0025 cm−1 K) and abnormal phonon hardening (close to −0.0042 cm−1 K). Phonon anomalies also occur in the frequency ranges of 320−345 and 410−430 cm−1, and the amount of phonon hardening is reduced with increased frequency. In contrast, the highfrequency phonons in the ranges of 500−550 and 2205− 2240 cm−1 have normal phonon response under temperature field. The calculated temperature dependence of phonon frequency can give a prediction for the experimentally observed diverse phonon response under temperature field by use of infrared spectroscopy or Raman spectroscopy. These phonon anomalies in low-frequency ranges reveal the potential possibilities of phase instability in Zn[Au(CN)2]2. Actually, the overall thermal characteristic of the materials thermal expansion or contraction at specific temperature originates from the competition of excited phonons. The PTE can be present when phonons with positive contributions prevail over those with negative contributions to thermal expansion. In the case of Zn[Au(CN)2]2, the overall volume expansion is attributed to the suppression of phonons with negative contributions to thermal expansion by phonons with positive contributions. Specific Framework Topology. Framework materials with specific topologies such as wine rack, honeycomb, helices, molecular gears, torsion springs, and so on9,10,34−36 possess a high degree of mechanical anisotropy, in which anomalous physical properties including axial-NTE and NLC are favored by framework hinging. For Zn[Au(CN)2]2, the framework structure consists of six Zn2+ centers in tetrahedral nodes connected by six interpenetrating almost linear dicyanoaurate (−N−C−Au−C−N−) linkages. The 2D configurations of Zn[Au(CN)2]2 viewed along the [110] and [001] directions are shown in Figure 6a,b, respectively, where Au, C, N, and Zn atoms are represented by balls of different colors. The covalent

Figure 4. Calculated a axis (a) and c axis (b) in Zn[Au(CN)2]2 with variations in temperature at five different pressures.

Figure 5. (a) Calculated phonon dispersion of Zn[Au(CN)2]2 along a line of reciprocal space (at T = 0 K and P = 0 GPa). These dispersion curves represented by different colors are distributed in five frequency regions. (b) Calculated temperature dependence of phonon frequency curves. The parameter S is defined as S = −dω/dT. The S values corresponding to phonon modes of five different frequency ranges are shown separately with the colors corresponding to the frequency ranges in (a).

infrared-active modes, E1 is both the Raman-active and infraredactive modes, and B1 and B2 are silent. E1 and E2 are 2-fold degenerate. All phonon modes are distributed over five frequency ranges (0−225, 320−345, 410−430, 500−550, and 2205−2240 cm−1), which are distinguished by different colors in Figure 5a. To explore the phonon response in temperature field, the parameter S, defined as S = −dω/dT (ω is the phonon frequency and T is the temperature), is used to describe the change of phonon frequencies with temperature within the framework of QHA. As insulator materials, the physical properties of metallocyanide framework materials are mainly affected by phonons. The frequencies ω of vibrational modes decrease (or “soften”) with increasing temperature, prompting a positive S and volumetric expansion, whereas the vibrational modes whose frequencies ω increase (or “harden”) as the temperature increases will have negative S and tend to give negative contributions to the overall thermal expansion. The emergence of phonon hardening is indicative of abnormal physical properties of materials.20 Figure 5b displays the curves of S along the high-symmetry point in the Brillouin zone. The S

Figure 6. (a) Vibrational mode of lowest-frequency optical phonon at the zone center from its eigenvector in a view down the [110] direction. Zn[Au(CN)2]2 is composed of interpenetrating Zn−N−C− Au−C−N−Zn linkages. The springlike topology can be derived from the structural features. Au, C, N, and Zn atoms are represented by balls of different colors. (b) 2D structural model of β-quartz-like nets (2 × 2 × 1 supercell) in Zn[Au(CN)2]2 in a view from the [001] direction. (c) Schematic representations of the deformations of the springlike framework. The effects of a compressed spring on heating and an extended spring under hydrostatic pressure are presented. 15105

DOI: 10.1021/acs.inorgchem.7b02416 Inorg. Chem. 2017, 56, 15101−15109

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Inorganic Chemistry

compressibility in LnFe(CN)6, in contrast, results from a competition between two relatively strong effects of spring and gear.35 Phonon Group Velocity. The specific geometric framework of Zn[Au(CN)2]2 can lead to abnormal mechanical behavior in uniform temperature and hydrostatic pressure field, while a question arises: why can the c axis be the carrier of abnormal behaviors? As we know, phonons always exist in a crystal. Here the phonon group velocity as an important parameter determines the propagation speed of phonon energy. Therefore, phonon group velocities are calculated in temperature and hydrostatic pressure field to explore the difference in energy transfer speed along the a axis and the c axis. The corresponding results are present in Figure 7; it is seen that the

dicyanoaurate (−N−C−Au−C−N−) linkages function geometrically as the framework edges. Different from the specific geometries of wine rack in Ag3[Co(CN)6] and KMn[Ag(CN)2]3,9,10,36 as well as specific geometries of molecular gears and torsion springs in LnFe(CN)6,35 the topology in Zn[Au(CN)2]2 is a hybrid of the honeycomb and wine-rack nets, where rows of edge-sharing hexagons are connected vertically by rhombuses. By virtue of the increased Zn−Zn separation allowing 6-fold interpenetration, the topology in the Zn[Au(CN)2]2 framework can be identical with that of βquartz-like networks, and this highlights the cross-bracing effect of helical aurophilic chains running perpendicular to the c axis.22,23 Goodwin et al. reported experimentally that very low energy (typically 0−2 THz) phonon modes were closely associated with the NTE behavior in cyanide-containing framework materials.37 Recent work by Gupta et al. showed that the phonon excited by the lowest energy had the most negative Grüneisen parameter.33 Our calculated abnormal phonon response under temperature field in Figure 5b is consistent with these reports. Among these many contributing phonons, the lowest-frequency optical phonon mode at the zone center has the lowest excitation energy and the most weighting. Here, the vibration (denoted by arrows) corresponding to the lowest-frequency optical phonon mode at the zone center from its eigenvector in a view down the [110] direction is exhibited in Figure 6a. The deformation of the Zn[Au(CN)2]2 framework can be driven by the movement sideways of the nearly rigid dicyanoaurate linkages. A schematic frame diagram derived from the 2D geometric motif of Zn[Au(CN)2]2 in a view along the [110] direction is illustrated in Figure 6c. The analogy of a spring model can be adapted to describe the response of Zn[Au(CN)2]2 on heating and under hydrostatic compression. Compression of the spring may be thermally activated, leading to an increase in Au−Au distances and a lattice parameter and a decrease in c lattice parameter expected for metallophilic interactions (Au−Au interactions): namely, uniaxial-NTE behavior. However, Zn[Au(CN)2]2 under hydrostatic pressure will give rise to an effect of an extended spring, where repulsive Coulomb interactions can be more prevalent than Au−Au attractive interactions. This enables a shorter Au−Au distance and hence a shortening of the a axis and a elongation of the c axis: i.e., the NLC behavior. The effect of hydrostatic compression in Zn[Au(CN)2]2 is conceptually similar to that in Ag3[Co(CN)6]; the network hinge (angle) is involved in preference to the deformation of the covalent network (edge length) in both cases.23 Generally speaking, the crystal structure with a loose packing tends to have more lattice flexibility and have a more pronounced effect on compressive rather than expansive flexibility; an example is zeolite, which subsequently shows NTE.38 Here, some characteristics in Zn[Au(CN)2]2 can contribute to the appearance of abnormal properties, such as flexible framework, low-density structure, extreme machanical anisotropy, unusual phonon response, and specific topology. Two closed-shell metal cations (such as Au, Ag, Cu, and so on) are normally prone to attractive interactions when the metal−metal connections are shorter than the sum of the van der Waals radii, which is known as metallophilic interactions (metal−metal interactions).39 The metallophilic interactions in a specific topology motif play an important role in uniaxialNTE and NLC in metallocyanide frameworks. For example, the weak Ag−Ag interactions are considered to drive the abnormality in Ag3[Co(CN)6].19 The mechanism for extreme

Figure 7. Temperature (a) and hydrostatic pressure (b) dependence of phonon group velocities along the a axis and c axis in Zn[Au(CN)2]2.

response of phonon group velocity along the c axis is very intense in comparison to the moderate variation of phonon group velocity along the a axis under temperature field. Under hydrostatic compression, there is a reverse trend of phonon group velocities along both axes. It is noted that phonon group velocities along the c axis have larger values at low temperatures and low pressures, indicating that the c axis is more sensitive facing an outer field. Quicker propagation of phonon energy along the c axis can facilitate the c axis itself to have a unique role in abnormal uniaxial-NTE and NLC behaviors in Zn[Au(CN)2]2. Elasticity Analysis. Our five calculated independent elastic constants, i.e. elements of the stiffness tensor, in hexagonal Zn[Au(CN)2]2 are given in Table 2 with other recent Table 2. Five Calculated Independent Elastic Constants in the Hexagonal Crystal Zn[Au(CN)2]2 elastic constant (GPa) this work ref 33

C11

C12

C13

C33

C44

48.9 36.6

25.4 29.7

56.3 60.6

94.3 126.8

12.0 12.1

calculations.33 We observe that the values of C11 and C33 in our calculations are larger and smaller, respectively, than those in calculations by Gupta et al. The other elastic constants (C12, C13, C44) are almost identical. No experimental measurements are presently available in the literature for comparison purposes. Our calculated elastic constants are definitely positive and satisfy the Born stability criteria (C44 > 0, C11 > |C12|, (C11 + 15106

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Inorganic Chemistry

Figure 8. Spatial dependence of (a) Young’s modulus, (b) linear compressibility, (c) shear modulus, and (d) Poisson’s ratio of Zn[Au(CN)2]2. For the Young’s modulus and linear compressibility, green and red plots represent positive and negative values, respectively. For the shear modulus and Poisson’s ratio, the surface representing g(θ,φ) enclosing that representing f(θ,φ) is denoted by a blue plot. g(θ,φ) is represented by green lobes for positive values and red lobes for negative values.

2C12)C33 > 2C132), verifying the mechanical stability of the Zn[Au(CN)2]2 system. From the calculated elastic constants, we perform a full tensorial analysis of Zn[Au(CN) 2 ] 2 containing the Young’s modulus E(u), linear compressibility β(u), shear modulus G(u,n), and Poisson’s ratio v(u,v). Except for the linear compressibility β(u) and Poisson’s ratio v(u,v) that we introduced earlier, other mechanical behaviors are derived by key quantities: (i) the Young’s modulus E(u) represents the uniaxial stiffness of materials in the direction of unit vector u and (ii) the shear modulus G(u,n) characterizes the resistance to shearing of the plane normal to n in the u direction.21,32 These directional elastic tensors rather than their averages are visualized in Figure 8. Note that the Young’s moduli have positive values in all projection planes (green plot) but linear compressibility, however, has negative values in (xz) and (yz) projection planes (red plot), indicating NLC behavior

in the z direction (c axis). For the 2D projection of shear modulus and Poisson’s ratio, the surface representing g(θ,φ) encloses that representing f(θ,φ), denoted by the blue plot. The f(θ,φ) surface is green lobes for positive values and red lobes for negative values. Different from the positive values in all projection planes of shear modulus, negative values of the Poisson’s ratio in all projection planes are presented. The outof-flatness in the crystal configuration can cause tensile abnormalities: for instance, the reentrant-honeycomb lattice rather than conventional-honeycomb lattice is considered to be a special motif for targeting low-dimensional materials with NPR, such as single-layer black phosphorus.40 Here, the crossbracing effect of helical chains in the crystal structure of Zn[Au(CN)2]2 creates favorable conditions for the occurrence of NPR behavior. Similar results on NPR in all projection planes have also been observed in the Ag3[Co(CN)6] system.21 15107

DOI: 10.1021/acs.inorgchem.7b02416 Inorg. Chem. 2017, 56, 15101−15109

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Inorganic Chemistry Since the typical topology motif, such as wine rack, β-quartz (similarly as a hybrid of the honeycomb and wine-rack nets), molecular gears, torsion springs, helices, and so on, can induce some abnormal uniaxial-NTE and NLC behaviors in metallocyanide framework materials, can the NPR can be widely found in these unusual materials? This needs to be further confirmed experimentally.

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CONCLUSION The uniaxial-NTE and NLC properties of Zn[Au(CN)2]2 have been investigated by DFT calculations. Within the framework of QHA, phonon spectra are obtained. In addition, the temperature dependence of phonon frequencies exhibits abnormal phonon hardening. The unusual phonon response in temperature field can be indicative of the uniaxial-NTE and NLC properties of Zn[Au(CN)2]2. In combination with the analysis of vibrational mode of the lowest-frequency optical phonon at the zone center, it is found that a springlike framework as a specific topology may result in the coexistence of uniaxial-NTE and NLC behaviors. The effects of a compressed spring on heating and an extended spring under hydrostatic compression can occur. The distinguishing phonon group velocity along the a axis and c axis facilitates different responses for both axes under temperature and hydrostatic pressure field. The elastic constants are calculated to analyze the spatial dependence of elastic tensors. It is found that negative values of the Poisson’s ratio are present in all projection planes due to the specific topology.

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AUTHOR INFORMATION

Corresponding Author

*E-mail for C.W.: [email protected]. ORCID

Lei Wang: 0000-0003-3597-0974 Ying Sun: 0000-0003-0590-372X Cong Wang: 0000-0002-4100-4222 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (NSFC) (Nos. 51502010, 51572010, 51732001, and 51401227).

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REFERENCES

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