Softening of breathing elastic mode and trigonal elastic mode in disordered pyrochlore magnet NaCaCo 2 F 7
aa r X i v : . [ c ond - m a t . s t r- e l ] F e b Softening of breathing elastic mode and trigonal elastic mode in disordered pyrochloremagnet NaCaCo F T. Watanabe , ∗ H. Kato , Y. Hara , J. W. Krizan , and R. J. Cava Department of Physics, College of Science and Technology,Nihon University, Chiyoda, Tokyo 101-8308, Japan National Institute of Technology, Ibaraki College, Hitachinaka 312-8508, Japan and Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA (Dated: February 24, 2020)Cobalt pyrochlore fluoride NaCaCo F is a disordered frustrated magnet composed of Co ionswith an effective spin- magnetic moment and exhibits spin freezing below T f ∼ F . The temperaturedependence of the bulk modulus (the breathing elastic mode) exhibits Curie-type softening uponcooling below ∼
20 K down to T f , which is suppressed by the magnetic field. This Curie-type soften-ing should be a precursor to the enhancement of the strength of exchange disorder via the spin-latticecoupling, which causes the spin freezing. In contrast to the magnetic-field-suppressed Curie-typesoftening in the bulk modulus, the trigonal shear modulus exhibits softening with a characteristicminimum upon cooling, which is enhanced by the magnetic field at temperatures below ∼
20 K. Thismagnetic-field-enhanced elastic anomaly in the trigonal shear modulus suggests a coupling of thelattice to the dynamical spin-cluster state. For NaCaCo F , the observed elastic anomalies revealan occurrence of magnetic-field-induced crossover from an isostructural lattice instability towardthe spin freezing to a trigonal lattice instability arising from the emergent dynamical spin-clusterstate. PACS numbers:
I. INTRODUCTION
The antiferromagnetic (AF) pyrochlore lattice ofcorner-sharing tetrahedra is a canonical geometricallyfrustrated lattice [ ]. The prototypical examples for thislattice system are the spinel oxides with the general for-mula AB O and the pyrochlore oxides with A B O ,where, whereas only the B site of AB O forms the py-rochlore lattice, both the A and B sites of A B O in-dependently form the pyrochlore lattice. Many spineland pyrochlore oxides studies have respectively been de-voted to the 3 d transition-metal and the 4 f rare-earthoxides, where the intersite magnetic interaction in the3 d transition-metal oxide is stronger than that in the 4 f rare-earth oxide [ ].The recently discovered pyrochlore fluorides NaSr B F ( B = Mn, Fe, and Co) and NaCa B F ( B = Fe, Co,and Ni) provide a new platform for the study of 3 d transition-metal frustrated magnets, where the large sin-gle crystals are available for experimental studies [ ].In this family, despite the AF Weiss temperature θ W ∼−
140 K (NaCaCo F ) to −
73 K (NaCaFe F ), the spinfreezing occurs at low temperatures below T f ∼ F ) to 3.9 K (NaCaFe F ), indicating thepresence of strong frustration with | θ W /T f | ∼
19 to58 [ ]. For NaSr B F and NaCa B F , whereas themagnetic B ions uniformly occupy the pyrochlore B -sites of the A B F structure, the nonmagnetic Na + and Sr /Ca ions are randomly distributed on the py-rochlore A sites. Thus, the magnetism of NaSr B F andNaCa B F should be influenced by the inherently presentexchange disorder, which can lead to spin freezing [ ].The cobalt pyrochlore fluoride NaCaCo F is a disor- dered frustrated magnet with θ W ∼ −
140 K and T f ∼ . | θ W /T f | ∼
58 among the py-rochlore fluorides NaSr B F and NaCa B F [ ]. Forthis compound, inelastic neutron scattering (INS) experi-ments revealed that the single ion ground state of Co isa Kramers doublet with an XY-like effective spin- mag-netic moment, which is generated by spin-orbit coupling[ ]. Furthermore, the INS experiments in NaCaCo F revealed the formation of static short-range AF clusterswith XY character below T f , and the persistence of a dy-namical spin-cluster state above T f [ ]. The presenceof strong dynamic correlations above T f in NaCaCo F was also evidenced by NMR and ESR studies [ ]. Ad-ditionally, above T f , the ESR experiments suggested thecoexistence of a gapless excitation mode of a cooperativeparamagnetic state and a low-energy gapped excitationmode in the order of sub meV, arising from the strongfrustration [ ].In this paper, we present ultrasound velocity mea-surements of the cobalt pyrochlore fluoride NaCaCo F ,where we determine the elastic moduli of this compound.The sound velocity or the elastic modulus is a usefulprobe enabling symmetry-resolved thermodynamic infor-mation to be extracted from a crystal [ ]. Further-more, as the ultrasound velocity can be measured witha high precision of ∼ ppm, its measurements can sensi-tively probe elastic anomalies driven by phase transition,fluctuations, and excitations [ ]. For the frustrated mag-nets, the ultrasound velocity measurements have provento be a useful tool for studying not only the ground statebut also the excited states [ ]. In the present study onNaCaCo F , we find two different types of elastic anoma-lies at low temperatures above T f , which reveal, respec-tively, an isostructural lattice instability toward the spinfreezing at T f and a trigonal lattice instability arisingfrom the emergent dynamical spin-cluster state. Addi-tionally, the magnetic field dependence of the observedelastic anomalies reveals an occurrence of magnetic-field-induced crossover of the dominant lattice instability fromisostructural to trigonal. II. EXPERIMENTAL
Single crystals of NaCaCo F were prepared by theoptical floating-zone method [ ]. The ultrasound veloci-ties were measured utilizing the phase-comparison tech-nique, where the ultrasound velocity or the elastic modu-lus can be measured with a high precision of ∼ ppm. Forthe measurements, the longitudinal and transverse ultra-sounds at a frequency of 30 MHz were generated anddetected by LiNbO transducers attached on the paral-lel mirror surfaces of the single-crystalline sample. Wemeasured the ultrasound velocities in all the symmet-rically independent elastic moduli in the cubic crystal,specifically, compression modulus C , tetragonal shearmodulus C − C ≡ C t , and trigonal shear modulus C .From C and C t data, we also obtained the bulk mod-ulus C B = C +2 C = C − C t . The respectivemeasurements of C , C t , and C were performed us-ing longitudinal ultrasound with propagation k k [100] andpolarization u k [100], transverse ultrasound with k k [110]and u k [1¯10], and transverse ultrasound with k k [110] and u k [001]. The sound velocities of NaCaCo F measuredat room temperature (300 K) are ∼ C , ∼ C t , and ∼ C . III. RESULTS AND DISCUSSION
Figures 1(a)–1(c) respectively present the temperature( T ) dependence of the elastic moduli C B ( T ), C t ( T ), and C ( T ) with zero magnetic field ( H = 0) in NaCaCo F .Here C B ( T ) = C ( T ) − C t ( T ) is obtained from C ( T )[Fig. 5(a)] (Appendix) and C t ( T ) [Fig. 1(b)]. In Figs.1(a)–1(c), all the elastic moduli exhibit monotonic hard-ening upon cooling down to about 10 −
20 K, as is usuallyobserved in solids [ ]. However, at low temperaturesbelow about 10 −
20 K, the elastic moduli exhibit elastic-mode-dependent unusual softening upon cooling [the in-sets in Figs. 1(a)–1(c)]. Fig. 1(d) compares the softeningmagnitudes in C B ( T ), C t ( T ), and C ( T ) with H = 0.Here it is evident that the softening magnitude in C B ( T )( ∆ C B C B ∼ C t ( T )( ∆ C t C t ∼
800 ppm) and C ( T ) ( ∆ C C ∼
120 ppm).Figures 2(a) and 2(b) respectively depict C B ( T ) and C t ( T ) with H || [110] below 20 K in NaCaCo F . Here C B ( T ) = C ( T ) − C t ( T ) is obtained from C ( T ) [Fig.5(b)] (Appendix) and C t ( T ) [Fig. 2(b)]. At H = 0, C B ( T ) and C t ( T ) exhibit softening upon cooling below H = 0 C ( G P a ) T (K) H = 0 (c) C H = 0 C t ( G P a ) H = 0 (b) C t H = 0 C B ( G P a ) H = 0 (a) C B C B C C t H = 0 (d) D C G / C G T (K) FIG. 1: (Color online) (a)–(c) Elastic moduli of NaCaCo F as functions of T with H = 0. (a) C B ( T ), (b) C t ( T ), and(c) C ( T ). The insets in (a)–(c) respectively depict the ex-panded views of C B ( T ), C t ( T ), and C ( T ) with H = 0 below30 K. (d) Comparison of the relative shifts of C B ( T ), C t ( T ),and C ( T ) with H = 0 below 20 K. ∼
20 K and ∼
15 K, respectively, but turn to hardening be-low ∼ T f ∼ ]. Thus, the softening-to-hardening turn-ing at ∼ C B ( T ) and C t ( T ) with H = 0 shouldbe a result of the spin freezing, and the softening above ∼ H ,as shown in Figs. 2(a) and 2(b) with the dotted ar-rows, suppresses the softening as well as the softening-to-hardening turning in C B ( T ) and C t ( T ), which shouldarise from the suppression of the spin-freezing behaviorby H . At temperatures above ∼
20 K, C B ( T ) and C t ( T )are independent of magnetic field (not shown).For NaCaCo F , taking into consideration that themagnitude of the softening at H = 0 in C B ( T ) is muchlarger than that in C t ( T ) and C ( T ) [Fig. 1(d)], the pre-cursor softening to the spin freezing above T f should becharacterized as a softening in the bulk modulus C B ( T ).As seen in Fig. 2(a), the softening in C B ( T ) with H = 0 behaves as C B ( T ) ∼ − /T . In magnets, such a H || [110] (b) C t T (K) ~
800 ppm ( H = 0) C t ( G P a ) T f T f H = 0 7 T 5 T 3 T 1 T H = 0 H || [110] Eq. (1) (0) C B = 76.9 GPa T c = 0.45 K q = 0.44 K 76.976.876.776.676.5 C B ( G P a ) ~ 4300 ppm ( H = 0) (a) C B FIG. 2: (Color online) (a) C B ( T ) and (b) C t ( T ) ofNaCaCo F with H || [110] below 20 K. The dotted arrowsin (a) and (b) are guides to the eye, indicating the variationsof C B ( T ) and C t ( T ) with increasing H . T f in (a) and (b)indicates the spin-freezing temperature determined from thedc/ac magnetic susceptibility and specific heat measurements[ ]. The solid curve in (a) is a fit of the zero-field experimental C B ( T ) to Eq. (1) below 20 K. The values of the fit param-eters are also listed in (a). The inset pictures in (a) and (b)respectively illustrate schematics of the volume strain in C B and the tetragonal strain in C t . The solid double arrows atthe right side of (a) and (b) respectively indicate the softeningmagnitudes in the zero-field C B ( T ) and C t ( T ). Curie-type softening emerges as a precursor to a struc-tural transition, which is driven by the coupling of the lat-tice to the electronic degrees of freedom [ ].A mean-field expression of the Curie-type softening in thetemperature dependence of the elastic modulus C Γ ( T ) isgiven as C Γ ( T ) = C (0)Γ T − T c T − θ , (1)where C (0)Γ is the background elastic constant, T c is thesecond-order critical temperature for elastic softening C Γ →
0, and θ is the intersite strain-sensitive magneticinteraction. In Fig. 2(a), a fit of the experimental C B ( T )with H = 0 to Eq. (1) for T > T f is drawn as a solidblack curve, which, with the fit parameter values listed inthis figure, excellently reproduces the experimental data.For NaCaCo F , the INS experiments confirmed thatthe single-ion ground state of Co is a Kramers dou-blet with the effective spin- magnetic moment, which isgenerated by spin-orbit coupling [ ]. Thus, one possibleorigin for the Curie-type softening in NaCaCo F is the Jahn–Teller effect, which is driven by the coupling of thelattice to the degenerate single-ion state (the quadrupole-lattice coupling) [ ]. In this scenario, the Curie-typesoftening is a precursor to the lowering of the lattice sym-metry, which lifts the degeneracy of the single-ion groundstate, and this precursor softening should occur in thesymmetry-lowering elastic mode such as, in the cubic lat-tice, C t ( T ) and C ( T ). Thus, as a possible origin for theCurie-type softening in the symmetry-conserving “breath-ing” elastic mode C B ( T ) of NaCaCo F , the Jahn–Tellereffect is ruled out.Consequently, the Curie-type softening in NaCaCo F is most probably explained by assuming a coupling ofultrasound with the magnetic ions through the magne-toelastic coupling acting on the exchange interactions,where the exchange striction arises from an ultrasoundmodulation of the exchange interactions [ ]. Thatis, the softening in NaCaCo F should be driven bythe pseudospin-lattice coupling. One possible origin forsuch a softening is the so-called spin Jahn–Teller ef-fect [ ]. In this scenario, the Curie-type soften-ing is a precursor to the magnetostructural transition,where the spin-lattice coupling lowers the crystal sym-metry resulting in the release of frustration [ ]. Ad-ditionally, this precursor softening should occur in thesymmetry-lowering elastic mode but not in the breathingelastic mode C B ( T ), which is similar to the quadrupolarJahn–Teller effect mentioned in the preceding paragraph.Thus, the Curie-type softening in the breathing elasticmode C B ( T ) of NaCaCo F is uniquely different fromthe Curie-type softening originated from the spin Jahn–Teller effect.For NaCaCo F , the observation of the Curie-typesoftening in the breathing elastic mode C B ( T ) indicatesthe presence of an isostructural lattice instability, whichis a precursor to the spin freezing. Taking into con-sideration the inherent presence of exchange disorder inNaCaCo F due to the random occupation of Na + andCa on the pyrochlore A sites of the A B F struc-ture, the most natural explanation for the isostructurallattice instability is a precursor to the enhancement ofthe strength of exchange disorder via the spin-lattice cou-pling, which causes the spin freezing. Such a unique mag-netoelastic effect in the disordered frustrated magnet isexpected to occur in not only NaCaCo F but also otherfamilies of the disordered pyrochlore fluorides, namely,NaSr B F ( B = Mn, Fe, and Co) and NaCa B F ( B =Fe and Ni), which all exhibit spin freezing at a temper-ature T f much lower than the Weiss temperature | θ W | , | θ W /T f | ∼
19 to 58 [ ].In addition to the above-mentioned Curie-type soft-ening in C B ( T ), for NaCaCo F , we also find anotherkind of intriguing elastic anomaly in the trigonal shearmodulus C ( T ). At zero magnetic field, as already men-tioned in conjunction with Fig. 1(d), the magnitude ofthe softening in C ( T ) is much smaller than that in C B ( T ). However, the application of H enormously en-hances the softening in C ( T ). Figure 3 depicts C ( T ) H = 0 T (K) H || [110] C ( G P a ) T f ~
120 ppm ( H = 0) (b) C ~
140 ppm (1 T) H = 0 1 T 3 T 5 T 7 T H || [110] C ( G P a ) (a) C ~ FIG. 3: (Color online) C ( T ) of NaCaCo F with H || [110]below 20 K. (a) H = 0 ∼ H = 0 and 1 T. Thedotted arrows in (a) and (b) are guides to the eye, indicatingthe variation of C ( T ) with increasing H . The inset picturein (a) illustrates a schematic of the trigonal strain in C . T f in (b) indicates the spin-freezing temperature determinedfrom the dc/ac magnetic susceptibility and specific heat mea-surements [ ]. The solid double arrows at the right side of (a)and (b) respectively indicate the softening magnitudes in the7 T, 1 T, and zero-field C ( T ). The inset in (b) depicts theexpanded view of the zero-field and 1 T C ( T ) in 2 K < T < C ( T ). with H || [110] below 20 K. As shown in Fig. 3(a), the soft-ening in C ( T ) is enhanced by H below ∼
20 K, whichis opposite to the suppression of the softening by H in C B ( T ) and C t ( T ) [Figs. 2(a) and 2(b)]. The soften-ing magnitude in the 7 T C ( T ) ( ∼ C ( T ) ( ∼ H -enhanced soften-ing in C ( T ) should have an origin different from theprecursor to the spin freezing that is the origin of the H -suppressed softening in C B ( T ). At temperatures above ∼
20 K, C ( T ) is independent of magnetic field (notshown).Here, we look more closely at the observed H varia-tion of the softening in C ( T ). Figure 3(b) presents theexpanded view of C ( T ) with H = 0 and 1 T below20 K. In Fig. 3(b), whereas C ( T ) with H = 0 andthat with 1 T identically harden upon cooling down to ∼
12 K, the starting temperature of the softening in the1 T C ( T ) is lowered compared to that in the zero-field C ( T ), which indicates the suppression of the softeningby H [the dotted arrow in Fig. 3(b)]. This H -suppressedsoftening component should correspond to a precursorto the spin freezing, namely, the Curie-type softening, which is similar to the softening in C B ( T ) and C t ( T )[Figs. 2(a) and 2(b)]. However, in Fig. 3(b), the soft-ening in the 1 T C ( T ) below ∼ C ( T ), and the softening magni-tude in the 1 T C ( T ) ( ∼
140 ppm) is larger than thatin the zero-field C ( T ) ( ∼
120 ppm), which indicates thepresence of the H -enhanced softening component. Thus,the comparison of the zero-field and 1 T C ( T ) in Fig.3(b) indicates the presence of not only H -enhanced butalso H -suppressed softening components in C ( T ), andthe H -enhanced component should become dominant athigher H , as is seen in the 3 T, 5 T, and 7 T C ( T ) inFig. 3(a).As seen in Fig. 3(b), C ( T ) with H = 0 and 1 Texhibits its minimum at ∼ C ( T ) in 2 K < T < C ( T ) with H = 0 and that with 1 T are ∼ ∼ T f ∼ C ( T ) should be caused by anorigin other than the spin freezing. At ∼ C ( T )with H = 0 and 1 T exhibits a small slope change [thesolid and open arrows in the inset in Fig. 3(b)], whichshould arise from the spin freezing. Consequently, the H -enhanced softening component in C ( T ) with H = 0and 1 T should be characterized as a nonmonotonic soft-ening, which exhibits the elasticity minimum at around ∼ H -enhanced component becomes dominant in the 3 T, 5 T,and 7 T C ( T ), where the minimum point temperatureis lowered with increasing H .Similar to the H -suppressed Curie-type softening in C B ( T ), the H -enhanced softening with minimum elas-tic anomaly in C ( T ) should also arise via the exchangestriction mechanism, but it should have an origin otherthan the precursor to the spin freezing. The most prob-able origin for the softening with minimum in C ( T ) isthe coupling between the correlated paramagnetic stateand the acoustic phonons. For NaCaCo F , the INS [ ],NMR [ ], and ESR [ ] studies revealed the presenceof short-range dynamical magnetic correlations above T f ∼ T f [ ]. Thus, thesoftening with minimum in C ( T ) should originate from the coupling of the lattice to the dynamical short-rangeXY clusters. It is noted that, similar to NaCaCo F , the softeningwith minimum in C Γ ( T ) is also observed in the frustratedspinel oxides, the origin of which is considered to be thecoupling of the lattice to the spin-cluster excitations viathe exchange striction mechanism [ ]. This spin-cluster-driven elastic softening is generally explained asthe presence of a finite gap for the excitations, which issensitive to strain [ ]. In the mean-field approximation, C Γ ( T ) in the spin-cluster system is written as [ ] C Γ ( T ) = C (0)Γ − G N χ Γ ( T )[1 − K Γ χ Γ ( T )] , (2)where C (0)Γ is the background elastic constant, N is thedensity of spin clusters, G Γ = | ∂ ∆ /∂ǫ Γ | is the couplingconstant for a single spin cluster measuring the strain( ǫ Γ ) dependence of the excitation gap ∆, K Γ is the inter-spin-cluster interaction, and χ Γ ( T ) is the strain suscepti-bility of a single spin cluster. From Eq. (2), when C Γ ( T )strongly couples to the excited state at ∆, this elasticmode exhibits softening upon cooling roughly down to T ∼ ∆ but recovery of the elasticity (hardening) roughlybelow T ∼ ∆; C Γ ( T ) exhibits a minimum roughly at T ∼ ∆.We now analyze the softening with minimum in C ( T )in NaCaCo F using Eq. (2). In the frustrated spineloxides, the observation of the spin-cluster excitations bythe INS experiments revealed that the number of mag-netic ions, shape, and symmetry of spin clusters varyfrom compound to compound depending on the domi-nant exchange path [ ]. For NaCaCo F , the INSstudy revealed that the correlation length of the dynam-ical short-range XY clusters in the paramagnetic phase( ∼ A ) is larger than the length scale of the single Co tetrahedron ( ∼ A ) [ ]. Additionally, the Q -space INSpattern in the paramagnetic phase of NaCaCo F lookslike that of the AF seven-spin clusters (spin heptamers)proposed in the chromite spinel MgCr O with the iso-morphic magnetic pyrochlore lattice as the Co sites inNaCaCo F [ ]. Thus, although the exact shapeof the spin cluster in NaCaCo F has not yet been iden-tified, we assume the AF spin heptamer for the analysis,which consists of two corner-sharing Co tetrahedra [theinset picture in Fig. 4(a)]. We note here that, from thesymmetry point of view, the spin-heptamer excitationsshould couple more sensitively to the trigonal lattice de-formations, which is compatible with the selective ob-servation of the softening with minimum in the trigonalshear modulus C ( T ) in the present study.Figure 4(a) depicts the experimental data of C ( T )with H || [110] in 2 K < T < F , the densityof the spin heptamers is assumed to be N = 2 . × m − [Fig. 4(b)(i)], which is one-seventh of the density ofCo ions in NaCaCo F . The fittings are performed attemperatures below 5 K because, among the experimen-tal data of C ( T ) with H || [110] [Figs. 3(a) and 3(b)],the lowest starting temperature of the softening is ∼ C ( T ) [Fig. 3(b)]. Whereas the lower limitof the temperature range of the fitting is 2 K for the 3T, 5 T, and 7 T C ( T ), that for the zero-field and 1T C ( T ) is the slope-change temperature of T f ∼ C ( G P a ) T (K) 7 T H = 0 5 T 3 T1 T Eq. (2) (a) Heptamer (0) C = 33.5 GPa N = m -3 G ( K ) (ii) G K ( K ) (iii) K m H (T) D ( K ) (iv) D (b) Eq. (2)(i) FIG. 4: (Color online) (a) C ( T ) of NaCaCo F with H || [110] in 2 K < T < C ( T ),where the solid and open arrows respectively indicate theslope-change temperatures in the zero-field and 1 T C ( T ).(b) Fit parameter values for the fit curves in (a). (i) C (0)44 and N , (ii) G , (iii) K , and (iv) ∆ in Eq. (2). The values of G , K , and ∆ are respectively plotted in (ii), (iii), and (iv) asfunctions of H . The dotted lines in (ii)–(iv) indicate linearextrapolations of the plots above µ H = 1 T to H = 0. are in excellent agreement with the experimental data,reproducing the softening with minimum in C ( T ).Figures 4(b)(ii)–(iv) respectively display plots of the fitvalues of G , K , and ∆ in Eq. (2) for the fit curves in Fig.4(a) as functions of H . These plots in Figs. 4(b)(ii)–(iv)respectively exhibit monotonic H variations at µ H = 1T ∼ H = 0 deviate from the re-spective monotonic H variations. These deviations at H = 0 are probably due to the presence of the Curie-typesoftening in the zero-field C ( T ) in addition to the soft-ening with minimum, as was mentioned in conjunctionwith Fig. 3(b). The “correct” values of G , K , and ∆at H = 0 are expected to follow the monotonic H varia-tions at µ H = 1 T ∼ H = 0 and 1 T.As seen in Fig. 4(b)(ii), the intra-heptamer coupling G is enhanced with increasing H , which corresponds to theenhancement of the softening magnitude in C ( T ) by H [Fig. 3(a)]. Accordingly, as seen in Fig. 4(b)(iii), the ap-plication of H also enhances the inter-heptamer antifer-rodistortive interaction K ( < H variations of G and K indicate that the application of H enhances thetrigonal lattice instability, which is driven by the couplingof the lattice to the dynamical spin-cluster state. Here wenote again that, in contrast to the H -enhanced trigonallattice instability [Fig. 3(a)], the application of H sup-presses the spin freezing and its precursor of the isostruc-tural lattice instability, namely, the Curie-type softeningin the breathing elastic mode C B ( T ) [Fig. 2(a)]. ForNaCaCo F , the present study reveals an occurrence of H -induced crossover from an isostructural lattice insta-bility toward the spin freezing to a trigonal lattice insta-bility arising from the emergent dynamical spin-clusterstate. As seen in Fig. 4(b)(iv), the excitation gap of the sin-gle heptamer ∆ decreases with increasing H . From this H variation, assuming the H -linear decrease of ∆, it isexpected that a continuous transition from the gappedground state to the gapless state occurs at ∼
19 T. Onthe other hand, assuming the isolated spin heptamers, itis expected from the gap value of ∆ ∼ H = 0 [thedotted extrapolation line in Fig. 4(b)(iv)] that the tran-sition from the gapped ground state to the gapless stateoccurs at ∼ H dependence of ∆ shownin Fig. 4(b)(iv) indicates the presence of inter-heptamerinteraction, which is compatible with the enhancementof the magnitude of K with increasing H [Fig. 4(b)(iii)].The entire family of the disordered pyrochlore fluoridesNaSr B F ( B = Mn, Fe, and Co) and NaCa B F ( B =Fe, Co, and Ni) exhibits spin freezing, which should be aresult of the inherent presence of exchange disorder [ ].However, as the single-ion state of the pyrochlore B site varies from compound to compound, the correlatedmagnetic state of NaSr B F and NaCa B F should varyfrom compound to compound. For instance, unlike thespin- XY magnet NaCaCo F , the nickel pyrochlore flu-oride NaCaNi F is considered to be a spin-1 Heisenbergmagnet and is suggested to be a three-dimensional quan-tum spin liquid candidate [ ]. Thus, for NaSr B F and NaCa B F , the bulk modulus C B ( T ) of the entirefamily is expected to exhibit the Curie-type of precur-sor softening to spin freezing, but the elastic anomalydriven by the correlated magnetic state is expected tovary from compound to compound, which in the case ofNaCaCo F is the softening with minimum in C ( T ). IV. SUMMARY
Ultrasound velocity measurements of NaCaCo F re-vealed elastic anomalies in the bulk modulus C B ( T ) andthe trigonal shear modulus C ( T ), which are respec-tively suppressed and enhanced by the magnetic field attemperatures below ∼
20 K. These anomalies indicatedthe occurrence of the magnetic-field-induced crossoverfrom the isostructural to the trigonal lattice instabil-ity. The isostructural lattice instability indicated thatthe spin freezing in NaCaCo F is driven by the en-hancement of the strength of exchange disorder via the spin-lattice coupling, which is unique to the disorderedfrustrated magnet. The trigonal lattice instability sug-gested the coupling of the lattice to the dynamical spin-cluster state. It is inferred from the present study thatthe isostructural lattice instability emerges in the entirefamily of disordered pyrochlore fluorides NaSr B F ( B = Mn, Fe, and Co) and NaCa B F ( B = Fe, Co, andNi), which gives rise to the spin freezing. Furthermore,taking into consideration that the single-ion state of thepyrochlore B site in NaSr B F and NaCa B F variesfrom compound to compound, the correlated magneticstate of NaSr B F and NaCa B F should vary from com-pound to compound, which is expected to give rise tocompound-dependent lattice instabilities. V. ACKNOWLEDGMENTS
This work was partly supported by Grant-in-Aid forScientific Research (C) (Grant No. 17K05520) fromMEXT of Japan and by Nihon University College of Sci-ence and Technology Grant-in-Aid for Research. Mate-rials synthesis at Princeton was supported as part of the H = 0 H || [110] C ( G P a ) T (K) ~ H = 0) (b) C T f H = 0 C ( G P a ) T (K) (a) C H = 0 FIG. 5: (Color online) C ( T ) of NaCaCo F with (a) H = 0below 200 K and (b) H || [110] below 20 K. The inset in (a)depicts the expanded view of C ( T ) with H = 0 below 30K. The dotted arrow in (b) is a guide to the eye, indicat-ing the variation of C ( T ) with increasing H . T f in (b)indicates the spin-freezing temperature determined from thedc/ac magnetic susceptibility and specific heat measurements[ ]. The solid double arrow at the right side of (b) indicatesthe softening magnitude in the zero-field C ( T ). Institute for Quantum Matter, an Energy Frontier Re-search Center funded by the U.S. Department of Energy,Office of Science, Basic Energy Sciences under Award No.de-sc0019331.
VI. APPENDIX: RESULTS OF C ( T ) Figures 5(a) and 5(b) respectively depict the compres-sion modulus C ( T ) of NaCaCo F with H = 0 below 200 K and that with H || [110] below 20 K. As seen in Fig.5(a), C ( T ) exhibits monotonic hardening upon coolingdown to ∼
20 K, as is usually observed in solids [ ]. How-ever, at low temperatures below ∼
20 K, C ( T ) exhibitsunusual softening upon cooling [the inset in Fig. 5(a)].As seen in Fig. 5(b), this softening is suppressed by theapplication of H . At temperatures above ∼
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