Negative thermal expansion near two structural quantum phase transitions Connor A. Occhialini1, Sahan U. Handunkanda1,2, Ayman Said3, Sudhir Trivedi4, G. G. Guzmán-Verri5,6, and Jason N. Hancock1,2 1Department of Physics, University of Connecticut, Storrs, Connecticut 06269 USA 2Institute for Materials Science, University of Connecticut, Storrs, Connecticut 06269 USA 3Advanced Photon Source, Argonne National Laboratory, Argonne, Illinois 60349, USA 4Brimrose Technology Corporation, Sparks, MD 21152-9201, USA 5Centro de Investigación en Ciencia e Ingenieŕıa de Materiales and Escuela de F́ısica, Universidad de Costa Rica, San José, Costa Rica 11501 and 6Materials Science Division, Argonne National Laboratory, Argonne, Illinois, USA 60439 (Dated: December 6, 2017) Recent experimental work has revealed that the unusually strong, isotropic structural negative thermal expansion in cubic perovskite ionic insulator ScF3 occurs in excited states above a ground state tuned very near a structural quantum phase transition, posing a question of fundamental interest as to whether this special circumstance is related to the anomalous behavior. To test this hypothesis, we report an elastic and inelastic X-ray scattering study of a second system Hg2I2 also tuned near a structural quantum phase transition while retaining stoichiometric composition and high crystallinity. We find similar behavior and significant negative thermal expansion below 100K for dimensions along the body-centered-tetragonal c axis, bolstering the connection between negative thermal expansion and zero temperature structural transitions. We identify the common traits between these systems and propose a set of materials design principles that can guide discovery of new materials exhibiting negative thermal expansion. Negative thermal expansion (NTE) is an emerging area of material behavior discussed in chemistry, engineering, and physics which challenges conventional notions of lat- tice dynamics. Two routes to realizing NTE have been realized: one recipe is via a broadened phase transition from some high-temperature phase to a higher-volume, lower temperature phase. Treatment with quenched disorder can smear the transition and achieve a grad- ual and tunable NTE evolution of lattice parameters. Examples of this type of NTE are found in InVar[1], anti-perovskites[2, 3], ruthenates[4], and charge-transfer insulators[5, 6]. In contrast, a second type of NTE is realized from intrinsic dynamical origins, also referred to as structural NTE (SNTE)[7], which is not obviously resultant from phase competition and does not require quenched disorder but seems to arise from intrinsic ge- ometrical modes with tendencies to draw in the lattice dimensions when thermally activated. Unlike the broad- ened phase transition type, SNTE appears in a wide variety of lattice systems[8, 9] without necessarily con- straining the magnetic or electronic phase diagram. This freedom allows one to envisage new multifunctional ma- terials with diverse mechanical, spin, orbital, thermal, electronic, superconducting, and more exotic order co- existing with NTE, potentially enabling the benefits of strain control to enable new types of order. ScF3 is prominent among SNTE systems, forming in the so-called “open” perovskite (ReO3-type) structure with a small, four-atom unit cell and cubic symme- try at all temperatures below the high melting point of 1800K (Fig 1a)[10–16]. The linear coefficient of thermal expansion (CTE) of this material is isotropic, strongly negative, and persistent over 1000K tempera- ture window[10]. Combined computational and inelastic scattering work[11] has described the configurational po- tential for R point distortions as having a nearly quar- tic form at small displacement, presenting an interest- ing limit of lattice dynamics. Further inelastic scatter- ing investigations on single crystals aimed at exploring the consequences of this unusual situation discovered an incipient soft-mode instability[12] implying the develop- ment of a structural instability with a small extrapo- lated critical temperature Tc<0. This result implies that while similar compounds like TiF3 realize a cubic-to- rhombohedral structural phase transition, the transition is never realized in ScF3, except under an extremely small <1kbar hydrostatic pressure at cryogenic temperatures. These special circumstances are contextualized in Fig- ure 1a, which shows the global structural phase diagram of 3d transition metal trifluorides BF3 as a function of the B+3 ionic radius, rB . The occurrence of the end- point of a structural phase boundary so near the ground state of a stoichiometric compound is extremely rare, as is the pronounced SNTE property of ScF3. The con- fluence of these unusual circumstances raises the broad question of whether SNTE can arise as unusual behav- ior above structural quantum phase transitions (SQPTs) associated with transverse shifts of linking units between volume-defining vertices. To directly address this issue, we have explored the thermal expansion behavior and lat- tice fluctuation spectra in optical/detector quality single crystals of a second stoichiometric SQPT material Hg2I2, known colloquially as protiodide[17]. Figure 1b summarizes the known structural phase di- agram of the mercurous halides Hg2X2 (X=Cl,Br,I) as a function of the X− ionic radius[17], rX . The high sym- metry structure in this case is body-centered tetragonal (BCT; Fig 1e) and can be described as a dense packing ar X iv :1 71 2. 01 44 6v 1 [ co nd -m at .m tr l- sc i] 5 D ec 2 01 7 2 T c (K ) 0 500 1500 1000 0.60 0.65 0.70 0.75 (a) T c (K ) 1.8 1.9 2.0 2.1 2.20 200 (b) X-site Ionic Radius rX (Å)B-site Ionic Radius rB (Å) (e) (f)Cubic trifluoride Rhombohedral trifluoride BCT Mercury Halide Ortho Mercury Halide (c) (d) VF3 FeF3 0.60 0.65 0.70 0.75 0 500 1000 1500 2000 B-site Ionic Radius (Α) Te m pe ra tu re (K ) ScF3TiF3CrF3 Cubic trifluoride Rhombohedral trifluoride Ba aa F F F F F Fluid 1.8 1.9 2.0 2.1 2.2 0 100 200 300 400 500 600 700 X-site Ionic Radius (Α) Te m pe ra tu re (K ) Gas Hg2Cl2 Hg2Br2 Hg2I2 BCT Mercury Halide Orthorhombic Mercury Halide aa Hg Hg c X X X X X X X X 2000 Fluid 400 600 â b̂ ĉ â b̂ ○○○○○○○ 0.60 0.65 0.70 0.75 0 500 1000 1500 2000 B-site Ionic Radius (Α) Te m pe ra tu re (K ) ○ ○ ○ ○ 1.8 1.9 2.0 2.1 2.2 0 100 200 300 400 500 600 700 X-site Ionic Radius (Α) Te m pe ra tu re (K ) FIG. 1: Structural phase diagrams of (a) 3d transition metal trifluorides BF3 and (b) mercurous halides Hg2X2. Open circles represent solid solutions Sc1−xTixF3[18] and Hg2(Br1−xIx)2[19], respectively. Insets show the basic volume-defining polyhedral units: (a) the BF6 octahe- dron and (b) the elongated square dipyramid (ESD). (c-f) Schematic structures of the (c) cubic trifluoride (d) rhombohe- dral trifluoride (e) body-centered tetragonal mercury halide, and (f) orthorhombic mercury halide. The lower panels in (e) and (f) show views down the 001 axis, showing the shift pattern of the Hg dimer across the structural transition and gray line show the BCT unit cell and the black diamond in (f) shows the orthorhombic unit cell. of X-Hg-Hg-X linear molecules oriented along the tetrag- onal c axis[20]. The basic structural unit that defines the high-temperature unit cell is an elongated square dipyra- mid (ESD) formed as a cage with parts of 10 X− ions surrounding a Hg dimer oriented along c, shown in Fig- ure 1b inset and Figure 4b. The structural transition to the orthorhombic phase can be described as a freez- ing of the Hg dimer transverse fluctuation in a staggered pattern (Fig. 1f) at the X point of the BCT Brillouin zone. Single crystals of Hg2I2 and Hg2Br2 were prepared from purified materials using physical vapor deposition as previously reported[17]. Diffraction and inelastic X-ray scattering (IXS) data were collected using the HERIX spectrometer in sector 30 of the Advanced Pho- 0 100 200 300 400 500 0.97 0.98 0.99 1.00 1.01 Temperature (K) R el at iv e un it ce ll di m en si on ScF3 TiF3 (cH) TiF3 (aH) (a) a (T )/ a (5 0 0 ) Temperature (K) 0.98 1.00 0 100 200 300 400 500 Fr ee e ne rg y Staggered angle High symmetry Low symmetry IFE(e) ScF3 0 200 400 600 800 1000 1200 0.998 0.999 1.000 1.001 1.002 1.003 1.004 1.005 Temperature (K) R el at iv e un it ce ll di m en si on 0 200 400 600 800 1000 1200 -15 -10 -5 0 Temperature (K) R el at iv e un it ce ll di m en si on 0 200 400 600 800 1000 1200 -15 -10 -5 0 Temperature (K) R el at iv e un it ce ll di m en si on 0 200 400 600 800 1000 1200 -15 -10 -5 0 Temperature (K) R el at iv e un it ce ll di m en si on a (T )/ a (5 0 0 ) C TE (ppm /K) 0.998 1.000 1.002 1.004 0 400 1200800 0 -5 -10 -15 Temperature (K) Bond length Fr ee e ne rg y (f) ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲▲▲ ▲ ▲ ▲▲ 0 50 100 150 200 250 300 0.9995 0.9996 0.9997 0.9998 0.9999 1.0000 0 50 100 150 200 250 300 -4 -2 0 2 Temperature (K) c( T )/ c( 30 0) ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲▲▲ ▲ ▲ ▲▲ 0 50 100 150 200 250 300 0.9995 0.9996 0.9997 0.9998 0.9999 1.0000 Hg2I2 c C TE (ppm /K) 0 50 100 150 200 250 300 -4 -2 0 2 0.9998 1.000 0.9996 0 2 -2 -4 0 50 100 150 200 300250 (d) ◆ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ ◆ ◆◆◆◆◆◆◆◆◆◆◆ ◆ ◆◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆◆ ◆ ◆ ◆◆◆◆◆◆◆ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ 0 50 100 150 200 250 300 0.985 0.990 0.995 1.000 a (T )/ a (3 0 0 ) 0.995 1.000 0.990 0.985 Hg2I2 a Hg2Br2 a,b (g) 0 50 100 150 200 300250 Temperature (K) c( T )/ c( 30 0) Hg2I2 c ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲▲ ▲ 0 50 100 150 200 250 300 0.95 0.96 0.97 0.98 0.99 1.00 Hg2Br2 c 0 50 100 150 200 300250 Temperature (K) 0.99 1.00 0.98 0.97 0.96 0.95 (b) THREESUB_FIG2 0 200 400 600 800 1000 1200 -15 -10 -5 0 (c) FIG. 2: Temperature-dependent lattice parameters of (a,c) ScF3 and TiF3 from references [10, 12, 21], and (b,d,f) Hg2I2, and Hg2Br2. The mercurous halide c axis parameters were collected from the 004 and 008 reflections and the a,b param- eters are derived from the 300 and 030 twin reflections. The panel (e) shows a schematic free energy landscape of high, low, and transitioning systems near the SQPT. (f) Shows an intermolecular potential appropriate for bond-stretch coordi- nates. The mean separation is indicated by a dashed curve, illustrating the positive thermal expansion effect of this type of excitation. Horizontal arrows in (e) and (f) show the fluc- tuation domain in each case. (g) Planar lattice parameters for Hg2Br2 and Hg2I2. ton Source, Argonne National Laboratory. Figure 2 shows the lattice parameters of ScF3[10] and Hg2I2 along with data from their nearest realized structural tran- sitions in TiF3[18] and Hg2Br2, respectively, for com- parison. In the case of TiF3 (Fig 2a), the cubic-to- rhombohedral transition has profound effects on the lat- tice dimensions, showing a signature step of a first or- der transition and subsequent continuous order parame- ter development, shrinking ∼3% between the structural transition temperature Tc and base temperature[21]. The hexagonal c axis on the other hand displays SNTE sim- ilar in magnitude to the negative CTE observed along the cubic a axis of ScF3 in the same temperature range. The lattice expansion of the realized transition in Hg2Br2 bears remarkable resemblance, with a significant reduc- tion in tetragonal c axis lattice parameter below the tran- sition, changing 4.5% between 150K and base tempera- ture. The transition appears to be nearly continuous, consistent with prior work[22]. The c axis of Hg2I2 how- ever shows a strongly negative CTE below 100K shown in Figure 2d, reaching a significant low-temperature value of -5ppm/K. While negative values of the c-axis compress- ibility have been noted in mercurous halides[23], SNTE in Hg2I2 has not been reported to our knowledge. The similarities to the fluorides are striking - both real- 3 ized transitions TiF3 and Hg2Br2 show a strong positive CTE below their respective transitions while the systems tuned near SQPTs show SNTE in high symmetry di- rections aligned with a linkage undergoing strong trans- verse fluctuations at the lowest temperatures. Beyond the phase diagram and structural motifs, we note other gross similarities between these two material classes: they are ionic insulators with no reported magnetism or free carriers and retain high symmetry structures at all temperatures below their respective melting temper- atures. Furthermore, both systems are driven to lower symmetry phases at modest pressure and ambient tem- perature (ScF3 cubic-to-rhombohedral at pc=7kbar and 300K[10]; Hg2I2 body-center-tetragonal-to-orthorhombic at pc=9kbar and 300K[24]) or very low pressure at cryo- genic temperatures (estimated pc∼1kbar for both sys- tems at 4K[10, 25]). We note also analogous structural motifs: the bridging F ion in ScF3, which occurs in all three spatial directions and the bridging Hg dimer, which is oriented along the c axis in the mercurous halides. Figs 2c,d show remarkable similarities in the functional form of the CTE of Hg2I2 and ScF3, which strengthen significantly at low temperature, implying the relevant lattice modes lie at extremely low energy. Apart from exceptional cases[26, 27], the influence of higher-energy bond-stretch excitations are expected to provide positive thermal expansion influences which generically compete with SNTE due to the short-range hardening and long- range softening of central force interionic potentials (Fig. 2f)[28, 29], which explains the overwhelming prevalence of positive thermal expansion in materials at ambient temperature. For Hg2I2, the magnitude of the c-CTE (-5ppm/K) is about one third of the maximum a-CTE in ScF3 (-14ppm/K) and the range of thermal persis- tence is also reduced about 10-fold. This suppression can be partially explained by the much larger mass in the iodide case (ScF3 density 2.53g/cm3; Hg2I2 density 7.7g/cm3), which significantly reduces the phonon band- width (∼140meV in ScF3[15] and ∼25meV in Hg2I2[24]) and activation energy for bond-stretch dynamics which compete with and ultimately overtake the SNTE ef- fect. We point out further that quenched disorder is known to compete with SNTE[30], and while we have restricted our attention to pure stoichiometric systems, the halide system is likely more disordered than the flu- oride based on comparison of the rocking curve width (0.002◦ for ScF3 and 0.13◦ for Hg2I2; see Supplemental Materials). A recent study of a SQPT achieved through chemical substitution shows prominent elastic and ther- modynamic anomalies near the SQPT in solid solution LaCu6−xAux[31] but does not report SNTE near the crit- ical composition. In contrast, both ScF3 and Hg2I2 lie very close to the critical endpoint of a structural phase boundary without the additional detrimental effects of quenched disorder. Figure 3 provides an experimental basis to demon- strate proximities to SQPTs in ScF3 and Hg2I2, show- ing inelastic X-ray scattering spectra at the momentum IXS surface plot Energy Transfer (eV) IX S S ig n al (a b u ) 150 �2 0 2 R R ! 0 -22 200 100 0 T (K) ScF3 Hg2I2 ! X (a) (b) (c) (d) -50 0 50 100 150 200 250 300 0 2 4 6 8 10 12 Temperature(K) ω 2 ( m eV 2 ) with error 0 50 100 150 0.0 0.1 0.2 0.3 0.4 0.5 Temperature(K) ω 2 ( m eV 2 ) S(R,\omega) (arb) ScF3 Hg2I2 12 8 4 0 0 50 100 150 200 250 300 0 50 100 150 0.2 0 0.4R X Tc = �39K T (K) T (K) Tc = �23K (~ ! )2 (m eV 2 ) (~ ! )2 (m eV 2 ) ~!(meV ) 100 50 0 T (K) 1500 -55 ~!(meV ) S(R,!)(arb) S(X,!)(arb) FIG. 3: (a) Dynamical structure factor for ScF3 collected using inelastic X-ray scattering at the (2.5,3.5,0.5) recipro- cal vector, corresponding to the R point of the simple cubic Brillouin zone, shown as an inset. (b) Dynamical structure factor for Hg2I2 collected using inelastic X-ray scattering at the (2.5,3.5,0) reciprocal vector, corresponding to the X point of the body-centered tetragonal Brillouin zone, shown as an inset. (c) Soft mode frequency squared determined from fits to the data in (a), showing the incipient ferroelastic state un- derlies the SNTE effect in ScF3. (d) (d) Soft mode frequency squared determined from fits to the data in (b), showing the incipient ferroelastic state also underlies the SNTE effect in Hg2I2. points corresponding to the soft mode instabilities. The trifluoride low-temperature phase can be described as a staggered tilt of octahedra around the 111 axis (Figure 1f). This fluctuation has the (π,π,π) spatial texture of the R point in the simple cubic Brillouin zone, shown in Figure 3a. Also shown is a surface plot of the dynamical structure factor S(R,ω) obtained using IXS[12]. At high temperature, a Stokes and anti-Stokes mode at low fre- quency of 3.4 meV softens considerably, approaching an extrapolated transition temperature Tc'-39 K, as shown in Figure 3c. This singular point in the response function is suggestive of a flattening of the free energy landscape in an approach to an unrealized structural phase transi- tion and strongly supports our identification of ScF3 as near a SQPT. In the halide case, starting from the BCT phase, the relevant distortion to the low-temperature orthorhom- bic phase is a staggered shift of the Hg-Hg dimer from the ESD central axis in the 110 direction of the basal plane (Figure 1f) and the structural transition then cor- responds to condensation of the transverse acoustic wave at the X point of the BCT Brillouin zone[22, 32, 33]. 4 Figure 3b,d shows the evolution of S(X,ω) with temper- ature and an avoided condensation of a soft mode at the X point of the body-centered tetragonal Brillouin zone in Hg2I2. S(X,ω) also shows Stokes/anti-Stokes mode pairs that imply a putative transition at Tc'-23 K. This is fully consistent with previous studies on single crystals using energy-integrated diffuse X-ray scattering showing Tc ∼-20 K[34]. We explore more detailed analysis of the influences driving these incipient structural transitions in each case below. For the realized structural transitions in the mercurous halides X=Cl,Br[22, 34] Tc >0 and the T=0 energy land- scape consists of four minima corresponding to the pos- sible saturated shifts of the Hg dimers (Fig 4d). For larger ionic radius X=I, these minima flatten and co- alesce to the central axis, as no symmetry breaking is observed. The flattening of the energetic landscape how- ever induces strong temperature-dependent fluctuations of the staggered shift, as is observed experimentally from X-ray diffraction data as large transverse dimensions of the Hg thermal ellipsoids, approaching 2 √ U11=0.38Å at T=150 K for Hg2I2[35]. To gain further insight into the origin of the avoided transition in Hg2I2, we analyze trends among the bond lengths of the ESD, which is not regular, but rather com- pressed (Fig 1b inset), defined by X-X bond lengths of three varieties: the long X-X bonds (green) lying in the basal plane, the short X-X bonds (blue) oriented along the tetragonal c axis, and apical X-X bonds (red). Figure 4b shows reported literature values of the lengths of the three types of X-X bond distance[20, 35–39]. The basal planar (green) X-X bonds cluster well[62] with a clear trend far in excess of the diagonal dashed line that indi- cates simplest expectation based on sphere packing. The apical (red) bonds trend with ionic radius and stay near the X ionic diameter, indicating the apical half octahe- dron is flattened relative to a regular ESD but roughly satisfies simple packing conditions. On the contrary, the c-oriented (blue) X-X bond is near the ionic diameter only for X=Cl (calomel) and deviates significantly for X=Br and even more so for the SQPT material X=I, sug- gestive that the compression of the X-X bond plays a role in the energy flattening behind the SQPT and the NTE we report here. These short halide-halide bonds along c are in a state of compression due to forces provided by the rest of the framework[63]. The natural source of this compression is a net tension in the X-Hg-Hg-X linear molecule along the c axis through the ESD center. We propose that the origin of the coalescing energy minima can be viewed as an effect of two competing forces: the compressive stress on the c-oriented X-X bonds and the tensile stress on the X-Hg-Hg-X linear molecule by its en- vironment. For X=I, these forces have a near-canceling effect which stabilizes the high-symmetry BCT structure with large transverse dimer fluctuations inside the ESD and the corresponding c-axis SNTE. Our identification of the compressive/tensile balance in Hg2X2 constitutes a context for the 1D “tension effect”[8, 14, 40, 41], which 3.5 4.0 4.5 5.0 2.5 3.0 3.5 4.0 4.5 5.0 2×Effective Ionic Radius (Å) In te ra to m ic D is ta nc e (Å ) (c) Hg-Hg X-X(c) X-X(apex) X-X(a) d=2rX 3.5 4.0 5.04.5 4.5 5.0 4.0 3.5 3.0 2.5 In te ra to m ic d is ta nc e d (Å ) 2rX (Å) d=2rHg d=2(rF+rB) d=2(rF+rB) (Å) In te ra to m ic d is ta nc e d (Å ) 4.5 4.0 3.5 3.0 2.5 3.9 4.0 4.1 4.23.8 (d) d=2rF F-F F-B-F ●● ●●● ■■■■■ 3.8 3.9 4.0 4.1 4.2 2.5 3.0 3.5 4.0 4.5 2×Effective Ionic Radius (Å) In te ra to m ic D is ta nc e (Å ) (a) (b)ScF3TiF3CrF3 Hg2Cl2 Hg2Br2 Hg2I2 100 001 110 001 FIG. 4: (a) Development of the structures of the transition metal trifluorides, with purple spheres representing F− and silver spheres representing B+3 ions. (b) Development of the structures of the mercurous halides with purple spheres rep- resenting X− and silver spheres representing Hg− ions. B+3 or X− ionic radius increases from left to right in (a) and (b) respectively. Overlayed lines show the ESD of Figure 1b inset, where solid lines lie in the plane of the page and dashed lines show bonds that protrude out of the plane. (c,d) Compari- son of the observed bond distances and expectations based on hard sphere packing (dashed lines) for (c) transition metal tri- fluorides in the cubic phase[49] and (d) mercurous halides in the BCT phase. Symbols in (d) are from corresponding refer- ences: triangles[38], squares[39], pentagons[37], hexagons[36], septagons[50], and circles[35]. has been observed in cyanides[42, 43] in 1D and 2D. The Hg linking dimer is distinct from the linking CN complex in that it does not set up orientational order known to exist in a broad class of cyanides[44]. Crossing the SQPT from the high symmetry side can be viewed as an increase in the number of zero modes as the potential landscape flattens in certain directions. In the language of structural mechanics, a change in zero mode count must be accompanied by a reduction in the number of states of self stress (SSS) in the ESD poly- hedral unit[45–47], a condition which increases its sta- bility against deformation[48]. For the critical composi- tion X=I, a critically-tensioned linear molecule inside the ESD unit exhibits large transverse fluctuations of the Hg dimer which exert tension on all bonds in the c direc- tion and leads to the observed negative thermal expan- sion along this axis which increases as the temperature is lowered (Fig. 2d). This size-induced stiffening is also apparent in the lowering of the melting temperature and triple points as the X ion grows, since the entropy of the BCT phase is reduced by the stiffening of the ESD, enabling the fluid phase to persist at lower temperature. 5 In a similar set of considerations, we analyze the evo- lution of the basic trifluoride octahedral subunit with rB , used as a baseline for comparison of relative size effects. Figure 4a shows that to a good approximation, the unit cell dimension F-B-F ionic distance (a lattice parame- ter) trends with rF +rB while the nearest F-F distance grows to values significantly in excess of 2rF . We note further that for rB>0.75Å in the rare-earth class and be- yond, trifluorides take on other crystal structures with B site coordination larger than n=6 in the orthorhom- bic tysonite (n=8), and hexagonal (n=9)[51] structures. Together, these observations suggest an effective reduc- tion of the overall stiffness of the BF6 octahedron for large B ions through reduced interaction of the anions situated at the octahedral vertices. In this case, a large number of SSSs are removed as the octahedral unit loses integrity with increasing rB . Notably, fluctuations of the F− position transverse to the B-F-B bond are very large in ScF3, approaching 2 √ U33=0.24Å[11, 14, 15] at T=150 K, consistent with this picture. In contrast to the mech- anism in the mercurous iodide, we expect that the onset of pliancy of octahedral molecules stabilizes the cubic phase, as many states are available with the average cu- bic structure. This situation is also manifest in the high melting point of ScF3, where the high entropy of trans- verse bond fluctuation competes with the fluid phase as high as 1800K. We note that incipient lattice instabilities and broadly systems tuned near SQPTs have recently attracted re- newed interest[52] in light of their use to develop a 50% increase[53] in superconducting transition temperature of Nb-doped SrTiO3 in an exceptional limit of the strong coupling theory[54], while the recent observation of elec- tronic coupling to a substrate phonon in FeSe films raises questions regarding the possible role of substrate lat- tice fluctuations in stabilizing film superconductivity[55]. The common appearance of incipient soft modes also in SNTE materials potentially opens promising areas for future work combining SNTE and superconductivity to realize new emergent phases enabled by extremal strain conditions. Our results here suggest renewed importance of spectroscopic studies of known high-symmetry NTE materials such as ZrW2O8[56, 57], Zn(CN)2[58, 59], and Ag3[Co(CN)6][60]. The transition metal trifluoride and mercurous halide materials bear strong similarities besides the unusual strengthening of the SNTE effect at low temperature. In both cases, molecular units form a high-symmetry struc- ture whose bonds are on average straight and are situated so as to define the linear dimensions of the crystal, a in ScF3 and c in Hg2I2. On approach to zero temperature, high-energy bond-stretch excitations become frozen out quickly according to the Bose factor, while the soft mode angular fluctuations reduce more gradually due to the observed ω ∝ √ T dependence. These competing influ- ences lead to trend of NTE strengthening at the low- est temperatures and conventional expansion at elevated temperatures and the soft mode is crucial to boost the weak NTE influence. From this point of view, ScF3 has extremely strong and thermally-persistent NTE behav- ior due both to the stiff bond-stretch, three dimensional lattice system, and proximity to the SQPT. The incipi- ent nature of the transition is vital to this condition to avoid a staggered strain symmetry breaking which dis- rupts the coupling of angle to dimension. We generalize this understanding in a proposal that candidate SNTE materials may be identified in systems which have (i) structural instability associated with a transverse linkage shift between-volume-defining vertices, (ii) soft modes and large fluctuations of the near the SQPT, (iii) low quenched disorder and stoichiometric composition, and (iv) relatively stiff bond-stretch excitations. Acknowledgments Work at the University of Connecticut was provided by National Science Foundation Award No. DMR-1506825 with additional support from the U.S. Department of En- ergy, Office of Science, Office of Basic Energy Sciences, under Award No. de-sc0016481. C.A.O. acknowledges support from the Treibick family scholarship, managed by the Office of Undergraduate Research at the Univer- sity of Connecticut. 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[62] The tight clustering for the X-X (a) bonds is due to the simple relation to structure - this bond distance is sim- ply the a lattice parameter and can be determined from a single reflection and Bragg’s law. The distances for the trifluoride show little scatter for the same reasons. The other internal distances require refinement of many al- lowed Bragg reflections over a larger angular range so show more scatter between reports. http://www.tandfonline.com/doi/abs/10.1080/00150198008207072 http://www.tandfonline.com/doi/abs/10.1080/00150198008207072 http://www.tandfonline.com/doi/abs/10.1080/00150190208260606 http://www.tandfonline.com/doi/abs/10.1080/00150190208260606 http://www.ncbi.nlm.nih.gov/pubmed/22346788 http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=PMC3274835 http://scripts.iucr.org/cgi-bin/paper?S1600536811056339 http://www.ncbi.nlm.nih.gov/pubmed/22346788 http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=PMC3274835 http://scripts.iucr.org/cgi-bin/paper?S1600536811056339 http://www.ncbi.nlm.nih.gov/pubmed/22346788 http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=PMC3274835 http://scripts.iucr.org/cgi-bin/paper?S1600536811056339 http://www.ncbi.nlm.nih.gov/pubmed/22346788 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http://apps.webofknowledge.com/full{_}record.do?product=UA{&}search{_}mode=GeneralSearch{&}qid=36{&}SID=2Fv4pNcCJ8qjseiAchW{&}page=1{&}doc=8 http://scripts.iucr.org/cgi-bin/paper?S0108768102006948 http://scripts.iucr.org/cgi-bin/paper?S0108768102006948 8 [63] This distance is much shorter than typical X-X distances obtained through searches of the international crystal database[61]. Acknowledgments References