Negative thermal expansion in ZnF 2
NNegative thermal expansion in ZnF Tapan Chatterji, Mohamed Zbiri, and Thomas C. Hansen
Institut Laue-Langevin, B.P. 156,38042 Grenoble Cedex 9, France (Dated: October 25, 2018)We have investigated temperature dependence of the lattice parameters and the unit cell vol-ume of ZnF by neutron diffraction and have discovered negative thermal expansion (NTE) at lowtemperature. To understand why this simple compound exhibits NTE we performed first principlecalculations. These calculations reproduce qualitatively the experimental temperature dependenceof volume. PACS numbers: 61.05.fm, 65.40.De
The negative thermal expansion (NTE) in solids hasattracted the renewed attention of condensed matter sci-entists ever since Sleight and coworkers discovered thatZrW O contracts over a wide temperature range of morethan 1000 K. There are excellent review articles on theNTE of this type of so-called framework materials. How-ever NTE is known and has been studied experimentallyand theoretically for a long time. Among these Si and Geand other tetrahedrally bonded crystals at low tempera-ture are classic examples . The NTE is however limitedin the low temperature range in these materials, whereasin higher temperature range they exhibit normal positivethermal expansion. There exists a more general reviewarticle covering all types of materials that exhibit NTE.Here we report observation of NTE in diamagnetic ZnF with the simple rutile-type structure. We also report theresults of our ab-initio calculations that reproduce qual-itatively the observed NTE in ZnF .The transition-metal difluorides MF (M = V, Cr,Mn, Fe, Ni, Cu, Zn) with the rutile-type or distorted FIG. 1: (Color online) (Left panel) Rutile-type crystal struc-ture of ZnF . The grey spheres are Zn and the green spheresrepresent F atoms. (Right panel) The stacking of ZnF octa-hedra in ZnF . I n t e n s it y ( a r b . un it s )
2q (deg.) peak positions I obs I calc I obs -I calc ZnF T=1.8K
FIG. 2: (Color online) Rietveld refinement of the ZnF crystalstructure at T = 1.8 K. rutile-type crystal structure form an important class ofmaterials with interesting magnetic and magneto-opticproperties. In order to entangle magnetic effects fromthe lattice effects, the last of this series viz. the non-magnetic ZnF has often been used to study the back-ground lattice effects. However even this non-magneticsolid showed anomalies in elastic constants at low tem-peratures not expected for ZnF with no temperature-induced phase transition. The observed softening of C and C s = ( C − C ) / had been interpreted as an incipient ferroelectrictransition but has also been contested .Figure 1 shows schematically the rutile-type crystalstrucure of ZnF that crystallizes with the space group D h or P /mnm . The unit cell is tetragonal with latticeparameter a = 4 . c = 3 . T = 296 K and itcontains two formula units Z = 2. The two Zn ions arelocated at positions (0 , ,
0) and (1 / , / , /
2) whereasfour F − ions are located at ( x, x, − x, − x, / − x, / x, / / x, / − x, /
2) with the positionalparameter x = 0 . ions are surrounded bysix F − ions to form slightly distorted octahedra. Theoctahedra are edge linked along the c -axis and corner-linked along < > crystallographic directions.Neutron diffraction experiments were done on ZnF on the high intensity powder diffractometer D20 of theInstitute Laue-Langevin in Grenoble. The 115 reflectionfrom a Ge monochromator at a high take-off angle of 118 ◦ gave neutron wavelength of 1.868 ˚A. Approximately 5 g a r X i v : . [ c ond - m a t . m t r l - s c i ] A p r a ( Å ) Temperature (K)ZnF c ( Å ) ZnF Temperature (K)69.2569.3069.3569.4069.4569.5069.5569.600 50 100 150 200 250 300 350 U n i t c e ll v o l u m e ( Å ) Temperature (K)ZnF a ( Å ) Temperature (K)ZnF c ( Å ) ZnF Temperature (K)69.2969.3069.3169.3269.3369.340 20 40 60 80 100 120 140 U n i t c e ll v o l u m e ( Å ) Temperature (K)ZnF FIG. 3: (Color online) Temperature variation of the latticeparameters a , c , and the unit cell volume V of ZnF plotted onthe left panels. The red curves in these figures represent thelattice parameter and the unit cell volume obtained by fittingthe data by fifth order polynomials. On the right panels onlythe low temperature data are shown. ZnF powder samples was placed inside an 8 mm diame-ter vanadium can, which was fixed to the sample stick ofa standard He cryostat. We have measured the diffrac-tion intensities from ZnF as a temperature in the range1 . −
320 K. The Rietveld refinement of the diffrac-tion data was done by the Fullprof program . The re-finement results from ZnF at T = 1.8 K is shown inFig. 2. The agreement factors R (not corrected for back-ground) for pattern of this refinement were R p = 3 . R wp = 4 .
13. The corresponding conventional Ri-etveld R-factors were R p = 14 . R wp = 10 .
5. Thegoodness of the fit as given by χ was χ = 2 . a , c , and the unit cell volume V of ZnF on the left panels. The red curves in these figures rep-resent the lattice parameters and the unit cell volumeobtained by fitting the data by fifth degree polynomials.Attempts to fit the data by Debye or Einstein functionsin Gr¨uneisen approximation failed for a and V because ofthe negative thermal expansion at low temperaure. Fifthdegree polynomial function fit the low temperature suc- -0.020-0.0100.0000.0100.0200.0300.0400 20 40 60 80 100 120 140 obs.calc. Δ V / V ( % ) Temperature (K)-5051015 0 20 40 60 80 100 120 140 calc.obs. α V x ( K - ) Temperature (K)
FIG. 4: (Color online) (Upper panel)Temperature variationof ∆
V /V . The blue circles are the data points and the dot-ted red curve is the DFT calculation of ∆ V /V .(Lower panel)Temperature variation of the thermal expansion coefficient α V = V dVdT . The blue and the dotted red curves correspondto the observed and calculated volume thermal expansion co-efficient α V . The experimental values were determined fromthe differentiation of the fitted data with a fifth-order poly-nomial function. cessfuly. On the right panels only the low temperaturedata are shown. The lattice parameter a and the unitcell volume V exhibit minima at about 75 K.The upper panel of Fig. 4 shows the experimentalnormalized volume change given by∆ VV = V ( T ) − V V (1)where V ( T ) is the unit cell volume at temperature T and V is the volume at T = 0 along with the values calculatedwithin the density functional theory framework (DFT).The lower panel of Fig. 4 shows the experimental andcalculated volume thermal expansion coefficients α V inthe low temperature range given by α V ( T ) = 1 V dVdT . (2) x (r l . u . ) ZnF Temperature (K)2.0102.0152.0202.0252.0302.0352.0402.0452.050 0 50 100 150 200 250 300 d1(Å)d2(Å) Z n - F bond d i s t an c e s d1 , d2 ( Å ) Temperature (K)ZnF FIG. 5: (Color online) Temperature variation of the positionalparameter x of the F atom and the two Zn-F bond distances d d . In order to check whether there exists any indicationof incipient ferroelectric phase transition in ZnF atlow temperature we refined the neutron powder diffrac-tion data by the Rietveld method and determined thepositional parameter x and the two Zn-F bond distances d and d as a function of temperature. Fig. 5 showsthese quantities. The absence of any anomalies suggestthat apart from the negative thermal expansion (NTE) no further structural changes take place in ZnF low tem-perature.We have done calculations to check whether we can re-produce NTE in ZnF using first-principles DFT. SinceDFT is a T = 0 K approach, the finite temperature de-pendence is introduced within the framework of phononsby applying the so-called quasiharmonic approximation.The anharmonic effects are included uniquely via the vol-ume dependence of the phonon frequencies. For a set ofvolumes around the equilibrium one, the procedure con-sists of evaluating the total Helmhotz free energy givenby F ( V, T ) = E el ( V ) + F ph ( V, T ) (3)where E el (V) and F ph ( V, T ) are the total ground-statetemperature-free energy at constant volume as obtaineddirectly from DFT and the phonon free energy extractedfrom subsequent lattice dynamical calculations for eachvolume, respectively. The free energy F ( V, T ) can bethen used to evaluate thermodynamics of the materialunder study.The starting geometry for the calculations was the ex-perimentally refined ZnF structure . Relaxed geome-tries, total energies, phonon frequencies and volume-dependent phonon free energies were obtained using sim-ilar computational procedure described previously .The underlying mechanism of NTE in ZnF seemsto be very similar to that for tetrahedral semiconduc-tors like Si, Ge, ZnS etc. with diamond and zincblendestructures . It is the excitations at low temperaturesof the low-energy phonon modes with negative Gr¨uneisenparameters that are responsible for NTE in ZnF . Thesemodes are likely connected with the rigid-mode vibra-tions of ZnF octahedra and their linkage along the aaxis shown in the right panel of Fig. 1.In conclusion we have done neutron diffraction studyof the temperature dependence of the crystal structure ofthe simple non-magnetic or diamagnetic transition metaldifluoride ZnF with rutile structure and have discoverednegative thermal expansion (NTE) at low temperature.Our first principle calculations reproduces qualitativelythis experimental result. T.A. Mary, J.S.O. Evans, T. Vogt and A.W. Sleight, Sci-ence , 90 (1996). J.S.O. Evans, T.A. Mary, T. Vogt, M.A. Subramanian andA.W. Sleight, Chem. Mater. , 2809 (1996). A.W. Sleight, Ann. Rev. Mater. Sci. , 29 (1998). J.S.O. Evans, J. Chem. Soc. Dalton Trans. 3317 (1999). D.F. Gibbons, Phys. Rev. , 136 (1958). P.W. Sparks, and C.A. Swenson, Phys. 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