Negative Thermal Expansion Properties and the Role of Guest Alkali

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C: Plasmonics; Optical, Magnetic, and Hybrid Materials

Negative Thermal Expansion Properties and the Role of Guest Alkali Atoms in LnFe(CN) (Ln = Y, La) from Ab Initio Calculations 6

Dahu Chang, Changqing Wang, Zaiping Zeng, Chunyan Wang, Fei Wang, Qiang Sun, and Yu Jia J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b01165 • Publication Date (Web): 21 May 2018 Downloaded from http://pubs.acs.org on May 22, 2018

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The Journal of Physical Chemistry

Negative Thermal Expansion Properties and the Role of Guest Alkali Atoms in LnFe(CN)6 (Ln = Y, La) from ab initio Calculations Dahu Changa,b,c, Changqing Wanga, Zaiping Zengb, Chunyan Wangb, Fei Wangc, Qiang Sunc, Yu Jiab,c* a

Department of Mathematics and Physics, Luoyang Institute of Science and Technology, Luoyang 471023, China

b

Key Laboratory for Special Functional Materials of Ministry of Education, and School of Physics and Electronics, Henan University, Kaifeng 475001, China

c

International Laboratory for Quantum Functional Materials of Henan, School of Physics and Engineering, Zhengzhou University, Zhengzhou 450001, China

Abstract Using the ab initio calculations within density functional theory combined with the quasiharmonic approximation (QHA), we have investigated the negative thermal expansion (NTE) properties of LnFe(CN)6 (Ln=Y, La) systems and the role of alkali as guest atoms being inserted in LnFe(CN)6. We have found that both the anti-prism mode originated from the torsional vibration of N atoms in LnN6 units and the transverse vibration of CN ligand mainly contribute to the magnitude of NTE in LnFe(CN)6, and the largest coefficients of negative thermal expansion (CNTE) are found to be approximately -39.3×10-6 K-1 at 180 K for YFe(CN)6 and -50×10-6 K-1 at 110 K for LaFe(CN)6, respectively. The transverse vibration modes of CN ligand are significantly enhanced when Y atoms are substituted by their La counterparts, which results a larger CNTE of LaFe(CN)6 than that of YFe(CN)6. Such NTE can be tuned to be smaller, up to zero or even to positive expansion by inserting one or more gust alkali ions in the vacancy of LaFe(CN)6 owing to the fact that the guest ions effectively oppose the anti-prism vibrational modes. Our study provides not only the understanding of the NTE mechanism but also an approach to tune the NTE properties of LaFe(CN)6 systems.

∗

Corresponding author: E-mail: [email protected]

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1. Introduction Negative thermal expansion (NTE) is a relatively rare phenomenon but it enables many important applications in modern material technology, therefore attracting considerable attention in recent years 1-3 The occurrence of NTE materials offers a promising possibility to tailor the thermal expansion coefficient to a desired value including particularly near zero thermal expansion. To the best of our knowledge, the underlying mechanism of the NTE materials can be classified into three categories, (i) flexible networks, (ii) atomic radius contraction and (iii) magnetovolume effect. The widely studied flexible open framework NTE materials include oxides, fluorides and cyanides, coordination polymers (or MOFs) etc., among which the most typical ones are ZrW2O8, ScF3, and Prussian blue analogues MX(CN)64-5 and nonporous M(eim)2 (M = Zn, Cd)6 and so on. It is well known that the NTE behavior of flexible open framework materials has been attributed to the existence of low-energy transverse vibrational modes of the bridging species such as O, F, and CN atoms 5, 7-10. Comparing to the other open framework materials, cyanide-bridged NTE systems exhibit larger coefficients of negative thermal expansion (CNTE) because of their more flexible bridging bi-atom species CN 8, 11-12. Therefore, Prussian blue series as an important member of this family have received increasing attention both theoretically and experimentally 7, 13-16. The Prussian blue analogues MX(CN)6 processes either cubic or hexagonal crystal structures, where M is divalent or trivalent metal element(s) and X is trivalent or tetravalent metal element(s). Their NTE properties have been widely studied experimentally. Duyker et al. 15 have studied the NTE behavior of LnCo(CN)6 (Ln=La, Pr, Sm, Ho, Lu, Y) and explored the underlying mechanisms. Similar work has been carried out by Chapman et al. 7 but for MII PtIV (CN)6 (M =Mn, Fe, Co, Ni, Cu, Zn, Cd) systems. The corresponding properties have also been studied in ErIII [CoIII (CN)6] and the effect of H2O molecules adsorption has been further discussed16. Contrary to most of the Prussian blue analogue MX(CN)6 materials with a large CNTE, Margadonna et al.17 found zero thermal expansion in FeCo(CN)6 by using X-ray pow-

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der diffraction method. A further study has been devoted to the NTE properties of the YFe(CN)6-based Prussian blue analogue18-19 and the effect of guest alkali metal atoms or molecules H2O has been experimentally investigated 13-14. For LnFe(CN)6 and KLnFe(CN)6 (Ln=Y, La), water molecules not only play a very important role on mechanical, thermal and electronic properties, but also could induce structural phase transitions20-21. The NTE properties of YFe(CN)6, which exhibit excellent elastic properties18-19, have been fairly well studied13. As experimentally confirmed in Ref. 13, the presence of guest ions or molecules H2O plays a critical damping effect on transverse vibrations, consequently inhibiting the performance of NTEs13. More interestingly, different numbers of water molecules absorbed in these open framework structures could tune the crystal structures. For example, Mullica et al.20 have found that after mono-dehydration, the ninecoordinated lanthanum atom in La[Fe(CN)6]·5H2O changes to give eight-coordination in the tetrahydrated form of La[Fe(CN)6]·4H2O being hexagonal with space group P63/m. Goubard et al.21 have studied the properties of a series of KMFe(CN)6·xH2O by thermal analysis, IR spectroscopy and X-ray diffraction. They have found that many kinds of complexes with different symmetries exist corresponding to different numbers of water molecules (x = 3, 3.5, 4), i.e. KYFe(CN)6·3H2O (Pbnm). For a given structure of LnFe(CN)6 with space group P63/mmc, we focus on two important aspects of NTE properties in the present study, (i) the underlying physical mechanisms in the open framework of YFe(CN)6 and the influence of La substitution, (ii) the role of K-atom being inserted in such open frameworks. However, we have not considered the effects of H2O molecules based on the aforementioned reasons. That is, the number of H2O molecules absorbed in the vacancies of YFe(CN)6 could dramatically change the crystal structure from P63/mmc to Cmcm. In some cases, both the absorption sites and the number of H2O molecules in YFe(CN)6 are temperature-dependent and the vibrational modes become complex. The absorption site and number of K-atoms in LnFe(CN)6 could be regarded as unchanged and rather simple. This could be easier for the analysis of the variation of the vibrational modes related to NTE. It therefore could be treated as a typical model for studying the 3



NET properties, which presents a clearer picture to understand the damping effects both for atoms and for molecules. In this contribution, we present a theoretical study of the NTE in LnFe(CN)6 (Ln = Y, La). We find that the transverse vibration of the CN ligand and the anti-prism mode derived from the torsional transverse vibration of N atoms in YN6 units have the dominant contribution to the NTE of LnFe(CN)6. The guest ions K could effectively mediate the vibration of the anti-prism mode leading to a normal thermal expansion. We further predict a larger CNTE for LaFe(CN)6 due to its more flexible structures which can significantly enhance the transverse vibration of CN ligand. 2. Computational methods All the calculations are carried out using first-principles method based on density functional theory (DFT) as implemented in the Vienna ab initio simulation package (VASP)22. The ion-electron interaction is depicted by projector augmented wave (PAW) method23, and the exchange and correlation effects are described by the generalized gradient approximation with PBE functional24. The wave functions are expanded by the plane waves up to an energy cutoff of 520 eV. Integrals over the first Brillouin zone (BZ) are approximated by a Monkhorst-Pack K-point meshes of 9 × 9 × 5. The total energy is calculated with high precision, converged to 10-8 eV/atom and the structural relaxation ends when the residual forces on atoms become less than 10-4 eV/Å. The Murnaghan equation of state is used to fit the total energies as a function of primitive cell volume, which delivers the equilibrium lattice constants and bulk modulus. Vibrational properties are calculated using the supercells method and post-processed by PHONOPY code25. The real-space force constants of the supercell are computed using the density-functional perturbation theory (DFPT) as implemented in VASP code. The thermal dynamics properties of LnFe(CN)6 (Ln= Y, La) are simulated through the quasi-harmonic approximation (QHA). In this approximation, the volume-dependent phonon frequencies at finite temperatures are introduced as a part of the anharmonic effect26. It has been proved that QHA is a good approximation if the temperatures are much lower than the melting point of materials9, 27-28. Within the QHA, the Helmholtz free energy F (V, T) has been calculated ac4


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cording to the following formula29 F (V , T ) = E (V ) + Fvib (V , T )

(1)

Here E0 is the total energy of MX(CN)6 materials at 0 K when all atoms are fixed at their bulk position. The Fvib stands for the vibration energy, which is given by Fvib (V , T ) = k BT ∑ log {2 sinh ( hωv (q,V ) 2k BT )}

(2)

v,q

where ω and g (ω ) are the phonon frequency and the phonon density of states, respectively. In our calculations, we have used a 2×2×1 supercell to compute these quantities. A further check about the accuracy is performed by using a 3×3×1 supercell and the same results have been obtained. To capture the relation of between temperature and volume, we have selected 10 different volumes near the optimized equilibrium. The internal atomic positions are optimized at each volume and the phonon dispersion curves are calculated. The isothermal F-V curves are fitted according to the Murnaghan equation of state in order to harvest the Helmholts free energy which is the minimum value of the thermodynamic function. The volume thermal expansion coefficient under the QHA can be further given by30 αV (T ) =（1 BV）∑ γ i CV (T )

(3)

i

i

where γ i = −∂ ln ωi / ∂ ln Vi and CVi (T ) are the Grüneisen parameter and specific heat of the ith vibrational state at temperature T, respectively.

3. Results and Discussion 3.1 Geometric Structures and Vibrational Properties of LnFe(CN)6 and KLnFe(CN)6 We first study the geometric structures and vibrational properties of LnFe(CN)6 and LnFe(CN)6 with one K atom being inserted into the vacancy of LnFe(CN)6, namely, KYFe(CN)6, and KLaFe(CN)6. YFe(CN)6 has hexagonal structures with space group P63/mmc and KYFe(CN)6 in trigonal structure P-31c13. They contain two formula units in one conventional unit. These Prussian blue analogues have special structures which consist of alternating LnN6 prisms and FeC6 octahedrons connected by CN ligand (see Figure 1 (a)-(d)). 5



The calculated lattice constants and bulk modulus together with the experimental results are listed in Table 1. The optimized lattice constants of YFe(CN)6 (a = b=7.49 Å and c = 13.11 Å) are in good agreement with the experimental data in the supplementary information of Ref. 13. The six N atoms and one Y atom in YFe(CN)6 form one YN6 prism unit while six C atoms and one Fe atom form one FeC6 octahedron unit. There are many vacancies among these prisms and octahedrons, which introduce flexibilities to analogues. For this reason, we have found a relatively small bulk modulus of 51.42 GPa. Comparing to YFe(CN)6 , LaFe(CN)6 which we take the same structure as YFe(CN)6 has larger lattice constants (a = b = 7.75 Å and c = 13.42 Å) and exhibits a smaller bulk modulus of 42.80 GPa. Concerning the geometric structure of KYFe(CN)6, inserting one K atom leads to the disappearance of one vacancy in the primitive cell. YN6 and KN6 polyhedral units connect through the edgings, which makes this structure more rigid-like as illustrated in Figure 1(c) and (d). We find that the bonds along Fe-C-N-Y chain direction of LnFe(CN)6 is a straight line while it becomes crooked in KLnFe(CN)6. This difference will play an important role in the thermal expansion properties discussed below.

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Figure 1. The perspective (a and c) and top views (b and d) of geometric structures of LnFe(CN)6 and KLnFe(CN)6, where the orange, blue and purple polygons represent the FeC6, LnN6 and KN6 polyhedral units, respectively. The first Brillouin zone of LnFe(CN)6 is shown in (e). We first present the phonon dispersion curves and phonon DOS of LnFe(CN)6 and KLnFe(CN)6, respectively (see Figure 2(a)-(c)), which are prerequisites for thermal expansion properties. It is shown that the phonon dispersion curves along the high-symmetry lines of the Brillouin zone (BZ) have no imaginary frequencies, indicating that LnFe(CN)6 and KLnFe(CN)6 are dynamically stable. The contribution to vibrational frequency below 200 cm1

is mainly from C and N atoms, which suggests that the vibrations associated with C and N

atoms are easy to be excited predominantly at low temperatures. It is found from the phonon dispersion curves and DOS of KLnFe(CN)6 that all the vibrational modes exhibit a dramatic shift towards high frequency, indicating KLnFe(CN)6 becoming more rigid along a or b-axis direction. Such shift is more pronounced for lower frequency parts. Moreover, the vibrational modes of acoustic and the lowest optic branch are very different from that of LnFe(CN)6. The 7



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vibrations of K atoms are found to mostly contribute to the low frequency part of phonon DOS, suggesting that these atoms will play an important role in the thermal expansion properties. Such differences of phonon characteristics between LnFe(CN)6 and KLnFe(CN)6 shall result in different thermal expansion properties.

Table 1. Comparison of optimized and experimental lattice constants a=b and c (Å), Volume (Å3), bulk modulus B (GPa). YFe(CN)6

KYFe(CN)6

LaFe(CN)6

KLaFe(CN)6

This work

Expt.13

This work

Expt.13

This work

This work

a

7.49

7.46

7.04

6.97

7.75

7.20

c

13.11

13.11

12.47

12.28

13.42

12.52

V

637.88

632.34

535.60

516.32

697.74

562.13

B

51.42

42.80

27.58

30.61

Figure 2. The calculated phonon dispersion curves and vibrational density of states (DOS) for (a) YFe(CN)6 , (b) KYFe(CN)6, (c) LaFe(CN)6, and (d) KLaFe(CN)6 respectively. The Bradley-Cracknell notation for the high-symmetry points of BZ is adopted and these points are Γ (0,0,0), M (0,1/2,0), K (-1/3,2/3,0) and A (0,0,1/2), respectively. 8


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The vibrational characteristics of phonon modes are very important for physical properties such as surface enhanced Raman scattering and NTE. For simplicity but without loss of generality, we therefore only focus on the four lowest-frequency optical modes to discuss the NTE mechanism of YFe(CN)6, which are explicitly shown in Figure 3. We find that the lowest optical vibration mode comes from the so called anti-prism motion, that is, three N atoms at the top of YN6 prism rotates clockwise while three N atoms at the bottom rotates anticlockwise (see Figure 3(a)). The corresponding vibrational frequency is 44.21 cm-1 at Γ point of BZ. The second lowest one corresponds to a local collective motion where YN6 prism rotates along the c-axis and the frequency is 54.58 cm-1, as shown in Figure 3(b). Beside these two modes, Figure 3(c) describes the rotational vibration of the side in YN6 prism. The corresponding vibrational frequencies are 59.43 cm-1. When K atom is added into YFe(CN)6, two main findings in the low frequency range of vibrational properties are observed. One is that the vibrational frequencies of aforementioned three modes increases from 44.21, 54.58, 59.43 cm-1 to 49.62, 61.11, 61.45, cm-1, respectively. The influence of the insertion of K atom in YFe(CN)6 on the vibrational properties is on two-fold. Firstly, it leads to the vibrational frequency of anti-prism motion shifted from 44.21 cm-1 to 49.62cm-1 due to the enhance interaction between K-atom and lower/upper planes of the prism. Most importantly, the vibrational amplitude of such modes (i.e. rotational angle between the upper and lower N-N planes) is reduced significantly for KYFe(CN)6. Consequently, the contribution of such modes on the NTE properties almost becomes vanishing. The other one is that the vibrational mode between C-N changes. Contrary to the parallel vibration of CN ligand in YFe(CN)6, they vibrate in various directions in KYFe(CN)6. This is why YFe(CN)6 and KYFe(CN)6 exhibit different thermal expansion properties experimentally13.

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Figure 3. The atomic displacement schemes for low frequency vibrational modes of YFe(CN)6 (upper panel) and KYFe(CN)6 (lower panel). (a)-(f) are for phonon frequencies of 44.21, 54.58, 59.43, 49.62, 61.11, 61.45 cm-1, respectively. Before computing the CNTE of LnFe(CN)6 and KLnFe(CN)6, we study the frequency shift of vibrational mode as a function of volume. In this way, one could easily predict whether such materials have NTE. The low frequency part of the phonon spectrum of LnFe(CN)6 is shown in Figure 4 for two different volumes (e.g., V0 and 0.99V0, where V0 is the volume at equilibrium). We find that the low-energy vibrational modes exhibit strongly softening phenomena when the volume is decreased. This is especially true for the lowest optical phonon modes at Γ point of the Brillouin zone. This frequency softening phenomena will lead to a negative Grüneisen parameters associated with the NTE properties. Contrary to LnFe(CN)6, such softening phenomena in low-energy vibrational modes almost disappear for KLnFe(CN)6. It therefore predicts that the thermal expansion properties will be different when K ions are inserted. This will further differentiate the NTE characteristics between LnFe(CN)6 and KLnFe(CN)6.

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Figure 4. The low frequency part of phonon spectrum of (a) YFe(CN)6, (b) KYFe(CN)6, (c) LaFe(CN)6, and (d) KLaFe(CN)6 with equilibrium lattice constants and volume compressed by 1%, respectively. 3.2 The thermal expansion properties of LnFe(CN)6 and KLnFe(CN)6. After understanding the geometric structures and vibrational characteristics, we turn to evaluate the thermal expansion properties of LnFe(CN)6 and KLnFe(CN)6. We first present in Figure 5 (a-d) the temperature dependence of cell volume and CNTE. It is found that the cell volumes at a finite temperature for both YFe(CN)6 and LaFe(CN)6 are smaller than that at 0 K, therefore exhibiting a typical NTE behavior. The NTE temperature window for YFe(CN)6 is relatively large, ranging from 0 K to ~1200 K, in comparison to other Prussian blue materials. Our theoretical results for YFe(CN)6 are in excellent agreement with the experimental results of Ref. 13. The largest CNTE is found to be approximately -39.3×10-6 K-1 at the temperature about 180 K. The corresponding average CNTE is about -31.52×10-6 K-1 for temperature ranging from 300 to 550 K, which agrees excellently with the experimental measurements (33.67 ×10-6 K-1) 13. For KYFe(CN)6, a normal thermal expansion behavior is observed in the temperature range of 0-500 K, which is in good qualitative agreement with the experimental data measured at temperature range of 300-500 K. We further predict that LaFe(CN)6 with the same crystal structure as YFe(CN)6 will have a much larger CNTE (about -50×10-6 K-1 at 110 K) but a smaller NTE temperature window (0-850 K) than that of YFe(CN)6 (see Figure 5(c)). A smaller NTE in KLaFe(CN)6 at low temperature can be found (see Figure 5(d)), suggesting

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that there will be an obvious CNTE if Na or Li atoms are inserted in these open framework structures. It should be noted that for YFe(CN)6, our simulated results are in excellent agreement with the experimental ones. This justifies that the anharmonic effects well describes the NTE properties of such materials. We believe that it could also predict the NTE properties of LaFe(CN)6, which has the same space group. For KYFe(CN)6, our ab inito results also qualitatively agree with the experimental measurements. However, relatively large differences could be noticed. This could be attributed to the higher-order an-harmonic effects.

Figure 5. Temperature dependence of the cell volume and CNTE of (a) YFe(CN)6 and (c) LaFe(CN)6 having hexagonal structure with space group P63/mmc, (b) KYFe(CN)6 and (d) KLaFe(CN)6 having trigonal structure with space group P-31c, respectively. The red stars in (a) and (b) are the experimental data from Ref. 13 for comparison.

3.3 The negative thermal expansion mechanism and the role of K atoms in LnFe(CN)6 To explore the underlying mechanism responsible for the NTE of LnFe(CN)6 and discover the role of K atoms, we have calculated the corresponding Grüneisen parameters. In general, the Grüneisen parameter reflects the anharmonic effect, i.e., the nonlinear vibrational modes of atoms. If the sum of negative Grüneisen parameters ( γ i ) for the excited modes is larger than that of the positive ones, the CTE of the material will be negative. The calculated results are shown in Figure 6 as a function of vibrational frequency at the Γ points in Brillouin zone. We find that the Grüneisen parameters of LnFe(CN)6 and KLnFe(CN)6 exhibit very dif12


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ferent features. The low-frequency vibrational modes (below 100 cm-1) of YFe(CN)6 have relatively larger negative Grüneisen parameters, which will contribute to the corresponding NTE. It should be noted that the largest three Grüneisen parameters in Figure 6(a) corresponds to the aforementioned three vibration modes as shown in Figure 3(a), (b) and (c). In contrast to YFe(CN)6, the Grüneisen parameters of KYFe(CN)6 are positive for most of the low vibrational modes, therefore leading to a normal thermal expansion behavior. The negative Grüneisen parameters of LaFe(CN)6 are larger (in absolute value) than that of YFe(CN)6, which therefore introduces a larger CNTE. Contrary to most studies which ascribe NTE only to the low frequency modes31-32, we find that LnFe(CN)6 in high frequency range of 300-500 cm-1 also have negative Grüneisen parameters, which should also contribute to the corresponding NTE. Such type of contribution has also been found earlier in TaVO5 33.

Figure 6. Calculated Grüneisen parameters of (a) YFe(CN)6, (b) KYFe(CN)6, (c) LaFe(CN)6, and (d) KLaFe(CN)6 respectively. The parameters in high frequency range are not shown for simplicity.

Since the NTE is closely related to the modes with relative large negative Grüneisen parameters, we have therefore studied these vibrations modes to further shed light on the mechanism of the NTE. The atomic displacement diagrams of the three modes with relative large negative Grüneisen parameters have been shown in Figure 3. For these types of vibrational modes, 13



the upper N atoms in the YN6 prism rotate out of phase with the lower N atoms along the caxis, which is named as anti-prism mode. This twist prism rotation will leads to a reduction in the distance between upper and lower planes (see Figs. 7 (a) and (b)). The anti-prism mode consequently contributes to most part of the NTE. On the other hand, the C and N atoms in the CN ligand, which vibrates perpendicular to the Y-N-C-Fe atom line, have two modes. One is parallel mode where C and N atoms vibrate in parallel (see Figure 7c) and the other one is anti-parallel mode where these atoms vibrate in an anti-parallel way (see Figure 7e). Both modes drive an out (or in)-of-phase rotation for YN6 and FeC6 units. The YN6 and FeC6 polyhedral units rotate rigidly in the two modes and can be viewed as rigid unit mode (RUM). Besides these anti-prism modes, it should be noted that the transverse vibration of C and N atoms, to some extent, could also contribute to NTE. Substituting Y atom with its La counterpart in LaFe(CN)6 increases the lattice constants. Consequently the interaction of La or Fe atoms with C or N atom becomes weaker. The antiprism mode and the CN transverse vibrational mode are easier to be excited, leading to a larger NTE. Concerning KYFe(CN)6, it can be viewed as one K atom being inserted into the vacancies of YFe(CN)6. The inserted K ion actually hinders the flexibility of the motions of these YN6 and FeC6 rigid units due to the strong interaction of K atom with YN6 and FeC6 units. This geometric structural property is therefore responsible for the absence of NTE in KYFe(CN)6. Such an absence can also be explained by the changes of phonon modes introduced by adding K atoms: (i) the aforementioned anti-prism mode disappears, (ii) the vibration of C and N atoms is no longer perpendicular to the Y-N-C-Fe line and the bonds along the line become crooked. Based on above analysis, the NTE properties of LaFe(CN)6 can be effectively tuned by inserting a K atom. It would be reasonable to expect that both the CNTE and the corresponding NTE temperature window can be tuned in a much larger range if atoms with smaller radii, such as Na and Li in the same element group as K, being inserted. This results in much weaker interaction between Na or Li and YFe(CN)6. In the other words, we believe that this might provide an effective way of tuning the CNTE of YFe(CN)6 or LaFe(CN)6 to a desired value by applying this strategy. 14


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In general, the relationship between the substituting atoms in YFe(CN)6 and the corresponding NTE properties could be understood as following. Concerning open framework NTE materials, large volume (or size) of vacancies is one of prerequisites of having a strong vibrational mode, which contributes to NTE. To be more specific, the substitution of Y with La in YFe(CN)6 would result in a lager vacancy because the atomic radius of La is slightly larger than Y atoms. Therefore, this might contribute to a larger NTE of LaFe(CN)6 than that of YFe(CN)6. On the other hand, another factor which might affect the NTE is the interaction between Y (La) with Fe and C-N atoms. Thus, the underlying relationship between the size of the vacancies and the NTE coefficient is complicated. As for inserting alkali atoms into the vacancies of YFe(CN)6, the relationship is clear. Inserting an atom with larger atomic radius would dramatically prohibit the vibrational modes contributing to NTE, and vice versa.

Figure 7. The anti-prism (upper panel) and transverse vibration modes (lower panel) contribute mostly to NTE in LnFe(CN)6. (a) shows the anti-phase-rotation of prism and (b) shows the length reduction of the prism. Red and blue balls represent the location of N atoms before and after the prism rotation, and ∆ is the length reduction. (c) and (e) show the in-phase and out-of-phase vibration of CN ligand, respectively, where the vibrational directions are high15



lighted by arrows. In the lower panel, the yellow hexagon and blue triangle denote the Fe(CN)6 and Ln(CN)6 units.

IV. Summary To conclude, the structural and the negative thermal expansion properties of YFe(CN)6 and LaFe(CN)6 are studied by using the ab initio calculations within density functional theory (DFT) combined with the quasi-harmonic approximation (QHA). The underlying mechanism is resolved by analyzing Grüneisen parameters and phonon vibrational modes. We find that the anti-prism and the transverse vibrational modes contribute dominantly to the NTE properties of LnFe(CN)6, that is, anti-prism mode leads to the length reduction purely along c-axis direction and the transverse vibrational modes of CN ligand contribute to the NTE along both in-plane (a-axis) and out-of-plane (c-axis) directions. We further find that the presence of guest ions (K+) effectively hinders the anti-prism mode, resulting in a normal positive thermal expansion of KYFe(CN)6. We believe that the present study will be useful for understanding the underlying NTE mechanism and will provide an efficient way of manipulating the NTE properties of Prussian blue analogue.

Notes The authors declare no competing financial interest

Acknowledgments This work is supported partly by the NSF of China (Grant No. 11774078), and partly by Innovation Scientists and Technicians Troop Construction Projects of Henan Province (Grant No. 10094100510025). References (1) (2) (3)

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Negative Thermal Expansion Properties and the Role of Guest Alkali

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