Novel Phase Transitions in the Breathing Pyrochlore Lattice: 7Li-NMR on LiInCr4O8 and LiGaCr4O8
Yu Tanaka, Makoto Yoshida, Masashi Takigawa, Yoshihiko Okamoto, Zenji Hiroi
aa r X i v : . [ c ond - m a t . s t r- e l ] N ov Novel Phase Transitions in the Breathing Pyrochlore Lattice: Li-NMR on LiInCr O and LiGaCr O Yu Tanaka, Makoto Yoshida, Masashi Takigawa, Yoshihiko Okamoto, ∗ and Zenji Hiroi Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan (Dated: October 15, 2018)We report Li-NMR studies on LiInCr O and LiGaCr O , in which Cr ions with spin 3/2 forma breathing pyrochlore lattice, a network of tetrahedra with alternating sizes. In LiInCr O withlarge alternation, the nuclear relaxation rate 1/ T shows an activated temperature ( T ) dependencedown to 18 K, indicating a singlet ground state with a spin gap. This behavior, however, is disruptedby an antiferromagnetic (AF) transition at 13 K, which is preceded by another, most likely structural,transition at 16 K. In contrast, LiGaCr O with small alternation shows no spin gap but exhibits afirst-order AF transition over a distributed T -range 13–20 K. Nevertheless, 1/ T of the paramagneticphase diverges toward 13 K, indicating proximity to a second-order transition. The results indicatethat LiGaCr O is located in the vicinity of a tricritical point in the phase diagram. PACS numbers: 75.47.Lx, 75.40.Gb, 76.60.-k
Geometrically frustrated spin systems have been ex-tensively studied from both experimental and theoreti-cal aspects because frustrating interactions prevent con-ventional magnetic order and may lead to an exoticground state. The pyrochlore lattice, a three dimen-sional network of corner-sharing tetrahedra, is a well-known example of frustrated geometry. The ground stateof Heisenberg spins on this lattice with an antiferromag-netic (AF) interaction between nearest neighbors is theo-retically predicted to be a spin liquid without long-rangemagnetic order [1–4]. In real materials, however, an or-dered state can be stabilized by various kinds of pertur-bations [5–12]. Therefore, definite observation of a spinliquid state has not been reported yet. Cr spinel oxides A Cr O ( A = Mg, Zn, Cd, Hg) where Cr ions with spin3/2 form a pyrochlore lattice, have been investigated asgood model materials [13]. Although magnetic orders inthese materials are largely depressed by frustration, si-multaneous AF and structural first-order transitions oc-cur at low temperatures [14–17]. The spin-Jahn-Tellereffect due to spin-lattice coupling has been proposed asthe mechanism for these transitions [6, 7, 18]. The spin-lattice coupling is considered to play important roles alsofor various phases in high magnetic fields including themagnetization plateaus [19–22].Recently, Okamoto et al. reported magnetic proper-ties of a new type of Cr spinel oxides LiInCr O andLiGaCr O [23]. The Li + and In /Ga ions in thesematerials alternately occupy the A -site of A Cr O [24].This results in alternation of the nearest Cr-Cr bondlength in all directions (see the inset of Fig. 1). TheCr spins, therefore, form a breathing pyrochlore latticewith alternation of large and small tetrahedra and twodistinct exchange couplings J and J ′ ( J > J ′ ). Thebreathing factor B f defined as B f = J ′ /J quantifies thedegree of frustration, which is estimated to be 0.1 forLiInCr O and 0.6 for LiGaCr O [23]. The two limitingvalues B f = 0 and 1 correspond to the cases of trivial isolated tetrahedra and strongly frustrated uniform py-rochlore lattice, respectively. Therefore, spin systems onthe breathing pyrochlore lattice with various values of B f provide unique opportunity to tune the frustration andpossibility to discover novel quantum phases. The mag-netic susceptibility χ of LiInCr O shows an activatedtemperature dependence, which can be fit reasonably wellto the result of an isolated tetrahedron with a spin gapof 57 K [23]. In contrast, χ ( T ) of LiGaCr O is similarto that of ZnCr O [23]. The heat capacity divided bytemperature C p /T exhibits a sharp peak at 15.9 K forLiInCr O and at 13.8 K for LiGaCr O , indicating aphase transition in both compounds [23].We have performed Li-NMR measurements on I n t e n s it y ( a . u . ) -400 -200 0 200 400 ∆ f (kHz) Cr J’J
FIG. 1. (Color online) Temperature dependence of the NMRspectra for LiInCr O obtained at 2 T. The vertical scaleis normalized by the peak intensity for each spectrum. Theorigin of the horizontal axis ∆ f = 0 corresponds to the centerof gravity of the spectrum. The inset shows the structure ofa breathing pyrochlore lattice. LiInCr O and LiGaCr O to get microscopic under-standing of these phase transitions. The nuclear spin-lattice relaxation rate 1/ T in LiInCr O exhibits anactivated temperature dependence down to 18 K, sug-gesting a singlet ground state with an excitation gap.This behavior, however, is disrupted by successive phasetransitions. On the other hand, LiGaCr O shows nosign of a spin gap and undergoes a first-order AF transi-tion. However, 1/ T of the paramagnetic phase exhibitsa critical divergence toward 12.8 K as if there were asecond-order magnetic transition. Our results indicatethat LiGaCr O is located in the vicinity of a tricriticalpoint in the phase diagram. This finding opens a newroute to study spin-lattice coupling on a pyrochlore lat-tice.The powder samples of LiInCr O and LiGaCr O were synthesized by the solid state reaction method [23].All the Li-NMR measurements were done in a magneticfield of 2 T. In both compounds, the NMR spectra con-sist of a sharp single line in the paramagnetic state withno quadrupole structure, consistent with the cubic localsymmetry 43 m at the Li site. The hyperfine couplingconstant was estimated to be 0.09 T/ µ B for LiInCr O from the magnetic shift and χ measured above 18 K and0.1 T/ µ B for LiGaCr O above 80 K. The NMR spectrawere obtained by Fourier transforming the spin-echo sig-nal. When the spectra became broad in the AF state, thewhole spectra were constructed by summing the Fouriertransformed spin-echo signals obtained at equally spacedfrequencies. 1/ T was determined by fitting the recov-ery curves of the spin-echo intensity I ( t ) as a functionof the time t after an inversion pulse to the stretched-exponential function I ( t ) = I eq − I exp " − (cid:18) tT (cid:19) β (0 < β ≤ , (1)where I eq is the intensity at the thermal equilibrium and β is the stretch exponent that provides a measure of inho-mogeneous distribution of 1/ T . The case of homogenousrelaxation corresponds to β = 1. For broad spectra, 1/ T was measured at the spectral center.We first discuss the results on LiInCr O . Figure 1shows the temperature dependence of the NMR spectra.The spectrum at 20 K has a sharp line of the param-agnetic phase. With decreasing temperature, the spec-trum broadens continuously below 14 K, indicating anAF transition. The upper panel of Fig. 2 shows the tem-perature dependence of 1/ T for LiInCr O comparedwith the C p /T data reported in [23]. The recovery curvescan be fit well to the single exponential function de-scribed by Eq. (1) with β = 1 above 16 K, indicatinghomogeneous relaxation. Below 16 K, β gradually de-creases and reaches 0.5 at 4.2 K. In the temperaturerange 18–48 K, 1/ T can be fit well to the activation law,1 /T ∝ exp ( − ∆/T ), with the energy gap ∆ = 31 K (the solid line). This result is consistent with the singlet for-mation expected for a breathing pyrochlore lattice withsmall B f . The value ∆ = 31 K is somewhat smaller than,but comparable to, ∆ = 56.8 K estimated by fitting χ ( T )to the model of an isolated tetrahedron [23]. However,1 /T increases suddenly below 16 K and shows a sharppeak at 13 K, indicating an AF transition with criticalslowing down with T N = 13 K.This value of T N is clearly different from the peak tem-perature T P = 15.9 K in C p /T . In general, NMR mea-surements are very sensitive to a magnetic phase transi-tion, but are not sensitive to a structural one if the sam-ple is polycrystalline. Therefore, it is most likely thatthe peak in C p /T corresponds to a structural transition,not a magnetic one, which changes the symmetry of thelattice and thereby releases the frustration. This thenenables a second-order AF transition at a slightly lowertemperature. A preliminary neutron scattering experi-ment indicates symmetry lowering from cubic to proba-bly orthorhombic in the low temperature phase [25], sup-porting our scenario. The small shoulder in C p /T around14 K could be attributed to this AF transition. We alsodiscuss this two-step transition from the viewpoint of thespectral line width in the Supplemental Material [26].We now turn to the results on LiGaCr O . Figure 3(a)shows the temperature dependence of the NMR spec-tra. A sharp line of the paramagnetic phase is observedabove 22 K with the line width comparable to that ofLiInCr O . In the temperature range 13.5–16 K, the / T ( s - )
10 100 T (K) T para1 Ga T AF1 I n t e n s it y ( a . u . ) time (s) C P / T ( J K - m o l C r - ) In FIG. 2. (Color online) Upper panel: Temperature depen-dences of 1/ T measured at 2 T for LiInCr O with a fit tothe activation law and the C p /T data at zero field taken from[23]. Lower panel: Temperature dependences of 1 /T para1 and1 /T AF1 measured at 2 T for LiGaCr O . The shaded area rep-resents the coexistence region determined by Fig. 3(c). Thesolid line is the fit to Eq. (3). The inset shows the recoverycurves at different temperatures. spectrum consists of a sharp paramagnetic line and abroad line originating from an AF phase, indicating co-existence of two phases. The paramagnetic componentcompletely disappears below 13 K. Although the spectraat 13 K and 8 K show a small peak near the center, it islikely from a minor non-magnetic impurity phase. Thebroad spectrum from the AF phase further consists oftwo components: a relatively narrow line whose FWHMis about 1 MHz and a much wider one (the solid anddotted arrows in Fig. 3(a)).Generally, the line width in an AF phase is propor-tional to the magnitude of the ordered moments. There-fore, temperature dependence of the line width providesinformation about how the AF moments develop withtemperature. For instance, the gradual line broadeningobserved in LiInCr O below 14 K (Fig. 1) indicates acontinuous growth of the AF moments associated witha second-order transition. Figure 3(b) shows the broadcomponents of the spectra for LiGaCr O below 16 K.We can clearly see that all the spectra show nearly iden-tical line shape except for small difference in the inten-sity of the wide component. This means that the AFmoments develop discontinuously and the magnitude ofthe ordered moments is independent of temperature oncethe transition has occurred, even though the macroscopicstate is inhomogeneous mixing of the two phases. There-fore, we conclude that the AF transition in LiGaCr O is first-order.The volume fraction of the paramagnetic phasechanges gradually over a rather wide temperature rangeas indicated in Fig. 3(c). Here, the integrated intensityof the sharp paramagnetic line multiplied by temperature I p T , which represents the volume fraction of the param-agnetic phase, is plotted against temperature (red solidcircles and blue open triangles). A small hysteresis is ob-served between the heating and cooling processes, sup-porting the first-order nature of the AF transition. Thegradual change of the paramagnetic volume fraction for13–20 K indicates distribution of the AF transition tem-perature, consistent with the macroscopic coexistence ofthe two phases. The distribution function of the tran-sition temperature obtained by differentiating I p T with T (red crosses) shows a peak near 14 K similar to C p /T reported in [23] (black dots).Next we discuss the 1/ T data for LiGaCr O . Therecovery curves I ( t ) measured at different temperaturesare displayed in the inset of the lower panel of Fig. 2.The single-exponential function described by Eq. (1) with β = 1 gives a good fit above 20 K, where only the para-magnetic component is present. On the other hand, I ( t )in the AF phase below 13 K can be fit to the stretched-exponential function with β ∼ I ( t ) cannot be fit byEq. (1) with any β . Since the paramagnetic and AFphases coexist in this temperature region, we determined1/ T for the paramagnetic phase (1/ T para1 ) and the AF C P / T ( J K - m o l C r - ) I p T ( a . u . ) T (K) (c) I n t e n s it y ( a . u . ) -2 -1 0 1 2 ∆ f (MHz) (a) -3 -2 -1 0 1 2 3 ∆ f (MHz) I n t e n s it y ( a . u . )
16 K 14 K 13 K 8 K (b)
FIG. 3. (Color online) (a) Temperature dependence of theNMR spectra for LiGaCr O obtained at 2 T. The verticalscale is normalized by the peak intensity. The origin of thehorizontal axis ∆ f = 0 corresponds to the center of gravity.(b) The AF components of the spectra below 16 K. The ver-tical scale is normalized by the top of the broad line for eachspectrum, ignoring the sharp paramagnetic component. (c)The solid red circles (open blue triangles) show the intensity ofthe paramagnetic line multiplied by temperature I p T , whichrepresents the paramagnetic volume fraction, measured withheating (cooling). The same data are expanded in the inset.The black dots show the C p /T data at zero field reportedin [23]. The red crosses show d ( I p T ) /dT obtained from thecentral difference of the discrete I p T data. phase (1/ T AF1 ) separately by using a sum of the single-exponential and the stretched-exponential functions, I ( t ) = I eq − I para exp (cid:18) − tT para1 (cid:19) − I AF exp " − (cid:18) tT AF1 (cid:19) β . (2)This equation gives a good fit to the recovery curve in thecoexistence region as displayed in the inset of the lowerpanel of Fig. 2.The lower panel of Fig. 2 shows the temperature de-pendences of 1 /T para1 and 1 /T AF1 for LiGaCr O . 1/ T para1 increases slightly with decreasing temperature at hightemperatures without any sign of a spin gap in contrastto the result in LiInCr O . With further decreasing tem-perature, 1/ T para1 shows a divergence near 13 K, whichindicates critical slowing down of magnetic fluctuations.Such a divergence is usually associated with a second-order magnetic transition and seemingly contradicts ourconclusion for the first-order transition based on the tem-perature dependence of the NMR spectra. The temper-ature dependence of 1/ T para1 can be fit to the function1 T para1 = A √ T − T N + const , (3)derived for unfrustrated antiferromagnets near the tran-sition temperature T N [27]. We obtain T N = 12.8 K asshown by the solid line in the lower panel of Fig. 2.In a pure spin system on a breathing pyrochlore withthe limiting value B f = 0 or 1, a second-order AF tran-sition is not expected. It is an interesting result thatLiGaCr O with B f = 0.6 shows an indication of asecond-order magnetic transition. However, the transi-tion at T N = 12.8 K is not realized, because the para-magnetic component completely disappears just before T N as shown in Fig. 3(c). The observed transition isfirst order, and is probably an AF transition with latticedistortion. Indeed, the recent neutron scattering mea-surements indicate symmetry lowering of the lattice inthe lower temperature phase [25].Our results indicate that LiGaCr O exhibits a first-order AF transition and at the same time, it is closeto a critical point of a second-order transition. Suchbehavior can be most simply understood if the systemis in the vicinity of a tricritical point, which separatesa critical line into continuous and discontinuous regionsin the phase diagram as schematically shown in Fig. 4.Here α represents a phenomenological tuning parameterof the system. For a microscopic spin Hamiltonian thatdescribes the real material, α should be a combinationof the parameters such as anisotropy, spin-lattice cou-pling, and multi-spin exchange. The distribution of theAF transition temperature (∆ T in Fig. 4) observed inour experiments indicates a distribution of α (∆ α ) in thesample as represented by the gray area in the phase di-agram. The origin of the distribution ∆ T or ∆ α is notunderstood yet. It could be due to extrinsic disorder inthe sample.The susceptibility of the order parameter χ m divergesat the tricritical point [28, 29], where 1 /T is expectedto show a divergence due to critical slowing down. Thedistribution ∆ α for LiGaCr O must be located on thefirst-order side. Even if α slightly deviates from the tri-critical point to the first-order side, χ m shows a diver-gence toward a slightly lower temperature than the first-order transition temperature [29]. Therefore, 1 /T is ex-pected to show divergence-like behavior just before thefirst-order transition. The coincidence between T N andthe temperature at which the paramagnetic componentdisappears can be described assuming that the tricriticalpoint is located at the lower boundary of the gray area asshown in Fig. 4. The location of the tricritical point justat the boundary of the distribution may not be a merecoincidence but has some physical mechanism.Strong spin-lattice coupling in A Cr O ( A = Zn, Cd)causes a first-order transition involving simultaneous AFspin order and lattice distortion at 12.5 K (ZnCr O )and 7.8 K (CdCr O ) [15]. The spin-Jahn-Teller effecthas been proposed as the mechanism for these transi-tions. However, the experimental information has beenlimited due to the first-order nature of these transitions.Our results indicate that LiGaCr O also shows a simul-taneous AF and structural phase transition due to strong α T AF para ∆ α ∆ T FIG. 4. Schematic phase diagram for LiGaCr O . On thevertical axis, α represents a phenomenological tuning param-eter of the Hamiltonian. The region ∆ α shows the distribu-tion of α in our sample. The dashed and double lines in-dicate first- and second-order transitions, respectively, whichare separated by the tricritical point indicated by the opencircle. spin-lattice coupling. Furthermore, the transition is veryclose to second-order unlike A Cr O . If we can tune thecontrol parameter α , for example, by applying pressureor chemical doping and make the system cross the con-tinuous critical line, it will allow us to study novel criticalphenomena of a spin-lattice coupled transition in highlyfrustrated magnets. To explore such possibility is a chal-lenging future subject.In summary, we performed Li-NMR measurements ontwo breathing pyrochlore spin systems and showed thatcompletely different magnetic properties are realized de-pending on the degree of breathing. In LiInCr O withlarge breathing, spin gap behavior with ∆ = 31 K wasobserved in 1/ T at high temperature, which, however,is followed by a second-order AF transition at 13 K. TheAF transition is likely to be assisted by a structural tran-sition at a slightly higher temperature that changes thesymmetry of the lattice and releases frustration. In con-trast, LiGaCr O with smaller breathing does not showspin gap behavior but undergoes a first-order magnetictransition with a distribution in the transition temper-ature. A critical divergence of 1/ T in the coexistingparamagnetic phase indicates that LiGaCr O is in thevicinity of a tricritical point.We thank H. Tsunetsugu for stimulating discussionand useful comments on the interpretation of the exper-imental results. The work was supported by Grant-in-Aids for JSPS KAKENHI (B) (No. 25287083). Y. 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SUPPLEMENTAL MATERIALLine width of LiInCr O By taking a closer examination of Fig. 1 in the maintext, we can see that the spectrum broadens in two stepswith decreasing temperature. First only the tails on bothsides of the spectrum become broad with cooling whilethe main peak remains sharp down to 14 K. The width ofthe main peak then increase only below 14 K. Figure S1shows the temperature dependence of the full width halfmaximum (FWHM) and the square loot of the secondmoment M obtained by M = Z ( f − f ) I ( f )d f, (S1)where f is frequency, f is the center of gravity, and I ( f )is the NMR spectrum normalized as R I ( f )d f = 1. Withdecreasing temperature, FWHM and 2 √ M start to grownear 14 K and 18 K, respectively. 2 √ M is very sensitiveto growth of the spectral tails. This result suggests thata two-step transition occurs in LiInCr O , although wecould not determine whether the increase of 2 √ M near18 K arises from a structural transition or inhomogeneousspin freezing. L i n e w i d t h ( k H z ) T (K) √ M FWHM
FIG. S1. (color online) Temperature dependence of the linewidth of NMR spectra of LiInCr O at 2 T. Blue trianglesand red circles represent the FWHM and the 2 √ M2