Negative thermal expansion and magnetoelastic coupling in the breathing pyrochlore lattice material LiGaCr4S8
G. Pokharel, A. F. May, D. S. Parker, S. Calder, G. Ehlers, A. Huq, S. A. J. Kimber, H. Suriya Arachchige, L. Poudel, M. A. McGuire, D. Mandrus, A. D. Christianson
NNegative thermal expansion and magnetoelastic coupling in the breathing pyrochlorelattice material LiGaCr S ∗ G. Pokharel,
1, 2
A. F. May, D. S. Parker, S. Calder, G. Ehlers, A. Huq, S. A. J. Kimber, H.Suria Arachchige,
1, 2
L. Poudel,
1, 2, 5, 6
M. A. McGuire, D. Mandrus,
1, 3, 7 and A. D. Christianson
3, 2, 11
Department of Physics & Astronomy, University of Tennessee, Knoxville, TN 37996, USA Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA Materials Science & Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA Neutron Technologies Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA Department of Materials Science & Engineering, University of Maryland, College Park, MD 20742, USA NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA Department of Material Science & Engineering, University of Tennessee, Knoxville, TN 37996, USA (Dated: April 17, 2018)The physical properties of the spinel LiGaCr S have been studied with neutron diffraction, X-raydiffraction, magnetic susceptibility and heat capacity measurements. The neutron diffraction and syn-chrotron X-ray diffraction data reveal negative thermal expansion (NTE) below 111(4) K. The magneticsusceptibility deviates from Curie-Weiss behavior with the onset of NTE. At low temperature a broadpeak in the magnetic susceptibility at 10.3(3) K is accompanied by the return of normal thermal expan-sion. First principles calculations find a strong coupling between the lattice and the simulated magneticground state. These results indicate strong magnetoelastic coupling in LiGaCr S . I. INTRODUCTION
Breathing pyrochlore lattices have emerged as an im-portant structural motif for the realization of novel quan-tum phases of matter . A breathing pyrochlore lat-tice is an alteration of the pyrochlore lattice consist-ing of alternating large and small tetrahedra (see Fig.1). The modulation of the size can result in dramati-cally different exchange interactions connecting the mag-netic atoms within the differently sized tetrahedra. Re-cently studied examples include Ba Yb Zn O andLi(In,Ga)Cr O . These materials exist in the oppositelimits of the breathing pyrochlore lattice: Ba Yb Zn O is in the noninteracting limit where the individual tetra-hedra are uncoupled, whereas in Li(In,Ga)Cr O the indi-vidual tetrahedra are strongly coupled. The investigationof model systems between these limits is a current chal-lenge.One starting point for the realization of a breathingpyrochlore lattice is the chromium containing chalco-genide compounds, ACr X (X = S, Se). These ma-terials crystallize into a spinel structure with cen-trosymmetric space group Fd ¯ and provide a remark-ably versatile playground to study the physics emerg-ing from frustrated magnetic interactions . In thesematerials, the metal ions A + occupy the tetrahedralsites forming a diamond like structure and the mag-netic ions Cr + occupy the octahedral sites forming apyrochlore lattice with corner sharing Cr tetrahedra.These materials are often multifunctional and exhibitinteresting phenomena such as multiferrocity , largemagnetocapacitance/magnetoresistance , negative ther-mal expansion , helical magnetism , strong magne-toelastic coupling , spin nematics , and spin/orbitalglasses . This rich physical behavior is derived fromthe interplay of spin, charge and lattice degrees of free- FIG. 1. Crystal structure of LiGaCr S depicting the breathingpyrochlore Cr sublattice. The exchange interactions within thesmall and large tetrahedra are labeled J and J (cid:48) respectively. J denotes the next nearest neighbor exchange interaction. dom .Of particular interest here, are Cr-based spinels witha non-magnetic A-site cation which often exhibit strongmagnetoelastic effects . For example, the sourceof structural instability in chalcogenide spinels such asZnCr S and ZnCr Se has been identified as an ef-fect of competing ferromagnetic (FM) and antiferromag-netic (AFM) exchange interactions due to the presenceof strong bond frustration. Additionally, in CdCr O ,and ZnCr Se , the presence of strong magnetoelasticcoupling drives a region of negative thermal expansion(NTE). Hence, one of the motivations of this paper is to a r X i v : . [ c ond - m a t . o t h e r] A p r determine if similar physics is present in structurally-related quaternary spinels that also possess a breathingpyrochlore lattice of the magnetic species.A breathing pyrochlore lattice can be realized throughsubstitution of inequivalent cations on the A-site of theaforementioned family of ternary spinels. This path hasbeen recently explored in Cr-spinel oxides, ACr O , bysubstituting two inequivalent metal ions with differentoxidation states such as Li + and Ga + /In + . When thesubstituted cations are ordered the resulting breathingpyrochlore lattice is described by non-centrosymmetricspace group F ¯ . For example, in LiGaCr O thecations Li + and Ga + alternatively occupy the A-sites ofthe ACr O spinel structure and exert an unequal lo-cal chemical pressure on the Cr tetrahedra . Thus, abreathing pyrochlore lattice forms by the arrangementof larger and smaller Cr tetrahedra in a corner shar-ing network. Oxide breathing pyrochlores have beenreported to order antiferromagnetically at low tempera-ture with complex magnetostructural order and mul-tistage symmetry breaking . Related S-based mate-rials including LiGaCr S were studied by H.L. Pinchet.al who reported that they order antiferromagneti-cally with Neel temperature between 6 K and 31 K, yetmany of the basic physical properties remain unknown.Recently, additional studies of LiInCr S , LiGaCr S andCuInCr S have been reported further demonstratingthe rich physics of breathing pyrochlore lattice systems.In this paper, we study the chalcogenide spinelLiGaCr S , where the A site of spinel structure is oc-cupied by the two metal ions Li + and Ga + . As in theoxide-based spinel LiGaCr O , the metal atoms Li and Gaare ordered and occupy the tetrahedral sites alternatelywith a diamond like arrangement and Cr ions form abreathing pyrochlore lattice as shown in Fig. 1. The pres-ence of two unequally sized Cr tetrahedra yields a dis-tinct nearest neighbor (NN) exchange interaction for each(here denoted by J and J (cid:48) to allow for comparison to nor-mal spinel counterparts). One means of understandingthe degree to which the different sizes of the tetrahedramay modify the physical behavior is through the breath-ing ratio ( B r ), defined as the ratio of the Cr-Cr bond lengthwithin the larger and smaller Cr tetrahedra, B r = d’/d ≥
1. For the A-site ordered quaternary spinels, the breath-ing ratio can be determined from the x -coordinate of theCr + ion via B r = | (2 x − x − | . Importantly, thisshows that the breathing ratio is independent of the lat-tice parameter.We use neutron and X-ray diffraction to study the struc-tural properties of polycrystalline LiGaCr S . We find a B r of 1.077(2) which represents a modest increase overthe value of 1.035(1) for LiGaCr O . Below 111(4) K,we find a region of NTE which extends down to 10 K.This behavior is similar to that observed in the spinelZnCr Se however a similar region of NTE has not beenobserved in related oxide-based breathing pryochloressuch as LiGaCr O . The changes in lattice expansion areaccompanied by the departure from Curie-Weiss behavior ( a ) Y o b s Y c a l Y o b s - Y c a l B r a g g p o s i t i o n
Q ( Å - 1 ) Intensity (arb. unit)
T = 3 0 0 K
Intensity (arb. unit)
T = 3 0 0 K
I n t e n s i t y × 1 0 ( b )
I n t e n s i t y × 1 0
Q ( Å - 1 ) T = 1 . 5 K ( c )
FIG. 2. (a) Synchrotron X-ray diffraction data with λ = S can be comparedto the intensity of the main phase. The impurity phase is in-dicated by arrows. The axis labels are the same as the mainpanel, but note the logarithmic y-axis scale. Neutron diffractiondata collected with POWGEN with 0.167 ≤ λ ≤ F ¯ is indicated by the solid line through thedata. of the magnetic susceptibility and a peak in the magneticsusceptibility and specific heat at 10.3(3) K. This indicatesthat magnetic and lattice degrees of freedom are stronglycoupled in LiGaCr S . Additionally, first principles calcu-lations presented here also find strong coupling betweenmagnetism and the lattice. II. EXPERIMENTAL DETAILS
Polycrystalline samples of LiGaCr S were synthesizedby solid state reaction. Stoichiometric amounts of Ga(99.999%), Li S (99.9%), Cr (99.95%), S (99.9995%), pur-chased from Alfa Aesar, were ground together inside aglove box and then pressed into a 0.5 inch diameter pel-let. The resulting pellet was heated to 1175 K for 3 days.This process was repeated until laboratory X-ray diffrac-tion patterns indicated a phase pure sample.Synchrotron X-ray diffraction measurements were per-formed at 11-BM the Advanced Photon Source (APS)at Argonne National Laboratory (ANL) using X-rays ofwavelength λ = 0.4146 Å. For the measurements, a finelyground polycrystalline sample was packed inside a Kap-ton tube with a diameter of 0.8 mm, which was mountedon the cold finger of an Oxford helium cryostat. The sam-ple was spun at the frequency of 50 Hz to ensure properpowder randomization. The Rietveld refinement pack- H B 2 A d i f f r a c t i o n P O W G E N d i f f r a c t i o n a (Å)
T ( K ) D a (T) / a ( ) T ( K )
FIG. 3. Lattice parameter of LiGaCr S as a function of temper-ature obtained by neutron diffraction. Negative thermal expan-sion is observed below 111(4) K. The inset shows the estimatedtemperature dependence of ∆ a ( T )/ a determined as explainedin the text. ages FullProf and GSAS/EXPGui were used to refinethe crystal structure against the X-ray diffraction data.Neutron diffraction data were collected using the HB-2A powder diffractometer at the High Flux Isotope Re-actor at Oak Ridge National Laboratory (ORNL). Mea-surements were made at HB-2A in the temperature range1.5 K to 300 K using incident neutrons with wavelength, λ = 1.5396 Å. Additional neutron diffraction measure-ments were made with POWGEN at the Spallation Neu-tron Source, ORNL using a band of incident neutrons with0.167 ≤ λ ≤ and GSAS/EXPGui were used to refine thecrystal structure against the neutron diffraction data.Field cooled (FC) and zero field cooled (ZFC) dc sus-ceptibility measurements with applied magnetic fields H ranging from µ O H of 0.01 to 5 T in the temperature range2 K to 300 K were carried out using a Quantum Designmagnetic property measurement system (MPMS). ACmagnetic susceptibility measurements were performed ina Quantum Design physical property measurement sys-tem (PPMS). These measurements were performed witha static field of H dc = 0, using an ac amplitude of H ac =5 Oe. The frequency f dependence of the in-phase compo-nent χ (cid:48) was examined by performing measurements as afunction of f . Data were collected upon cooling from 30 to2 K. Heat capacity was measured using a Quantum De-sign PPMS in the temperature range 2 K to 300 K. Mea-surement of the electrical resistivity was carried out ona sintered sample with a Quantum design PPMS usingthe four-point measurement technique with gold wires at-tached to the sample using silver paint. T ( K )
FWHM 2 q (deg.) FIG. 4. Temperature dependence of peak width (Full WidthHalf Maximum of a standard Lorentzian function) of selecteddiffraction peaks determined from the laboratory X-ray diffrac-tion data. All peaks broaden with cooling at low temperature..
III. RESULTSA. Neutron Diffraction and X-ray Diffraction
We begin with a discussion of the crystallography andlattice behavior of LiGaCr S . Neutron and synchrotronX-ray diffraction patterns are shown in Fig. 2. Unlikethe diffraction pattern corresponding to a normal spinelstructure with space group Fd ¯ (No. 227), additionalreflections, such as 002 are observed as expected for spacegroup F ¯ (No. 216) . The space group F ¯ isa subgroup of space group Fd ¯ . The number of sym-metry operations are reduced by half, including the lossof inversion symmetry in F ¯ compared to the parentspace group. The diffraction patterns are consistent withthose of the oxides of related breathing pyrochlore lat-tice materials LiGaCr O and LiInCr O and studies ofLiGaCr S in Ref. . We find only weak impurity phases:Cr S and an even less significant unidentified phase (seeInset of Fig. 2(a)). We estimate the impurity phase frac-tion to be less than one percent of the main phase basedupon taking ratios of Bragg peak intensities. On the otherhand Ref. 25 finds impurity phases of less than a few per-cent of Cr S and an additional unknown phase.Rietveld refinement of the structural model against theX-ray data, as well as the neutron diffraction data dis-cussed below, confirms that the metal ions Li + and Ga + occupy the A-sites of the spinel structure with occupan-cies close to the ideal values of equivalent sites in thestandard spinel structure (Fd ¯3m). Initial refinements as-sumed these 4a and 4d sites were fully occupied by Li andGa respectively. Better agreement with the data was ob-tained by allowing Li and Ga site interchange, and thisapproach was informed by the complementary nature ofneutron and x-ray diffraction data. The best refinementsindicate the Ga site is fully occupied by Ga and in finalrefinements the occupancy of this site was fixed. TheLi site was found to have some Ga, however, and refine-ments suggest a ≈
5% deficiency of Li in the samples stud-ied here. The refined composition of Li Ga Cr S is obtained from a combined refinement of both the neu-tron and synchrotron x-ray data, and the additional re-finement results are presented in Table I.Initial x-ray diffraction data indicated a region of NTE.To examine this behavior over a broader temperaturerange, neutron powder diffraction measurements wereperformed and the lattice parameter as a function of tem-perature was extracted. The variation of lattice param-eter with temperature is shown in Fig. 3. Cooling fromroom temperature, the lattice parameter decreases un-til 111(4) K. Further cooling results in lattice expansion(NTE) until ≈
10 K where the NTE terminates. Param-eters obtained by a combined refinement of the struc-tural model to the synchrotron X-ray and POWGEN neu-tron diffraction data are shown in Table I (a). The re-fined parameters at 10 K obtained from POWGEN neu-tron diffraction are also shown in Table I (b). No directevidence for a distortion from cubic symmetry was de-tected in any of our diffraction data, including the high-resolution synchrotron data. However, below ∼
200 K atemperature-dependent broadening of the Bragg peakswas observed in the x-ray diffraction data upon cooling(see Fig.4), likely indicating the development of micros-train.This behavior can be compared to other NTE materialssuch as ZnCr Se by comparing the relative change inlattice parameter as a function of temperature, ∆ a ( T )/ a .Here ∆ a ( T )/ a of LiGaCr S has been estimated using acubic spline interpolation of the data. For this estimation,the lattice parameter at 1.5 K is taken as the standard ini-tial value, a . ∆ a ( T )/ a increases up to ∼
12 K. ∆ a ( T )/ a exhibits NTE in the temperature range 12(2) - 111(4) K.The onset of NTE is reflected in the internal degreesof freedom of the unit cell. The effect is evident in bondlengths and bond angles involving Cr. The two Cr-Cr bondlengths show a weak anomaly near the onset of negativethermal expansion (Fig. 5). Changes in the two Cr-S-Crbond angles are also observed (Fig. 6). Both the Cr-Cr dis-tances and Cr-S-Cr bond angles are expected to play animportant role in the magnetic exchange interactions inLiGaCr S and the observations of changes in these quan-tities appear correlated with the changes in magnetic be-havior that are discussed in following sections.As described in the Introduction, the difference be-tween the two Cr-Cr bond lengths is driven by the in-equivalent ionic radii (59 pm and 47 pm ) of the Li + andGa + ions respectively. This difference in bond lengthsis quantified by B r . At 300 K the larger and smallerCr-Cr bond lengths d’ and d extracted from the neutrondiffraction data are found to be 3.655(2) Åand 3.394(2)Åyielding a breathing ratio B r = 1.077(2). The value of B r as a function of temperature is shown in Fig. 5. Thevalue B r in LiGaCr S is somewhat larger than the values TABLE I. (a) Parameters of LiGaCr S obtained from combinedrefinement of the structural model to the synchrotron X-ray andPOWGEN neutron diffraction data.T = 300 K, R p = 6.9 %, R wp = 4.2 % χ = 1.4, a = 9.9696(1) Åatom x = y = z B d (Å ) B nd (Å ) occupancyS ( ) 0.1342(1) 0.69(2) 0.08(2) 1S ( ) 0.616(1) 0.61(2) -0.06(2) 1Cr( ) 0.370(1) 0.60(1) 0.00(2) 1Ga ( ) 0.75 0.63(1) 0 1Li( ) 0 1.24(7) 0 0.953(1)Ga ( ) 0 1.24(7) 0 0.046(1)(b) Refined parameters of LiGaCr S obtained fromPOWGEN neutron diffraction.T = 10 K, R p = 5.8 %, R wp = 1.9 % χ = 1.6, a = 9.9633(3) Åatom x = y = z B d (Å ) B nd (Å ) occupancyS ( ) 0.1345(1) 0.33(2) 0.02(1) 1S ( ) 0.6167(1) 0.25(1) -0.02(1) 1Cr( ) 0.3708(1) 0.29(1) 0.04(1) 1Ga ( ) 0.75 0.23(2) 0 1Li( ) 0 1.10(8) 0 0.953Ga ( ) 0 1.10(8) 0 0.046 B d = Anisotropic diagonal thermal parameter B nd = Anisotropic non-diagonal thermal parameter T ( K )
Cr - Cr bond length ( Å ) Breathing ratio
FIG. 5. Temperature dependent Cr-Cr bond lengths in the smalland large Cr -tetrahedra (filled squares, left scale) and B r (filledcircles, right scale). These bond lengths and B r s are obtainedfrom the neutron diffraction data collected with POWGEN. of 1.035(1) and 1.051(1) for LiGaCr O and CuInCr O respectively . However, the breathing ratio is consid-erably less than in Ba Yb Zn O , where B r =1.90(2) isobserved . The value of B r and the thermodynamicmeasurements presented in the following sections placesLiGaCr S in the interacting limit of the breathing py-rochlore lattice. C r - S 2 - G a 1 S - L i - S , S - G a - S C r - S 2 - C r C r - S 1 - C r
Bond angle (deg.)
T ( K )
FIG. 6. Variation of selected bond angles with temperature inLiGaCr S . Cr-S-Cr bonds are close to 90 ◦ and are responsi-ble for the NN superexchange interaction. S-Li-S and S-Ga-S bonds are insensitive to the change in temperature whereasother bonds show a weak anomaly with the onset of NTE. B. Magnetic Properties
To study the magnetic properties of LiGaCr S ,temperature-dependent magnetization M measurementswere performed under applied fields µ O H ranging from0.01 to 5 T, and the results are summarized in Fig. 7. Acusp in the susceptibility χ = M / H is evident at 10.3(3) K(Fig. 7 (a)), indicating a magnetic transition. The zerofield cooled and field cooled curves bifurcate below thecusp, and the temperature where bifurcation ceases isinversely proportional to applied field (Fig. 7(b)). Thisbehavior is commonly observed in systems with glassyspin dynamics, and such spin freezing transitions are of-ten observed in the spinel family due to the strong frus-tration and competing interactions. The weak hystere-sis observed in the field-dependence of the magnetizationis also consistent with glassy-dynamics below ≈
10 K (in-set, Fig. 7(c)). To verify that the cusp near 10 K is as-sociated with a transition involving glassy spin dynam-ics, we performed a time-dependent measurement below10 K and observed that the remanent moment does in-deed decrease with increasing time. In addition, we haveperformed ac susceptibility measurements which show aweak frequency dependence of the peak in the in-phasepart of the susceptibility, χ (cid:48) (see Fig. 7(d)). Thus, the cuspin χ near 10 K is likely associated with some type of spinfreezing transition, the nature of which is still under in-vestigation.From 150 to 300 K, 1/ χ is linear in T (Fig. 7(c)) and thusthe Curie-Weiss law provides an excellent description ofthe data. However, at temperatures below 125 K, thesusceptibility does not increase as fast as expected fromCurie-Weiss behavior, and this may reflect the increasingimportance of antiferromagnetic correlations upon cool- - 6 - 3 0 3 6- 2 2 0 0- 1 1 0 001 1 0 02 2 0 0 ( d )( b ) Z F CF CF CZ F C m H = 0 . 1 T m H = 1 T m H = 5 T c (cm3/mol-Cr) c (cm3/mol-Cr) FC ( a ) (ii) c- (mol-Cr/cm3) T ( K ) m H = 0 . 1 T ( c )
T = 3 K
T = 2 0 K T = 8 0 K T = 1 5 0 K
M (emu/mol-Cr) m H ( T ) H a c = 5 O e H d c = 0 c ' (cm3/mol) T ( K )
FIG. 7. Magnetic susceptibility measurements of LiGaCr S . (a,b) Show the temperature dependent dc susceptibility, χ and (c)shows the temperature dependent inverse dc susceptibility, 1/ χ of LiGaCr S . (d) Displays the in-phase part of the ac suscepti-bility, χ (cid:48) around the magnetic transition, measured with H ac =5 Oe with frequencies 29 Hz, 127 Hz, 1129 Hz, 2336 Hz, 4832Hz and 10000 Hz. The arrow in (d) indicates the direction of in-creasing applied frequency. The inset of (c) displays M ( H ) loopsmeasured at selected temperatures. A peak in the susceptibil-ity occurs at 10.3(3) K and deviation from Curie-Weiss behavioris observed below 110 K. Small shift in the peak of χ (cid:48) towardshigher temperature is noticed with the increase in frequency.Weak hysteresis in the M vs. H data is observed at 3 K. ing. Importantly, the departure from Curie-Weiss behav-ior appears to coincide with the onset of NTE. Fitting thedata above 150 K to the Curie-Weiss law, χ = C /( T − Θ CW ),produces a Curie constant C that yields an effective mo-ment of µ ef f = 3.96(1) µ B /Cr, and a Weiss Temperature Θ CW = 19.5(1) K. The value of µ ef f is slightly larger thanthe theoretical value of 3.86 µ B for S=3/2. The positive Θ CW indicates ferromagnetic correlations dominate in thehigh- T regime. We note that including a diamagnetic con-tribution in the fitting procedure still results in a posi-tive Θ CW of ∼
10 K. In their studies of LiGaCr S , Ref.25 obtain a negative Θ CW of -20 K. The reason for thedifference with the results presented here is not readilyapparent but may be due to different impurity levels inthe samples or differing amounts of Li-deficiency. The re-sults presented here along with the presence of the shortrange or glassy magnetic order below 10.3 K, indicate sub-stantial competition between AFM and FM interactionsin LiGaCr S . CP ( J mol-1K-1)
T ( K ) ( a )
R l n 4
T ( K ) CP /T (J mol-1Cr-1K-2) S m + S n m C P / T ( b ) S (J mol-1Cr-1K-1) FIG. 8. Temperature dependent heat capacity, C P , of LiGaCr S at zero magnetic field. (a) Displays the C P from 2 to 200 K. (b)Shows C P /T and the estimated entropy, S, at low temperatures.The theoretical value of magnetic entropy for spin 3/2 system isindicated by the horizontal line in (b). The change in slope of C P around at 120 K and a broad hump at 10.9(4) K look coincidentwith the bounding temperatures of NTE. C. Heat Capacity
The temperature dependent specific heat, C P , ofLiGaCr S is displayed in Fig. 8. A broad hump in C P occurs with a maximum near 10.9(4) K. A very weakanomaly is also observed around 120 K. While a feature isvisible, and occurs near the change in thermal expansionbehavior, it is too weak to provide any additional infor-mation. This observation is consistent with the inabilityof our synchrotron diffraction data to detect a structuraltransition near the onset of the negative thermal expan-sion.The anomaly in the specific heat capacity that is asso-ciated with the magnetic transition at low T is best ob-served in a plot of C P /T vs. T (Fig. 8(b)). From these data,the entropy as a function of temperature is obtained byintegrating C P /T dT (8(b)) starting at T = 2 K. Note thatthis estimate of the entropy includes all contributions to C P and represents an upper limit to the magnetic contri-bution to the entropy, S mag (no baseline is utilized as anon-magnetic analogue is not present). In this context,it is noteworthy that the entropy determined is dramat-ically reduced from the value of RLn(4) expected from amagnetic order-disorder transition involving S=3/2 Cr + .This could be caused by the presence of residual entropydue to the glassy phase. However, a substantial portionof the entropy may be lost at higher temperatures, cor-responding to the likely interplay of magnetic and latticedegrees of freedom with an onset near 111 K. r = r o e x p ( - E g / ( 2 K B T ) ) r ( W -m) T ( K )
FIG. 9. Temperature dependent resistivity, ρ of LiGaCr S . Thefitting function is described in the text. D. Electrical Resistivity
The electrical resistivity, ρ of LiGaCr S measured inthe temperature range 195 to 300 K is shown in Fig 9.Below 195 K, the resistance was too large for the instru-ments utilized. From 300 K, the resistivity increases ex-ponentially with decreasing temperature indicating typi-cal semiconducting behavior. To estimate the band gap,E g , the equation ρ ( T ) = ρ exp(-E g /2k B T) is fitted to theresistivity data, where k B is the Boltzmann constant.This yields E g =0.37(3) eV. IV. FIRST PRINCIPLES CALCULATIONS -MAGNETOELASTIC COUPLING
To understand the observed behavior, first princi-ples calculations of the structure and magnetic behav-ior were performed, using the all-electron planewave den-sity functional theory (DFT) code WIEN2K . The gen-eralized gradient approximation of Perdew, Burke andErnzerhof was employed, with an RK max of 7.0. Here RK max is the product of the smallest muffin-tin radius(that for S) and the largest planewave expansion wavevec-tor. We assume an ordered structure with crystallograph-ically separate Li and Ga sites, generally consistent withthe experimental refinements showing only a few percentof Li and Ga mixing on these sites.The experimental results suggest a strong coupling ofmagnetism and structure in LiGaCr S , in particularwith the inverse susceptibility first deviating from linearbehavior at nearly the same temperature as the onset ofthe NTE. While we do not directly address the NTE here,these calculations also find evidence for magnetoelasticcoupling.First principles calculations were performed for twoconfigurations: a non-magnetic configuration, and a sim-ple ferromagnetic (FM) configuration. While the presenceof chromium, the highly electronegative sulfur, and thecomplex geometric frustration may argue against a sim-ple ferromagnetic ground state, for the purposes of dis-cussing magnetoelastic coupling this structure appears tobe sufficient. For both calculations, the internal coordi-nates were optimized, and a cubic lattice parameter of a = 9.9675 Åwas utilized.The optimized FM calculation finds a Cr coordinate of0.3791, which leads to a nearest-neighbor Cr-Cr distanceof 3.408 Å, in excellent agreement (within 0.5 percent)with the experimental value of 3.390 Å. However, theoptimized non-magnetic configuration finds a Cr coordi-nate of 0.3956, and thereby a nearest-neighbor Cr-Cr dis-tance of just 2.86 Å, which differs from the experiment bymore than 0.5 Å. This is more than an order of magni-tude beyond any possible error associated with the inher-ent approximations in DFT. Rather, it is suggestive thateven at 300 K that the magnetism is affecting and pos-sibly determining the structure. While it has long beenknown that signatures of magnetism often persist wellabove the ordering point, a factor of 30 here is unusual.This is likely due to the effective suppression of the order-ing point by geometric and magnetic frustration, as wellas the strength of the magnetic interactions involved. Wefind the ferromagnetic state (with a Cr moment of approx-imately 3 µ B ) to fall some ∼
500 meV per Cr below thenon-magnetic state.This is a relatively large energy compared with the300 K thermal energy k B T of just 25 meV, and speaksto the likelihood of “disordered local moments" persist-ing up to and even beyond room temperature. By thiswe mean that while long-range magnetic order is absentabove the ordering temperature, the individual Cr atomslikely carry a moment of approximately 3 µ B , but thesemoments are essentially randomly oriented in spatial di-rection, with relatively little moment direction correlationbetween neighboring Cr atoms. Detailed discussions ofthis scenario applied to iron can be found in Ref. . In thiscontext the observed anomalous structural behavior, suchas the NTE, can be considered to arise from the detailedtemperature dependence of the ordering energy (via com-peting interactions) and its interactions with the lattice. V. DISCUSSION
As described above, the magnetic properties ofLiGaCr S are strongly reflected in the behavior of thelattice. NTE appears when the susceptibility deviatesfrom Curie-Weiss behavior. The lattice continues to ex-pand with decreasing temperature until the magnetictransition at 10 K after which the lattice contracts tothe lowest temperatures measured (1.5 K). These obser-vations along with the first principles calculations pre-sented in Sec. IV are compelling evidence that magne-toelastic coupling is strong in LiGaCr S . As noted aboveneither the neutron diffraction data nor the synchrotronX-ray data provide direct evidence of a departure from cu- bic symmetry. This appears to be one of the distinctionsbetween LiGaCr S and LiGaCr O , where two differentphases either tetragonal or orthorhombic along with a cu-bic phase are reported below the ordering temperature of12 K .The existence of NTE in magnetically frustrated Cr-spinel compounds has been explained as a consequence ofmagnetoelastic coupling . This mechanism relies oncompeting AFM and FM exchange interactions . Thissituation is likely realized in LiGaCr S as the compe-tition between antiferromagnetic and ferromagnetic ex-change interactions is evident in the magnetic properties.The bond angle for the Cr-S-Cr nearest neighbor superex-change paths are close to 90 ◦ (see Fig. 6) which can yielda ferromagnetic exchange interaction. This is in accordwith the fitting of χ with the Curie-Weiss law in the para-magnetic region from 150 to 300 K, which yields a pos-itive Θ CW =19.5 K. This indicates that ferromagnetic in-teractions dominate the magnetic properties at high tem-perature. This is another distinction between the oxidecounterparts of LiGaCr S where antiferromagnetic cor-relations dominate at high temperature as evidenced bythe reported Θ CW values of -658.8 K and -331.9 K forLiGaCr O and LiInCr O respectively .At lower temperatures, antiferromagnetic correlationsbecome important in LiGaCr S . In particular, the mag-netic phase transition at ∼
10 K exhibits glassy behaviorthat would only be expected with the presence of antifer-romagnetic couplings, and the deviation from Curie-Weissbehavior near the onset of NTE can be considered to haveAFM-like character (relative decrease in χ ). We note thatthe glassy behavior could be promoted by the deviationfrom perfect LiGaCr S stoichiometry, which was demon-strated via diffraction data that revealed our samples pos-sess excess Ga.There are at least two possible sources for antiferro-magnetic exchange in Cr-spinel compounds . One isdirect Cr-Cr exchange, which can be relevant in oxideswhere significantly smaller Cr-Cr distances are found. InLiGaCr S the minimum Cr-Cr distance is 3.39 Å andthus direct exchange is not likely to be the dominatesource of antiferromagnetic interactions. A second andlikely more relevant set of interactions are between 2ndand 3rd nearest neighbors . In contrast to J and J (cid:48) where there are 3 NN, there are 12 NNN each for J and J . Inelastic neutron scattering experiments could be use-ful to provide greater details to the various terms in thespin Hamiltonian.Finally, we comment on the nature of the low tem-perature phase. Glassy dynamics likely plays an impor-tant role in the low temperature physical properties asevidenced from the following experimental observations:The bifurcation of the field cooled and zero field cooledsusceptibility and the weak frequency dependence of theac susceptibility. Likewise, the observed hysteresis in thefield-dependence of the magnetization is likely associatedwith time-dependent relaxation of the moments. Addi-tionally, the anomaly in the heat capacity is rather broadand analysis of this data result in a small fraction of theentropy expected for a S = VI. CONCLUSION
In summary, we find that the chalcogenide spinelLiGaCr S forms a breathing pyrochlore lattice with thetetrahedral A sites alternately occupied by metal ions Li + and Ga + . Negative thermal expansion is observed from10 to 110 K and appears along with the deviation fromCurie-Weiss behavior of the magnetic susceptibility. Atlower temperatures a magnetic transition to a phase withslow dynamics occurs at 10.3(3) K and is accompaniedby the return to normal thermal expansion. Togetherthese experimental observations along with first princi-ples calculations point to strong magnetoelastic coupling in LiGaCr S . ACKNOWLEDGMENTS
We thank C. Batista for useful discussions and S.Lapidus for help with the x-ray diffraction measure-ments. ADC, AFM, MAM and DSP were supported bythe U.S. Department of Energy, Office of Science, BasicEnergy Sciences, Materials Sciences and Engineering Di-vision. GP and DM acknowledge support from Gordonand Betty Moore Foundation’s EPiQS Initiative throughGrant GBMF4416. This research used resources at theSpallation Neutron Source and the High Flux Isotope Re-actor, a Department of Energy (DOE) Office of ScienceUser Facility operated by Oak Ridge National Laboratory(ORNL). Use of the Advanced Photon Source at ArgonneNational Laboratory was supported by the U. S. Depart-ment of Energy, Office of Science, Office of Basic EnergySciences, under Contract No. DE-AC02-06CH11357.
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