Formation of short-range magnetic order and avoided ferromagnetic quantum criticality in pressurized LaCrGe 3
Elena Gati, John M. Wilde, Rustem Khasanov, Li Xiang, Sachith Dissanayake, Ritu Gupta, Masaaki Matsuda, Feng Ye, Bianca Haberl, Udhara Kaluarachchi, Robert J. McQueeney, Andreas Kreyssig, Sergey L. Bud'ko, Paul C. Canfield
FFormation of short-range magnetic order and avoided ferromagnetic quantumcriticality in pressurized LaCrGe Elena Gati , , John M. Wilde , , Rustem Khasanov , Li Xiang , , Sachith Dissanayake ,Ritu Gupta , Masaaki Matsuda , Feng Ye , Bianca Haberl , Udhara Kaluarachchi , ,Robert J. McQueeney , , Andreas Kreyssig , , Sergey L. Bud’ko , , and Paul C. Canfield , Ames Laboratory, US Department of Energy, Iowa State University, Ames, IA, USA Department of Physics and Astronomy, Iowa State University, Ames, IA, USA Laboratory for Muon Spin Spectroscopy, Paul Scherrer Institute, Villigen PSI, Switzerland and Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, TN, USA (Dated: November 10, 2020)LaCrGe has attracted attention as a paradigm example of the avoidance of ferromagnetic (FM)quantum criticality in an itinerant magnet. By combining thermodynamic, transport, x-ray andneutron scattering as well as µ SR measurements, we refined the temperature-pressure phase dia-gram of LaCrGe . We provide thermodynamic evidence (i) for the first-order character of the FMtransition when it is suppressed to low temperatures and (ii) for the formation of new phases at highpressures. From our microscopic data, we infer that short-range FM ordered clusters exist in thesehigh-pressure phases. These results suggest that LaCrGe is a rare example, which fills the gapbetween the two extreme limits of avoided FM quantum criticality in clean and strongly disorderedmetals. PACS numbers: xxx
The fluctuations, associated with quantum-criticalpoints (QCP), i.e., second-order phase transitions at zerotemperature ( T ), have been considered as crucial forthe stabilization of intriguing phenomena, such as super-conductivity or non-Fermi liquid behavior . This moti-vates the search for novel states by tuning a magneticphase transition to T = 0 K by external parame-ters, such as physical pressure, p , or chemical substitu-tion. Whereas for antiferromagnetic (AFM) transitionsthere is a large body of experimental evidence that aQCP can be accessed in metals, e.g., in heavy-fermionsystems or in iron-based superconductors , the ferro-magnetic (FM) transition in clean, metallic magnets is fundamentally different. Generic considerations suggest that the QCP is avoided when a second-orderparamagnetic (PM)-FM transition in a clean, metallicsystem is suppressed to lower T (with the exceptionof non-centrosymmetric metals with strong spin-orbitcoupling ). The predicted outcomes are generally ei-ther (i) that the PM-FM transition becomes a first-orderquantum-phase transition, or (ii) that a new groundstate, such as a long-wavelength AFM state (denotedby AFM q ), intervenes the FM QCP. Experimentally, thefirst scenario was verified in a variety of systems ,whereas the second scenario has so far been discussedfor only a small number of systems. Among thoseare CeRuPO , PrPtAl , MnP , Nb − y Fe y andLaCrGe .For understanding the avoided criticality in clean metal-lic FM systems, LaCrGe turns out to be an impor-tant reference system . First, LaCrGe is a simple3 d electron system with simple FM structure at ambi-ent p . Second, the FM transition can be tuned by p tolower T without changing the level of disorder. Third, earlier studies suggested that the FM transition inLaCrGe becomes first order at a tricritical point ,but also indicated the emergence of a new phase above ≈ . q phase.Motivated by identifying the nature of the various phasesin LaCrGe across the avoided FM QCP region, wepresent an extensive study of thermodynamic, transport,x-ray diffraction, neutron scattering and muon-spin res-onance ( µ SR) experiments (see SI for experimentaldetails). We provide thermodynamic evidence that (i)as the FM transition is monotonically suppressed withincreasing p , the FM transition becomes first order at p tr ≈ T and T , thatare very close in T , emerge for higher p , signaling the oc-currence of new phases in the vicinity of the avoided FMQCP. We demonstrate that below T the magnetic vol-ume fraction is strongly T dependent. At the same time,our results indicate that even below T < T the full-volume magnetism is not long-range ordered and is char-acterized by a remanent magnetization. These resultsquestion the existence of a long-range ordered AFM q phase line emerging near the boundary of the first-orderFM transition line in LaCrGe . Instead, the resultingphase diagram shows features of a subtle interplay ofcompeting magnetic interactions and weak disorder closeto the avoided FM QCP.Figure 1 shows representative data sets of the anomalouscontribution to specific heat (∆ C/T ) (the term “anoma-lous” indicates that data were corrected for a backgroundcontribution, see SI ), the anomalous contribution tothe thermal expansion coefficient (∆ α i with i = ab, c ),the c axis resistance ( R c ), the integrated neutron inten-sity of the (1 0 0) Bragg peak ( I ) and the c lattice a r X i v : . [ c ond - m a t . s t r- e l ] N ov
051 0 0 2 0 4 0 6 0 8 0 1 0 0 1 2 05 . 7 45 . 7 5 T T ( f )( e )( d ) ( c )
0 G P a1 . 9 2 G P a D C / T (a.u.)
0 G P a T F M Da ab (a.u.) Da c (a.u.) d Rc /d T (a.u.) I (100) (a.u.) c (Å) T ( K ) ( b )( a ) FIG. 1. Thermodynamic, transport and diffraction data ofLaCrGe for low pressures (close to p ≈ p ≈ . T ;(a) Anomalous contribution to the specific heat, ∆ C/T ; (b,c)Anomalous contribution to the thermal expansion coefficientalong the ab axes, α ab , and the c axis, α c ; (d) T -derivative ofthe resistance along the c axis, d R c /d T ; (e) Integrated inten-sity of the (1 0 0) neutron-diffraction Bragg peak (nuclear andmagnetic contributions); (f) c axis lattice parameters from x-ray (0 GPa) and neutron (1.9 GPa) diffraction experiments.The arrows indicate the position of various anomalies at T FM , T and T . Insets in (b,c,d,f) show the high- p data sets onenlarged scales around T . parameter for p < p tr and p > p tr .For p ≈ − .
21 GPa < p tr , we find clear anomalies at T FM (cid:39)
90 K (see blue arrows) that are consistent withFM ordering with moments aligned along the c axis ,as suggested by the increase of the I intensity. Themean-field type thermodynamic signatures are consistentwith a second-order phase transition. Notably, the tran-sition is accompanied by sizable lattice changes, as evi-dent from the evolution of α i ( i = ab, c ) and the c latticeparameter. Specifically, the in-plane a axis (the out-of-plane c axis) decreases (increases) upon entering the FMstate.For p ≈ . > p tr , our collection of data showanomalies at three characteristic temperatures. Uponcooling, a clear anomaly occurs in ∆ C/T and d R c /d T at T (cid:39)
60 K, together with small, but resolvable changesof the lattice in a and c direction. Interestingly, theanisotropic response of the crystal lattice, α ab and α c , at T is similar to the one at T FM , albeit much smaller insize, i.e., we find a contraction (expansion) along the a ( c ) axis upon cooling through T . At T (cid:39)
50 K, another T T T ( K ) p (GPa) T FM ( p tr ,T tr ) α c (T,p) α ab (T,p) μ SR C ( T , p ) R ( T,p)I (100) c lattice parameter s t o r de r nd o r de r abc PMPM
FIG. 2. Temperature-pressure ( T - p ) phase diagram ofLaCrGe , constructed from specific heat, thermal expansion,resistance, neutron scattering and µ SR measurements. Linesare a guide to the eye. The blue-shaded region correspondsto the region of ferromagnetic (FM) order, which is schemat-ically depicted in the insets by spins (arrows) pointing alongthe crystallographic c axis. The rhombus marks the positionof the tricritical point at ( p tr , T tr ), at which the character ofthe FM transition changes from second order for low p to firstorder for high p . Black- and red-shaded regions correspond tonew phases that occur for p > ∼ . p region. For T > T > T , small clusters of varying size with FM orderare embedded in a paramagnetic (PM) matrix. For T < T ,these clusters fill the whole sample volume. anomaly of similar size in ∆ C/T is clearly resolvable,which however does not have any discernible effect in α ab and α c . Further cooling down to T FM (cid:39)
40 K resultsin a strong feature in α i and the c lattice parameter,which, given the increase of I , is associated with theformation of long-range FM order, but does not resultin a clear feature in ∆ C/T . In contrast to low p though,the symmetric and sharp shape of the anomaly in α i forboth directions is strongly reminiscent of a first-orderphase transition (cf. also the more step-like change of c and I at T FM ). This, together with a sizable thermalhysteresis (see SI ), is clear thermodynamic evidencefor the change of the character of the FM transitionfrom second order to first order at p tr .The positions of the various anomalies, which weinferred from the full T - p data sets up to ≈ ), are compiled in the T - p phase diagram inFig. 2. Upon suppressing T FM with p , the FM transitionchanges its character from second order to first orderat ( p tr , T tr ) = [1 . for thedetermination of the position). For p > ∼ p tr , anomaliesat T and T emerge. (Only the latter phase line was p = 1 . 9 G P a( 1 Counts (100 cts/5 min) q ((cid:176) ) ( b ) 6 K p = 2 . 5 5 G P a ( a ) 6 1 . 2 K5 6 . 5 K 5 3 . 4 K4 5 . 5 K1 0 K Asymmetry t ( m s ) FIG. 3. (a) µ SR spectra of LaCrGe in zero field at p =2 .
55 GPa. Symbols correspond to the measured data, solidlines correspond to fits by Eq. S6 (see SI ); (b) Angle-dependent neutron intensity around the (1 0 0) Bragg peakat p = 1 . identified in previous studies .) The T and the T lines do not only both emerge in immediate vicinity to( p tr , T tr ), but also closely follow each other in the phasediagram and are suppressed much more slowly by p than T FM . Altogether, this phase diagram highlights thecomplex behavior associated with the avoided FM QCPin LaCrGe .To discuss the nature of the phases below T and T that are so clearly delineated in Fig. 2 by multiplethermodynamic and transport measurements (see SI ),we turn to µ SR and neutron scattering measurementsunder pressure. Previous µ SR measurements under p showed a clear magnetic signal below ≈
50 K at2.3 GPa. To confirm this result and to refine the onsettemperature, we performed another µ SR study witha finer T data point spacing close to T and T at2.55 GPa. Figure 3 (a) shows selected zero-field µ SRspectra, that are in full agreement with the notionof some type of local (on the scale of µ SR) magneticorder in the new phases. To discuss this in more detail,we show in Fig. 4 the T dependence of the internalfield, B int , and the transverse relaxation rate, λ T , asa measure of the width of the field distribution, fromzero-field µ SR data. We also include the T dependenceof the magnetic asymmetry, A mag , as a measure of themagnetic volume fraction, as well as the relaxation rateof the pressure cell, λ PC , as a measure of the field in thepressure cell that is created by a sample with macro-scopic magnetization, from weak-transverse field µ SRdata (see SI ). The thermodynamic and transport datafor similar p in Fig. 4 are used to determine the positionsof T ≈
56 K and T ≈
49 K as well as T FM ≈
22 K. B int sets in between T and T and increases uponcooling, with low T values similar to the ones in theFM state (see SI ). λ T shows a strong increase uponcooling through T . However, upon further coolingthrough T , λ T remains at a relatively high, finite valueand decreases only slightly below T FM . A large λ T
01 0 0 02 0 0 03 0 0 001 0 02 0 00 . 00 . 10 . 00 . 1 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0 T F M T B int (Oe) T l T ( m s-1) A mag l PC ( m s-1) ( f ) p = 2 . 4 3 G P a( a ) p = 2 . 5 5 G P a( d ) p = 2 . 5 5 G P a( c ) p = 2 . 5 5 G P a( b ) p = 2 . 5 5 G P a D C / T (a.u.) ( e ) p = 2 . 3 8 G P a Da c (a.u.)d Rc /d T (a.u.) T ( K ) ( g ) p = 2 . 5 5 G P a FIG. 4. Comparison of several high-pressure data sets closeto a pressure, p , of 2.5 GPa as a function of temperature, T .(a) Internal field, B int , (b) transverse relaxation rate, λ T , (c)magnetic asymmetry, A mag , and (d) relaxation rate of thepressure cell, λ PC , from zero-field (a,b) and weak transversefield (wTF) (c,d) µ SR measurements; (e) Anomalous contri-bution to specific heat, ∆
C/T ; (f) Anomalous contributionto thermal expansion coefficient along the c axis, ∆ α c . Thelow- T and high- T data are plotted on different scales; (g) T -derivative of the c axis resistance, d R c /d T . Black dashed, reddotted and blue dashed-dotted lines indicate the position ofthe anomalies at T , T and T FM , respectively. implies a broad field distribution, characteristic for notwell-ordered systems . A mag indicates partial volumefraction for T < T < T and A mag ≈ .
12 for T ≤ T ,consistent with full volume fraction (see SI ). Last, λ PC is small above T and starts to increase just below T upon cooling. Below T , λ PC is finite and almost T -independent. In addition, we found strong indicationsfor the presence of a remanent field for T = 35 K < T (see SI ).In Fig. 3 (b), we compare the (1 0 0) Bragg peak inneutron diffraction for selected T at p = 1 . T = 68 K) corresponding to the nuclear contribution,and grows markedly below T FM (cid:39)
40 K due to the fer-romagnetic contribution at T = 38 K and 6 K. The mo-ment in the FM ground state is 1 . µ B , which wasdetermined from I relative to a set of nuclear Braggpeaks. For T FM < T < T , shown here by the 44 Kdata, the (1 0 0) Bragg peak is not distinguishable fromthe data in the PM phase [see also Fig. 1 (e)]. Further-more, we cannot resolve any magnetic Bragg peak in thethree-dimensional q space in the new phases (see SI ).Overall, we can thus exclude any type of long-range FMor c -axis modulated AFM order below T and T , i.e.,the previously-suggested AFM q type magnetic order ,with a moment larger than 0 . µ B and 0 . µ B , respec-tively. In addition, we can rule out the formation ofa charge-density wave or structural transition at high p from x-ray diffraction studies (see SI ).We now turn to a discussion of the nature of the p -induced phases that emerge for p > ∼ p tr and T < ∼ T tr .We start by focusing on the range T < T , for which the µ SR data suggest ≈ µ SR data indicate asimilar B int for the low- p FM state and the new phasebelow T at low T , it seems unlikely that the moment ofthe T -phase is so low that it falls below our sensitivityin neutron measurements. Following this argument, anobvious scenario, which would reconcile both µ SR andneutron results, would be that the magnetic order below T is only short-range. We note that the sizable λ T valuefor T < T is fully consistent with the notion of a short-range ordered state , in which magnetic clusters exist.To discuss the question whether the order within theseclusters is FM or AFM, we refer to the observations of afinite λ PC and a remanent magnetization below T from µ SR. This speaks in favor of FM order in each cluster,whereas the clusters might either align FM or AFM withrespect to each other (see inset of Fig. 2 for a schematicpicture). We speculate that at least some of the clustersalign AFM with respect to each other, since this wouldexplain the small, but finite λ PC . An estimation of thesize of such FM clusters can be inferred from the λ T valueas well as the data of the (1 0 0) Bragg peak. The largevalue of λ T between T and T FM yields an estimate forthe cluster size of 6 nm . For the neutron data, if weassume a similar moment size as in the FM state, as sug-gested by a similar B int , the absence of a clear magnetic(1 0 0) Bragg peak results in an estimate of the averagecluster size of less than 12 nm. This scenario of clus-ters would also naturally account for a small amount ofentropy release upon subsequent cooling through T FM ,consistent with the lack of a clear specific heat feature inour experiment (see SI ). Note that moment size andspatial size can change with decreasing T , as suggestedby a continuous change of B int , the c lattice parameterand α i .How is LaCrGe for T < T < T then characterized?Our results indicate that in this regime the magneticvolume fraction is strongly T -dependent and increasesfrom ≈ T upon cooling to ≈ T . Wealso recall our result of the lattice strain: (i) the lat-tice response upon cooling through T shows the samedirectional anisotropy as for the FM transition, but only smaller in size, and (ii) there are no pronounced latticeeffects at T . The latter result indicates that no strongmodification of the magnetic order occurs at T , since itwould likely result in an additional lattice strain. It thusappears likely, that the magnetic clusters start to form inthe range T < T < T , and either their number or sizeis strongly dependent on T (see inset of Fig. 2). The sizeand anisotropy of the observed lattice strains are fullyconsistent with the notion of small FM clusters, in whichmoments are primarily aligned along the c axis (a smalltilt away from the c axis is possible) and in which thepartial AFM alignment of the clusters with respect toeach other strongly reduces the lattice strain (in contrastto large FM domains in the low- p FM state, resulting inlarge strains). As an alternative proposal for the natureof the intermediate T phase, we refer to the theoreticalidea, that spin-nematic orders can be promoted by quan-tum fluctuations close to an avoided QCP . Experi-mentally, we cannot rule out this option, which by itselfwould certainly be exciting. However, if there would bea spin-nematic phase below T , then it does not couplestrongly to the crystalline lattice, since we do not observeany lattice symmetry change across the entire T range forhigh p (see SI ).Our main results on the avoidance of FM criticality inLaCrGe can be summarized as follows. We (i) pro-vided thermodynamic evidence for a change of the tran-sition character from second order to first order, typicallyconsidered a hallmark for the avoidance of the QCP inclean metallic FM systems and (ii) argued that short-range magnetic order rather than long-wavelength AFMorder exists for p ≥ p tr between T and T FM , which isusually associated with the effects of strong disorder .The main question is then what drives the formation ofshort-range order in LaCrGe : do the enhanced AFMinteractions, that are suggested by theory ,and the associated frustration between FM and AFMinteractions lead to a tendency towards short-range or-der, or does weak disorder promote short-range order?In fact, an earlier theoretical study pointed out thatthe tricritical point can survive up to a critical disorderstrength, whereas an amount of disorder smaller than thecritical disorder strength can cause a short-range spiralstate. So far, this scenario has only been considered tobe realized in the stochiometric compound CeFePO ,for which the interpretation is complicated by Kondophysics, and a tuning across the avoided QCP is lack-ing up to now. Interestingly, CeFePO and LaCrGe havea very similar residual resisitivity ratio of ≈ isa rare example, which fills the gap between two extremelimits of clean and strongly disordered itinerant FM sys-tems. Given that its p tunability allows for accessing themultiple phases without introducing additional disorder,and that its magnetic building block is a 3 d element,LaCrGe turns out to be a very promising candidate forthe comparison to theoretical concepts, that address theeffects of weak disorder and modulated AFM orders closeto an avoided FM QCP in metals.We thank A.I. Goldman, V. Taufour and D. Ryan foruseful discussions and S. Downing and C. Abel for thegrowth of single crystals. The authors would like to ac-knowledge B. Li, D. S. Robinson, C. Kenney-Benson, S.Tkachev, M. Baldini, and S. G. Sinogeikin and D. Popovfor their assistance during the x-ray diffraction experi-ments. We thank C. Tulk, A. M. dos Santos, J. Mo-laison, and R. Boehler for support of the high-pressureneutron diffraction study, and Y. Uwatoko for provid-ing us the palm cubic pressure cell. Work at the AmesLaboratory was supported by the U.S. Department ofEnergy, Office of Science, Basic Energy Sciences, Mate-rials Sciences and Engineering Division. The Ames Lab-oratory is operated for the U.S. Department of Energyby Iowa State University under Contract No. DEAC02-07CH11358. E.G. and L.X. were funded, in part, by theGordon and Betty Moore Foundation’s EPiQS Initiativethrough Grant No. GBMF4411. In addition, L.X. wasfunded, in part, by the W. M. Keck Foundation. A por-tion of this research used resources at the High Flux Iso-tope Reactor and the Spallation Neutron Source, U.S.DOE Office of Science User Facilities operated by the OakRidge National Laboratory. This research used resourcesof the Advanced Photon Source, a U.S. DOE Office ofScience User Facility operated for the US DOE Officeof Science by Argonne National Laboratory under Con-tract No. DE-AC02-06CH11357. We gratefully acknowl-edge support by HPCAT (Sector 16), Advanced PhotonSource (APS), Argonne National Laboratory. HPCAToperations are supported by DOE-NNSA under GrantNo. DE-NA0001974, with partial instrumentation fund-ing by NSF. Use of the COMPRES-GSECARS gas load-ing system was supported by COMPRES under NSF Co-operative Agreement Grant No. EAR-11-57758 and byGSECARS through NSF Grant No. EAR-1128799 andDOE Grant No. DE-FG02-94ER14466. Research of R.G.is supported by the Swiss National Science Foundation(SNF-Grant No. 200021-175935). [1] P. C. Canfield and S. L. Bud’ko, Rep. 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EXPERIMENTAL METHODS
Single crystal synthesis -
Single crystals of LaCrGe were synthesized by the flux-growth technique, as de-scribed in Ref. [1]. To this end, high-purity elements( > and sealed in a fused silica am-poule under argon atmosphere. Subsequently, the am-poule (with the growth material inside) was heated up to1100 ◦ C over 3 h and held there for 5 h. The growth wasthen cooled to 800 ◦ C over 125 h. The excess liquid wasdecanted at 800 ◦ C using a centrifuge in a final step. Theobtained single crystals of rod-like shape were character-ized by means of x-ray diffraction, resistance and magne-tization at ambient pressure prior to all measurements atfinite pressures. These results were well consistent withprevious reports in terms of the Curie temperature T FM as well as the residual resistivity ratio RRR . Specific heat measurements under pressure -
Specificheat under pressure was measured using the AC calorime-try technique, as described in detail in Refs. [5, 6]. Tothis end, a single crystal of LaCrGe was placed be-tween a heater and a thermometer. The heater wassupplied with an oscillating voltage, and the resultingtemperature oscillation of the sample, which is relatedto the specific heat of the sample, was recorded. Giventhe non-adiabatic conditions of the pressure-cell environ-ment, absolute values of the specific heat cannot be ob-tained with high accuracy; nonetheless, the techniqueof AC calorimetry allows for a decoupling of the sam-ple from the bath (i.e., the pressure medium and thecell), to a good approximation, by choosing the appro-priate measurement frequency (see Ref. [5] for details onthe procedure of the determination of the measurementfrequency). Thus, changes of the specific heat with pres-sure can be obtained reliably. Our implementation of thistechnique has proven to be particularly sensitive for thedetection of specific heat anomalies of varying size, re-sulting from different amounts of removed entropy, andover a wide range of phase transition temperatures. Thisis highly beneficial for the present study, where the pro-nounced specific heat anomaly at high temperatures close to 90 K at ambient pressure becomes suppressed to verylow temperatures and strongly reduced in size.The cryogenic environment was provided by a closed-cycle cryostat (Janis SHI-950 with a base temperatureof ≈ p ≈ − , thus ensuringhydrostatic pressure application over the available pres-sure range. Pressure at low temperatures was determinedfrom the shift of the superconducting transition temper-ature of elemental lead (Pb) , which was determined inresistance measurements. The error in the determina-tion of the low-temperature pressure is estimated to be0.01 GPa, and pressure changes in this particular cell by less than 0.04 GPa by increasing temperature up to100 K. Thermal expansion measurements under pressure -
Thermal expansion, i.e., the macroscopic length changeof a crystal of LaCrGe along a particular crystallo-graphic axis as a function of temperature, was measuredusing strain gauges, which are sensors whose resistance, R , changes upon compression or tension. For our mea-surements, strain gauges (type FLG-02-23, Tokyo SokkiKenkyujo Co., Ltd. with R ≈
120 Ω) were fixed rigidly tothe sample by using Devcon 5 minute epoxy (No. 14250),and the resulting resistance changes of the strain gaugeswere recorded and converted into length changes usingthe known gauge factor. In total, two strain gauges werefixed orthogonally on the same sample to measure theexpansion along the ab axes and the c axis simultane-ously. Since the strain gauge resistance varies not onlydue to the expansion of the crystal with temperature, butalso due to the intrinsic resistance change of the straingauge wire material, another set of strain gauges wasmounted on a sample of tungsten carbide. Given thattungsten carbide is a very hard material and has a com-paratively small expansion coefficient over a wide temper-ature range, and in particular no anomalous behavior, the resistances of the strain gauges mounted on tung-sten carbide are used to subtract the intrinsic resistancechange of the strain gauge from the measured resistancedata on LaCrGe . This subtraction was performed insitu by using two Wheatstone bridges (see, e.g., Ref. [12]for similar designs). To this end, in each bridge, onestrain gauge on the sample, one strain gauge on tung-sten carbide inside the cell and two thin-film resistorswith similar and almost temperature-independent abso-lute resistances of ≈
120 Ω, which were placed outside ofthe cell in the low- T environment, were used. The currentfor the bridge was supplied by a LakeShore 370 Resis-tance Bridge, which was also used to measure the voltageacross each bridge. The cryogenic environment, pressurecell, pressure medium and manometer were identical tothe one for specific heat measurements, see above. Resistance measurements under pressure -
Resistanceunder pressure was measured in a four-point configura-tion with current directed along the crystallographic c axis (Note that previously-published data were obtainedwith current in the ab plane). Contacts were made us-ing Epo-tek H20E silver epoxy. The AC resistance wasmeasured by a LakeShore 370 Resistance Bridge. Thecryogenic environment, pressure cell, pressure mediumand manometer were identical to the one for specific heatmeasurements, see above. High-energy x-ray diffraction measurements on single-crystals under pressure -
High-energy (100 keV) x-raydiffraction measurements were performed on single crys-tals at station 6-ID-D of the Advanced Photon Source,Argonne National Laboratory. The samples were pres-surized in diamond anvil cells (part of the Diacell BraggSeries, Almax easylab ) using He-gas as a pressure-transmitting medium. We used diamond anvils with600 µ m culets and stainless-steel gaskets preindented tothicknesses of ≈ µ m, with laser-drilled holes of diam-eter ≈ µ m. The wavelength of a fluorescence line ofruby was used for room-temperature pressure calibration.By measuring the lattice parameter of polycrystalline sil-ver, we determined pressure in situ at all temperaturesand pressures with an accuracy of 0.1 GPa. Large regionsof the ( H H L ) plane and the powder diffraction patternof silver were recorded by a MAR345 image plate po-sitioned 1.249 m behind the DAC while the DAC wasrocked 2.4 ◦ along two independent axes perpendicularto the incident x-ray beam. At ambient pressure, otherplanes of high-symmetry were also recorded outside ofa DAC. In addition, at ambient pressure high-resolutionmeasurements were taken of the Bragg peaks (16 0 0) and(0 0 16) with a Pixirad-1 detector positioned 1.210 m be-hind the sample while rocking around one axis perpen-dicular to the incident x-ray beam. Powder x-ray scattering measurements under pressure-
Powder x-ray diffraction measurements were performedunder pressure with 30 keV x-rays at station 16-BM-D ofthe Advanced Photon Source, Argonne National Labora-tory. The powder was made by crushing single crystals of LaCrGe and only powder of less than a micron sizewas loaded into the DAC (Diacell Bragg Series, Almaxeasylab ). The DAC was configured identically as forthe high-energy x-ray experiment described above withHe-gas as pressure-transmitting medium, but the wave-length of a ruby fluoresence line was used to measurepressure at all temperatures and pressures with an accu-racy of 0.1 GPa. Large regions of reciprocal space wererecorded on a MAR345 image plate positioned 0.412 mbehind the DAC. Individual crystallites of LaCrGe stillhad very sharp peaks and so the sample was rocked alongone axis perpendicular to the beam to obtain a betterpowder average. Neutron diffraction measurements at HB1 on singlecrystals at ambient and finite pressure -
Neutron diffrac-tion measurements were performed on single crystals us-ing the HB1 diffractometer at the High Flux Isotope Re-actor, Oak Ridge National Laboratory. For measure-ments taken at ambient pressure a single crystal wassealed in an Al can containing He exchange gas, whichwas then attached to the head of a He closed-cycle refrig-erator (CCR). We refer to this experiment as N0. Thebeam collimators placed before the monochromator, be-tween the monochromator and the sample, between thesample and analyzer, and between the analyzer and de-tector were 48’-80’-80’-240’, respectively. HB1 operatesat a fixed incident energy of 13.5 meV and contaminationfrom higher harmonics in the incident beam was elimi-nated using Pyrolytic Graphite (PG) filters.For measurements with p < atroom temperature. For the 1.9 GPa measurement, thesample was loaded together with a NaCl single crystal,which was used to measure the pressure within the cellbased on the lattice parameter changes before and af-ter applying pressure at room temperature with an ac-curacy of 0.1 GPa. For all other pressures, the pres-sure was determined with an accuracy of 0.2 GPa basedon a calibrated pressure-load curve measured at roomtemperature for that specific cell and was corrected fortemperature-induced reduction of pressure via previouscalibration measurements. The cell was then attacheddirectly to the head of a CCR.For measurements with p > anvils anda gasket made out of an Al-based alloy . We refer tothese experiments throughout the text as N2. A singlecrystal with volume of 0.9 x 0.9 x 1.5 mm was attachedto the bottom of a teflon capsule together with a 1:1mixture of Fluorinet FC70 and FC77 as the pressure-transmitting medium. Although this medium solidifiesclose to 1.1 GPa at room temperature, previous studieshave shown that the PCAC applies pressure almost hy-drostatically up to much higher pressures than the so-lidification pressure due to the three-dimensional anvildesign that allows to compress the medium simulta-neously along three orthogonal directions. The pressurewas determined with an accuracy of 0.3 GPa at roomtemperature based on a calibrated pressure-load curvefor that specific cell and was corrected for temperature-induced reduction of pressure via previous calibrationmeasurements. After applying pressure the cell was thenloaded into a high-capacity CCR with a base temperatureof approximately 3 K. Neutron elastic scattering measurements at CORELLIunder pressure -
Elastic scattering measurements usinga diamond anvil cell (DAC) were performed using thetime-of-flight diffractometer CORELLI at the SpallationNeutron Source, Oak Ridge National Laboratory. Werefer to these experiments throughout the text as N3.CORELLI allows for the simultaneous measurement oflarge sections of the three-dimensional reciprocal spaceby utilizing a white-beam Laue technique with energydiscrimination by modulating the incident beam with astatistical chopper . This allows CORELLI to efficientlyseparate the elastic and inelastic channel of the diffusescattering signal, thus identifying whether the observedcorrelation is static or not. By applying pressure in aDAC at CORELLI we were able to reach pressures from0.8 GPa to 3.2 GPa at base temperature of T ≈ . Mea-surements of MnP have shown that CORELLI is capa-ble of measuring moments as low as 0.25 µ B /˚ A within aDAC . The sample (sample thickness of ≈ µ m andsample crossectional area of 0.7 × ) was loadedonto one polycrystalline anvil with the PH15-5 steel gas-ket (500 µ m height, 3 mm culet size, 1.3 mm initial gas-ket hole) in place with deuterated glycerin as a pres-sure medium, which solidifies at ≈ , but remains soft providing close to hy-drostatic conditions up to 9 GPa . This was then sealedand pressurized at room temperature with a press. Notethat an initial experiment using 4:1 methanol:ethanol aspressure-transmitting medium did not succeed, becausethe pressure medium evaporated too quickly during seal-ing. In contrast, glycerin does not readily evaporate onair which ensures that the pressure-transmitting mediumis contained. Pressure was assigned to the one obtainedfrom a calibrated pressure-load curve for that specificcell and anvil/gasket set-up at room temperature withan accuracy of 0.5 GPa. After applying pressure the cellwas then loaded onto the head of a CCR. We took mea-surements with the well-focused incident beam passingthrough the steel gasket from the side. Both gasket andVersimax anvils only yield powder diffraction rings which can be readily distinguished from the single crystal sam-ple peaks. µ SR measurements under pressure -
Approximately100 small single crystals of LaCrGe (in total ≈ µ SR experi-ments, are made out of MP35N alloy . The maximumpressure of this cell is ≈ . The pressure atlow temperatures was determined from the shift of thesuperconducting transition of elemental indium , whichwas also placed in the pressure cell. We estimate the er-ror in this low-temperature pressure to be ≈ µ SR experiments in a separate He cryostat. Intotal, µ SR measurements were performed in zero mag-netic field and in several transverse magnetic fields (upto 6000 Oe) at 0.2 GPa as well as at the maximum pres-sure of 2.5 GPa. The measurements were performed ina He cryostat with base temperature of 2.2 K at the µ E1 beamline at the Paul-Scherrer-Institute in Villigen,Switzerland, by using the GPD spectrometer. Typically,5 − · positron events were counted for each datapoint. Pressure media and homogeneity -
For all experimentsunder finite pressure, the pressure media and cell de-sign was chosen such to ensure hydrostatic or close tohydrostatic pressure conditions in the pressure range ofinterest . SPECIFIC HEAT DATA UNDER PRESSURE
Specific heat data sets under pressure and procedure toobtain anomalous specific heat contributions -
Figure S1shows selected data sets of specific heat divided by tem-perature,
C/T , which were taken during this study, cov-ering the pressure range 0.46 GPa ≤ p ≤ ≤ T ≤
100 K. These data setswere used to extract the anomalous contributions to thespecific heat, i.e., the specific heat data corrected for anestimate of the background contribution, resulting from,e.g., phonons. A previous work demonstrated by com-parison to the specific heat of the non-magnetic analogueLaVGe that the specific heat of LaCrGe is dominatedby non-magnetic contributions at low temperatures andat temperatures higher than the ferromagnetic transitionat ambient pressure. Only close to the ferromagnetic C / T (J/(mol K2)) T ( K ) increasing p FIG. S1. Specific heat divided by temperature,
C/T , ofLaCrGe vs. T over a wide temperature range (5 K ≤ T ≤
100 K) for finite pressures (0.46 GPa ≤ p ≤ ) for clarity. phase transition, a substantial magnetic contribution tospecific heat was observed. Given that the AC specificheat technique used here does not allow to determine spe-cific heat values to a very high accuracy, we do not referto specific heat measurements of LaVGe for the back-ground subtraction for the LaCrGe data under pressure(shown in Fig. S1), and instead follow the procedure,which is illustrated in Fig. S2 for a subset of the data.Following the knowledge of the ambient-pressure study,we approximate the non-magnetic background contribu-tion by fitting the C/T data by a polynomial function ofthe order of three across a wide temperature range exceptin the immediate vicinity of any phase transition temper-ature T p (i.e., either T FM , T or T ). Typically, the range T p −
10 K ≤ T ≤ T p + 5 K was excluded from the fit, andoverall, the fit was typically performed down to T p −
20 Kand up to T p + 20 K. The so-obtained background curvesmanifest a shoulder in C/T at T ≈
80 K, which can alsobe seen in the the ambient-pressure specific heat dataon LaVGe when replotted as C/T vs. T . Note thatthis procedure of background subtraction leads to sig-nificant uncertainties in estimating the absolute size ofspecific heat (and thus, entropy) that is associated witheach phase transition. However, the conclusions, whichare presented in the main text, are solely based on theanalysis of the positions of anomalies in ∆ C/T , whichshould not be affected by the background subtractionprocedure.
Position of anomalies in specific heat data and criteriato determine transition temperatures -
The so-obtainedanomalous specific heat contributions, ∆
C/T , as a func-tion of temperature, T , are shown in Fig. S3 (a), togetherwith the temperature-derivative of the same data in (b),
50 60 70 80 901.161.181.201.221.241.261.281.30 50 60 70 801.121.141.161.181.201.221.2440 50 60
40 50 600.951.001.051.10 C / T ( J / ( m o l K )) T (K) p = 0.46 GPa C / T ( J / ( m o l K )) T (K) p = 0.98 GPa C / T ( J / ( m o l K )) T (K) p = 1.72 GPa C / T ( J / ( m o l K )) T (K) p = 2.15 GPa(a) (b)(c) (d) FIG. S2. Illustration of the procedure to obtain the anoma-lous specific heat contributions, that are associated with vari-ous phase transitions in LaCrGe , for selected pressures (a-d).The background (red dashed line) was obtained by fitting thespecific heat data (black line) well below and well above thephase transitions simultaneously with a polynomial of the or-der three (for details, see text). for 0 GPa ≤ p ≤ T FM (indicated by the blue ar-row) manifests itself in an almost mean-field-like jumpin the specific heat, the size of which becomes progres-sively reduced with increasing pressure (Note that a dis-cussion of the specific heat signature of the first-orderFM transition for p ≥ p = 1.39 GPa, a second, more subtle anomaly occurs onthe high-temperature side of the ferromagnetic specificheat anomaly for the first time. This result suggests thepresence of a new phase transition, which was denoted by T in the main text. Upon increasing the pressure slightlyto 1.53 GPa, these two specific heat anomalies at T FM and T , respectively, become more separated in temperatureand thus clearly distinguishable. For even higher pres-sures, the anomaly, which we associate with T FM , contin-ues to drop (see below), but we also observe two specificheat anomalies, which are separated by only ≈
10 K intemperature and almost similar in size. The positions ofboth of these anomalies are almost unchanged in temper-ature upon increasing pressure (compared to the strongsuppression of the T FM -line with pressure). We thusassign the lower-temperature specific heat anomaly for p ≥ T , which isdistinct from the ferromagnetic transition. Note that the T -line has previously been reported in literature , basedon electrical transport measurements, and was assignedto a new magnetic phase transition of likely modulatedAFM q . In the main text and also here in the SI, wepresent neutron and new µ SR data for high pressures,which strongly suggest a new interpretation of the mag-netic state of the phase below T .To determine the transition temperatures T FM , T and T from the presented specific heat data, the position of - 7 5- 5 0- 2 502 5 0 2 0 4 0 6 0 8 0 1 0 0 T T F M ( b )( a ) D C / T (J/mol/K2)
0 G P a 0 . 4 6 G P a 0 . 7 8 G P a 0 . 9 8 G P a 1 . 1 2 G P a 1 . 3 9 G P a 1 . 5 3 G P a 1 . 7 2 G P a 1 . 8 2 G P a 1 . 9 2 G P a 2 . 0 1 G P a 2 . 0 8 G P a 2 . 1 5 G P a 2 . 3 0 G P a 2 . 3 8 G P a T F M T T T d ( D C / T)/ d T (a.u.) T ( K ) FIG. S3. Anomalous contribution to the specific heat, ∆
C/T ,(a) and temperature-derivative of this data, d(∆
C/T )/d T ,(b) vs. temperature, T , for LaCrGe for finite pressures inthe range 0 GPa ≤ p ≤ T FM , T and T , respectively. In all panels, data were shiftedvertically for clarity. the minimum in d(∆ C/T )/d T was chosen [see position ofthe arrows, which are exemplarily shown for p = 0 GPaand 2.38 GPa in Fig. S3 (b)]. The application of this cri-terion yields transition temperatures that are very closeto that obtained by iso-entropic construction. Signature of ferromagnetic transition in specific heatmeasurements at and beyond 1.53 GPa -
In the discus-sion above as well as in the main text, we do not showany specific heat signatures of the ferromagnetic (FM)transition for p ≥ T and T , which likelyresults in only a small amount of entropy being released ( b ) d( D C / T )/d T (a.u.) T ( K ) ( a ) T F M T F M , a ( T , p ) T (K) p ( G P a ) T F M T T C ( T , p ) FIG. S4. (a) Derivative of the anomalous specific heat con-tribution, d(∆
C/T )/d T , vs. T for LaCrGe for selectedpressures 1.53 GPa ≤ p ≤ C/T ) was obtainedby subtracting the specific heat data set at 0.98 GPa, whichserves as a proxy for the background contribution for
T < ∼
50 Kfrom the measured data. The two large peaks for
T > ∼ T and T inthe main text. In addition, tiny and very broad anomalies, thepositions of which are indicated by the arrows, can be identi-fied at lower temperatures and are likely associated with thefirst-order ferromagnetic transition at T FM for p > ∼ , as constructedfrom specific heat measurements under pressure. Blue solidcircles, black solid squares and red solid triangles correspondto T FM , T and T , respectively. Open blue symbols indi-cate the position of the clear anomalies at T FM in the thermalexpansion coefficient data (shown below). on the subsequent cooling through T FM . In turn, thenwe expect that the specific heat feature becomes verysmall, likely below the limit below which we can sepa-rate it from the background. In addition, the transitionchanges its character close to 1.5 GPa (see discussion be-low), and the absolute value of d T FM /d p gets larger uponincreasing p . It is therefore reasonable to expect thatany specific heat feature above 1.5 GPa might be differ-ent in shape and broadened in temperature, making ithard to separate the feature from the (unknown) back-ground contribution. Nonetheless, even if no clear featurecan be observed in C/T (or ∆
C/T ), a potential featuremight show up more clearly in the temperature-derivativeof these data. For this reason, we show in Fig. S4 (a),the temperature-derivative of ∆
C/T data across a widertemperature range. For this plot, ∆
C/T was obtainedby ∆
C/T = ( C ( T, p ) − C ( T, p = 0 .
98 GPa))/ T ,with C ( T, p ) being the temperature-dependent specificheat data at the pressure of interest, and C ( T, p =0.98 GPa) the temperature-dependent specific heat dataat 0.98 GPa. This analysis is needed, since a simple poly-nomial fit (with order three) is not sufficient to describethe background over a very wide temperature range. Inour approach, we assume that the C ( T, p = 0 .
98 GPa),for which only a FM transition for high temperatures
T >
60 K occurs, can be used as a good proxy for thebackground contribution for
T < ∼
50 K, where we expectthe FM transition to occur for p ≥ and the non-magnetic ana-logue LaVGe , which showed that the specific heat attemperatures well below the FM transition, is dominatedby non-magnetic contributions. Indeed, as a result ofour analysis, very subtle and progressively broader min-ima can be observed in the so-obtained d(∆ C/T )/d T data, the position of which (see arrows) coincide well[see Fig. S4 (b)] with the positions of the sharp and clearanomalies in the thermal expansion coefficient (see maintext and Fig. S6). Thus, we assume that this subtle fea-ture in the specific heat is indeed related to the ferro-magnetic ordering, in accordance with our hypothesis ofsmaller entropy associated with the ordering and/or ad-ditional broadening of the feature. Nonetheless, from thespecific heat data alone, it would not be possible to inferthe T FM -line for p ≥ LATTICE PARAMETERS UNDER PRESSURE
Definition of physical quantities -
Since we discuss andcompare measurements from various techniques, whichall give insight into the change of the crystalline latticewith pressure and temperature, we first want to definesome of the measurement quantities here and elaboratewhich of the different experimental techniques yields in-sight into which quantity.The temperature-dependent relative length change (oralternatively, thermal expansion) of a macroscopic crys-tal along a crystallographic axis i , (∆ L/L ) i (with i = ab, c for a hexagonal system, such as LaCrGe ) is definedas (∆ L/L ) i ( T, p = const.)= L i ( T, p = const.) − L i ( T ref , p = const.) L i ( T ref , p = const.) , (S1)with L i ( T, p = const.) being the absolute length of acrystal in i -direction at any given temperature, T , and T ref , being any reference temperature. The thermal ex-pansion coefficient along a crystallographic axis i , α i , isthen defined as α i = 1 L i ( T, p = const.) ∂L i ( T, p = const.) ∂T , (S2)and is experimentally often determined to a very goodapproximation (since ∆ L i (cid:28) L i ) by α i = 1 L i (300 K , p = const.) d∆ L i ( T, p = const.)d
T , (S3) with ∆ L i = L i ( T, p = const.) − L i ( T ref , p = const.),and L i (300 K , p = const.) being the length of the crystalat room temperature, which can be determined in an in-dependent measurement. Note that due to the freedomin the choice of T ref , ∆ L i can only be determined up toa constant. Since α i , as defined above in Eq. S3, is di-rectly proportional to the temperature-derivative of ∆ L i ,the size of α i is independent of the choice of T ref . Notethat in Eq. S3, we set the normalization length in thedenominator to L i (300 K , p = const.), since ∆ L i (cid:28) L i .Experimentally, the relative length change, (∆ L/L ) i ,and the thermal expansion coefficient, α i , can be deter-mined from, e.g. capactive dilatometry at ambient pres-sure or the strain-gauge technique for finite pressures.From neutron and x-ray diffraction measurements at am-bient and finite pressures, the crystallographic lattice pa-rameters a = b and c (for a hexagonal crystal system,such as LaCrGe ) can be inferred at any measured tem-perature and pressure. Each of these measurement quan-tities are related by simple equations, and we will do thisexplicit comparison for our data collection of LaCrGe under pressure at the end of this section. Functionality of the strain-gauge technique for the de-termination of the thermal expansion and the thermal ex-pansion coefficient -
Prior to the discussion of our variousdata sets, taken under finite pressures, we first want todemonstrate the functionality of our strain-gauge-basedsetup by comparing the relative length change, (∆
L/L ) i ,and the thermal expansion coefficient, α i , obtained viathe strain-gauge technique (at a relatively low appliedpressure) to the data obtained by the technique of ca-pacitive dilatometry at ambient pressure (see Fig. S5).Capacitive dilatometry is a well-established technique forthe determination of the thermal expansion of solids andknown for its extremely high sensitivity . The capaci-tive dilatometry data, presented in Fig. S5, were obtainedby using a dilatometer, which was described earlier inRef. [27], in a Quantum Design PPMS, which providedthe low-temperature environment.Figures S5 (a) and (b) show the temperature ( T ) de-pendence of (∆ L/L ) i and α i for i = ab, c at ambientpressure, as obtained from using the technique of capac-itive dilatometry. (We use the notion of ab , since the a and b direction are equivalent in a hexagonal crystal sys-tem.) Upon cooling from 150 K, the crystal shrinks alongboth crystallographic inequivalent directions, as can beseen from a reduction of (∆ L/L ) i , corresponding to pos-itive values of α i (with small directional anisotropy). Be-low 90 K, an anomalous behavior of (∆ L/L ) i and α i canbe observed, which is a result of the well-known ferro-magnetic ordering at T FM ’
89 K . In more detail,upon cooling through this ferromagnetic transition, thelength along the ab axes shrinks rapidly, whereas the - 2- 1012 - 2- 10125 0 7 5 1 0 0 1 2 5 1 5 0- 6 0- 4 0- 2 002 0 5 0 7 5 1 0 0 1 2 5 1 5 0- 6 0- 4 0- 2 002 0 ( D L / L ) i (10-3) i = a b i = c i = a b i = c ( D L / L ) i (10-3) i = a b i = c a i (10-6/K) T ( K ) ( d )( c )( b ) c a p a c i t i v e d i l a t o m e t r y , a m b i e n t p i = a b i = c a i (10-6/K) T ( K )s t r a i n g a u g e , p = 0 . 2 1 G P a ( a ) FIG. S5. Comparison of thermal expansion data on LaCrGe ,obtained by capacitive dilatometry (a,b) and a strain-gauge-based method (c,d); (a,b) Relative length change, (∆ L/L ) i (a), and thermal expansion coefficient, α i (b), vs. temper-ature, T , along the crystallographic ab and c direction, ob-tained by utilizing a capacitive dilatometer at ambient pres-sure; (c,d) Relative length change, (∆ L/L ) i (c), and thermalexpansion coefficient, α i (d), vs. temperature, T , along thecrystallographic ab and c direction, obtained by utilizing astrain-gauge-based method at p = 0 .
21 GPa inside the pres-sure cell. Due to the freedom of choice in T ref , which causesthat the relative length change can be only determined up toa constant, the (∆ L/L ) i values at 150 K were matched to the150 K values from the capacitive dilatometry data. length along the c axis shows a very pronounced in-crease. This response of the crystal lattice to the fer-romagnetic order is consistent with a picture of magne-toelastic effects resulting from dipolar coupling of fer-romagnetically aligned spins with moments aligned alongthe crystallographic c axis. The described relative lengthchanges yield a positive anomaly in ∆ α ab ( T ) and a neg-ative anomaly in ∆ α c ( T ), with | ∆ α c ( T ) | ’ | ∆ α ab ( T ) | (∆ α i ( T ) corresponds to the anomalous contribution tothe thermal expansion coefficients after subtraction ofnon-magnetic background contributions, not shown inFig. S6). Given that the temperature dependence of α i ( T ) is closely related to the temperature dependenceof the specific heat, C ( T ), via the (uniaxial) Gr¨uneisenparameter, it can be expected that the anomalous contri-butions, ∆ α i ( T ), are similar in shape to anomalous con-tributions to the specific heat, ∆ C ( T ). Indeed, similarto the specific heat measurements, the thermal expansioncoefficients, α i ( T ), display almost mean-field like changesat the phase transition temperature T FM . For reasons ofconsistency with the specific heat data and the chosencriteria, the positions of the extrema in d α i /d T (i.e., theminimum in d α ab /d T and the maximum in d α c /d T ) werechosen to determine T FM = 89 . L/L ) i and α i for i = ab, c ,obtained from the strain-gauge technique, as describedin the section on experimental methods. The presenteddata were taken inside the pressure cell, which was closed hand-tight prior to the experiment without applying loadto the piston. This procedure caused that the lowest-pressure measurements were actually performed alreadyat a finite pressure of 0.21 GPa, as determined from thelow-temperature Pb manometer. Whereas this smallpressure leads to a small, but measurable shift of thetransition temperature, it does not compromise our com-parison, since LaCrGe still undergoes a ferromagnetictransition with very similar responses of the crystallinelattice (as demonstrated by our x-ray and neutron diffrac-tion data under pressure, see below). Again, upon cool-ing from high temperature, we find a decrease of thelength along both inequivalent directions (i.e., positive α i values). Consistent with our dilatometry data, wefind a strong decrease (large increase) of the length alongthe ab axes ( c axis) at T FM . The anomalies in α i ( T )with i = ab, c are also almost mean-field like. Apply-ing the same criterion for the determination of T FM asabove yields T FM (0 .
21 GPa) = 86 K (see Fig. S6 for thetemperature-derivatives of these data).This suppression of T FM with modest pressures is fullyconsistent with our analysis of the phase diagram fromspecific heat measurements. In terms of the absolute α i [and (∆ L/L ) i ], we find that the maximum value of α ab ,determined from the strain-gauge technique, is similar tothe one of the capacitive dilatometry data, whereas thevalue of α c is smaller by about 1 /
3. Reasons for this dis-crepancy can be manifold. First, the strain gauges arerigidly glued to the samples. However, the glue will nottransmit the strain perfectly, thus naturally leading tothe observations of slightly smaller length changes in thestrain-gauge measurements. If this was the case, then thefact, that the α ab values match better, suggests that thestrain transmission of the glue for the strain gauge of the ab axes was higher. Second, another option is related tothe expansion of the tungsten-carbide samples, which weuse for the subtraction of the intrinsic strain-gauge re-sponse. Strictly speaking, in our strain-gauge technique,we measure the length change of our sample relative tothe one of the tungsten carbide pieces. However, tung-sten carbide is known for its small expansivity , andthus, this scenario is highly unlikely. Anomalies in the thermal expansion coefficient underpressure and criteria for inferring phase transition tem-peratures -
Figures S6 (a) and (b) show the anomalouscontributions to the thermal expansion coefficients, ∆ α i with i = ab, c , of LaCrGe for finite pressures up to2.43 GPa. These ∆ α i data were obtained by subtract-ing a background contribution, which was obtained byfitting a data set at 2.60 GPa, for which the ferromag-netic transition T FM is suppressed to T <
10 K. We findthat the above-described pronounced thermal expansionanomalies at T FM , i.e., the positive anomaly in α ab and
051 01 52 0 0 2 0 4 0 6 0 8 0 1 0 0 1 2 0- 8 0- 6 0- 4 0- 2 00 0 2 0 4 0 6 0 8 0 1 0 0 1 2 04 0 5 0 6 0 7 0 8 0- 3 0- 2 0- 1 00- 2024 ( d )( c )( b ) Da ab (10-6/K) ( a ) Da c (cid:1) (10-6/K) T ( K ) d( Da ab )/d T (a.u.) T F M T F M d( Da c )/d T (a.u.) T ( K ) T F M ( f ) T ( K ) 1 . 9 4 G P a2 . 1 8 G P a ( e ) Da c (10-6/K) T T d( Da c )/d T (a.u.) Da ab (10-6/K) d( Da ab )/d T (a.u.) FIG. S6. Analysis of the thermal expansion anomalies; (a,b)Anomalous contribution to the thermal expansion coefficientalong the ab axes, ∆ α ab (a), and along the c axis, ∆ α c (b),vs. temperature, T , of LaCrGe for pressures 0.21 GPa ≤ p ≤ T FM ( T ). Note that the change in criterionfor T FM is related to the change of the character of the tran-sition from second-order to first-order at p tr ’ . ; (c,d) Derivative of the anomalous ther-mal expansion coefficients along the ab axes, d(∆ α ab )/d T , (c)and along the c axis, d(∆ α c )/d T (d) for p = 0 .
21 GPa and1.94 GPa. The blue arrows indicate the criteria to determine T FM from these data sets; (e,f) Enlarged view on ∆ α i (leftaxis) and d∆ α i /d T (right axis) for i = ab (e) and i = c (f) at p = 1 .
94 GPa around the phase transition tempera-ture T . The criterion, which was chosen to determine T , isindicated by the black arrows. In each panel, d(∆ α i )/d T at p = 2.18 GPa is included to demonstrate that, if present, anyfeature at T ’
49 K is distinctly smaller than the one at T . the negative anomaly in α c , are shifted to lower tem-peratures with increasing p . Importantly, in contrast tothe signature of the ferromagnetic transition in specificheat measurements, the feature in the thermal expan-sion remains clear and measurable over the full, inves-tigated pressure range, thus allowing us to reliably de-termine the T FM -line across wide ranges of the phasediagram. At the same time, we find that the shape ofthe expansion anomalies along both directions changesits shape upon increasing pressure. Specifically, the al-most mean-field-type ∆ α i , with i = ab, c , anomalies forlow pressures change into symmetric and sharp peaks forhigher pressures. These changes of the anomaly shapestrongly suggest a change of the character of the phasetransition from second-order to first-order upon increas-ing pressure, consistent with previous reports as well asthe generic avoidance of ferromagnetic criticality in itin-erant ferromagnets. The detailed determination of theassociated tricritical point at ( p tr , T tr ) from an analysisof the asymmetry and the width of the thermal expan-sion coefficient feature will be discussed below. Here, wewould only like to discuss the implications for the choiceof criterion to determine T FM from the present thermalexpansion coefficient data. For low pressures, the mean-field-type anomaly gives rise to a pronounced minimumin d(∆ α ab )/d T (maximum in d(∆ α c )/d T , as exemplarilyshown in Figs. S6 (c) and (d) for p = 0.21 GPa. We chosethe positions of these extrema to determine T FM for lowpressures. In contrast, the sharp anomaly in the ther-mal expansion coefficient for high pressures gives rise toan anomaly in d(∆ α i )/d T ( i = ab, c ) with pronouncedover- and undershoots on the low- and high-temperatureside, see, e.g., the p = 1 .
94 GPa data sets in Figs. S6 (c)and (d). Correspondingly, we chose the mid-point ind(∆ α i )/d T between the maximum and minimum valuesof the anomaly to determine T FM for high pressures (seeblue arrows).In addition to the FM anomaly, a closer look on the∆ α i ( T ) ( i = ab, c ) data reveal a smaller, but nonethe-less clear anomaly at T . To show this, we present inFigs. S6 (e) and (f) the ∆ α i ( T ) (left axis) as well as thed(∆ α i )/d T (right axis) for p = 1 .
94 GPa on enlargedscales around T . The anomalies in ∆ α i ( T ) can be seenwith bare eyes, but become very obvious in d(∆ α i )/d T ,where we observe a minimum in d(∆ α ab )/d T and a max-imum in d(∆ α c )/d T . As already pointed out in the maintext, this result implies that the lattice responds in thesame way to the phase transition at T as to the fer-romagnetic order, albeit smaller in size, i.e., the crystalshrinks in the ab plane and expands along the c axis uponcooling. The positions of the extrema in d(∆ α i )/d T wereused to infer T .However, the phase transition at T , which gives riseto a clear specific heat feature, does not result in apronounced feature close to 50 K in the thermal expan-sion coefficient. To demonstrate this, we also added the T T F M ( b )x - r a y p = 0 G P a a (Å) ( a ) c (Å) p = 1 . 9 G P a a (Å) T ( K ) x - r a y c (Å) FIG. S7. Lattice parameters a (left axis) and c (right axis) atambient pressure (a) and at p = 1 . data sets of ∆ α i ( T ) ( i = ab, c ) for p = 2 .
18 GPa inFigs. S6 (e) and (f), since the ferromagnetic transition issuppressed well below 50 K for this pressure. No clearfeature is discernible in either the ab axes or the c axisdata. Thus, we can conclude that the anomalous latticeeffects are distinctly larger at T than at T , however,both are distinctly smaller than the one induced by long-range FM ordering. Lattice parameters under pressure from x-ray and neu-tron diffraction -
In Fig. S7, we show the temperaturedependence of the crystallographic lattice parameters, a and c , determined from x-ray diffraction measure-ments on single crystals, at ambient pressure (a) and at p = 1.9 GPa (b). The c lattice parameters were deter-mined by measuring the position of the (0 0 16) Braggpeaks with the Pixirad-1 detector at ambient pressure,and the (0 0 4) Bragg peaks with the MAR345 detec-tor while under applied pressure in the DAC. The a lattice parameters were determined the same way us-ing the position of the (16 0 0) and (2 2 0) Bragg peaksfor ambient pressure and with applied pressure, respec-tively. At ambient pressure, the change of the latticeparameters is very consistent with the behavior foundin measurements of the thermal expansion via capac-itive dilatometry or the strain-gauge technique, whichwere discussed earlier in this section, i.e., the in-plane a lattice parameter shrinks and the c parameter increasesupon cooling through the ferromagnetic (FM) transition T FM ’
90 K. The positions of the arrows correspond to T T F M ( d )( c ) ( b ) n e u t r o n p = 0 . 6 G P a a (Å) ( a ) c (Å) n e u t r o n p = 1 . 4 G P a a (Å) c (Å) n e u t r o n p = 1 . 9 G P a a (Å) c (Å) T ( K ) n e u t r o n p = 2 . 5 G P a c (Å) FIG. S8. Lattice parameters a (left axis) and c (right axis)at p = 0 . p = 1.4GPa (b), at p = 1.9GPa (c),and at p = 2.5GPa (d), as determined from neutron diffrac-tion experiments [N1 (a-c), N2 (d)]. The positions of thearrows correspond to the transition temperatures determinedfrom our thermodynamic measurements and approximatelycoincide with the points where the behavior of the latticeparameters deviates from the extrapolated high-temperaturebehavior. the transition temperatures determined from our thermo-dynamic measurements and approximately coincide withthe points where the behavior of the lattice parametersdeviates from the extrapolated high-temperature behav-ior. The lattice parameters show a typical temperaturedependence, consistent with the second-order nature ofthe phase transition. At 1.9 GPa, the FM transition issuppressed to much lower temperatures ( T FM ’
40 K)and still results in a strong increase of the c lattice pa-rameter upon cooling, whereas the a lattice parametershows a discernible decrease. In addition, a very sub-tle, kink-like feature might be discernible at much highertemperatures at T ≈
60 K at 1.9 GPa in the a and c lat-tice parameters, respectively (see black arrow). If indeedpresent, this feature coincides with T . A clearer featurearound T on the basis of neutron diffraction data willbe presented below.Figure S8 shows the temperature dependence of the a c lattice parameters, inferred from neutron diffrac-tion measurements on single crystals at HB1, for var-ious pressures. Again, similar to the discussed otherlow-pressure data sets, the a ( c ) lattice parameters at p = 0 . T FM ( p ), which follow an order-parameter typeof behavior, and thus, are consistent with the notion of asecond-order phase transition. The c lattice parameter at1.4 GPa also shows an order-parameter type decrease at T FM ( p ). At 1.9 GPa, the FM transition is shifted to evenlower temperatures in our neutron and x-ray diffractiondata. In addition, the c lattice parameter shows a kinkat much higher temperatures T ≈
60 K, which is thuslikely associated with the phase transition at T (Thisaspect becomes much clearer from a direct comparisonof the lattice parameter data and the thermal expan-sion data, obtained by the strain-gauge data, which willbe presented in the upcoming section). This kink-likefeature in the c lattice parameter is accompanied by avery subtle change of the slope of the a lattice parame-ter. Increasing pressure even further to 2.5 GPa, we stillfind a kink-like feature in the c lattice parameter closeto T (as identified in our thermodynamic and transportdata), and upon further cooling c shows a increase in thelow-temperature region. This increase is likely associatedwith the (broadened) first-order FM transition, as it getssuppressed closer to zero. Again, we will provide furtherevidence for the underlying phase transition in the nextsection, when we compare the different lattice parameterand length change data sets. Comparison of lattice parameter data from x-ray andneutron diffraction measurements under pressure to rel-ative length change data, obtained from the strain-gauge-based technique under pressure -
After the presentationof the measured data of relative length change and thelattice parameters under pressure, we want to turn tothe explicit comparison of the various data sets, takenat very similar pressures. The result of this comparisonis shown in Fig. S9. [Note that for each panel the axesare scaled such that they correspond to the same relativelength changes, and thus, the overlap of different datasets demonstrate the agreement of the data even on aquantitative level.] For the majority of the data sets, inparticular for all taken at finite pressures p ≥ . c lattice parameter at very low tempera-tures indeed coincides with the increase of (∆ L/L ) c data,which can clearly be associated with a feature of the FMordering. Thus, the c axis increase at low temperaturesis also likely a result of the ferromagnetic order, which issuppressed to very low temperatures. ISOSTRUCTURAL NATURE OF ALL SALIENTPHASE TRANSITIONS
In this section, we show additional x-ray diffractiondata, which demonstrate that all salient phase transi-tions, which were observed in our thermodynamic anddiffraction data, only result in a change of the crystal-lographic lattice parameters, but are not accompaniedby any symmetry changes of the crystallographic struc-ture. This allows us to exclude any type of charge-densitywave or structural phase transition for the high-pressurephases in LaCrGe . High-energy x-ray diffraction of single-crystals -
High-energy x-ray diffraction data were taken on single crystalsof LaCrGe to search for any structural anomalies. Thesedata are shown as two-dimensional images of the ( H H L )plane in Fig. S11, and as longitudinal cuts through theBragg peak (2 2 0) in Fig. S12 for p = 1.5 GPa, 1.9 GPa,and 4 GPa for several temperatures. We note that theprevious study of LaCrGe suggested that at p = 4 GPaand base temperature, LaCrGe is in the new magneticphase (see Fig. S10). Therefore we expect to probe theproperties of the new phase at 4 GPa, well separated fromthe low- p ferromagnetism. From the data in Fig. S11we see no indication of additional Bragg peaks fromLaCrGe within a dynamical range of 10 , which in-dicates that there is no superstructure and no charge-density wave at any pressure. In addition, we show inFig. S12 that we did not observe splitting or broadeningof the Bragg peaks which would indicate a symmetry-lowering lattice distortion.At ambient pressure, LaCrGe was reported to adopta hexagonal perovskite structure with space group P / mmc . The single-crystal x-ray data indicatesthat the crystal structure remains hexagonal or trig-onal and with a c -glide plane parallel to the uniax-ial c -direction through all salient phase transitions atall pressures. Given these constraints and the parentspace group of LaCrGe , P / mmc , several struc-tural phases can occur, (i) P / mmc , (iia) P ¯62 c with a1 ( e ) ( e )( d )( c )( b )
0 G P a ( X ) a (Å) ( a ) - 2 . 0- 1 . 5- 1 . 0- 0 . 50 . 0 c (Å) T ( K ) - 0 . 50 . 00 . 51 . 0 - 2 . 0- 1 . 5- 1 . 0- 0 . 5 T ( K ) - 0 . 50 . 00 . 5 - 2- 10 T ( K ) - 0 . 50 . 00 . 51 . 0 ( D L / L )a (10-3) - 1 . 5- 1 . 0 T ( K ) - 0 . 50 . 00 . 51 . 0 T ( K ) - 0 . 6- 0 . 4- 0 . 20 . 0 ( D L / L )c (10-3) FIG. S9. Comparison of a and c lattice parameters of LaCrGe , determined from x-ray and neutron diffraction experiments,with the relative length change (∆ L/L ) a and (∆ L/L ) c of LaCrGe , determined via the strain-gauge technique, as a function oftemperature, T , for ambient and finite pressures. In each panel, the top shows the comparison of the a lattice parameter (leftaxis, symbols) with the relative length change (∆ L/L ) a (right axis, solid line), and the bottom shows the comparison of the c lattice parameter (left axis, symbols) with the relative length change (∆ L/L ) c (right axis, solid line). Note that for each panelthe left and right axes are scaled such that they correspond to the same relative length changes. X-ray data (abbreviated as X)were taken at ambient pressure (a) and 1.9 GPa (d), neutron data (in cells N1 and N2) at pressures p = 0.6 GPa (b), 1.4 GPa(c), 1.9 GPa (d) and 2.5 GPa (e). Relative length change data (abbreviated as SG) were taken at p = 0.21 GPa (a), 0.51 GPa(b), 1.39 GPa (c), 1.94 GPa (d) and 2.52 GPa (e). The larger error bar in (c) of the lattice parameter data is related to shortercounting time. T FM T T T ( K ) p (GPa) single-crystalXRDpowderXRD
130 K
FIG. S10. Temperature-pressure phase diagram of LaCrGe ,as obtained from the present study, showing the phase tran-sition lines at T FM , T and T . High-pressure data for T istaken from previous works . Purple (dark yellow) symbolsindicate the temperature/pressure combinations, for whichsingle-crystal (powder) x-ray diffraction data were taken withhigh statistics and which are shown in Figs. S11-S13. Dottedlines correspond to extrapolation and not actual data. rotation in the ab -plane of the triangularly arranged Geatoms as a new degree of freedom, (iib) P mc with ashift of the La planes in the c -direction relative to theCr and Ge planes, and (iii) P c which is a combina- tion of both. Phase transitions from the parent spacegroup, P / mmc , to P ¯62 c , P mc or P c would leavethe lattice symmetry and the reflection conditions un-changed, but could be differentiated from analysis of theBragg peak intensities. To investigate whether there is achange of Bragg peak intensities, we performed an x-raydiffraction study on powder samples under pressure, theresults of which will be discussed in the following. Powder x-ray diffraction in a DAC -
To measure thelargest number of Bragg peak intensities possible at oncewithin a DAC we recorded x-ray diffraction data on pow-der samples of LaCrGe down to 20 K. To minimize theeffect of the small sample size and create a better poly-crystalline average, individual scans at each temperaturewere measured on a MAR345 with the DAC at differentangles along an axis perpendicular to the incident beamwhile rocking the sample along another axis perpendic-ular to the incident beam. Each scan was azimuthallyintegrated and combined at each temperature and pres-sure measured. To reduce the impact of comparativelylarge single-crystal grains in the sample, very strong in-dividual Bragg peaks were excluded from the azimuthalintegration. The powder x-ray diffraction data processedthis way is shown in Fig. S13, and shows virtually nochange in the Bragg peak intensity as a function of tem-perature. The single-crystal and powder x-ray diffractiondata strongly indicate that all the phase transitions forLaCrGe are isostructural in nature.2 SilverGasket
FIG. S11. High-energy x-ray diffraction data on a LaCrGe single crystal measured at different temperatures and pressures.Image plots of the ( H H L ) plane are shown in each panel with intensities color coded to a log plot as indicated in the colorbars. The large, non-central, black circles are from masking the Bragg peaks from the diamond anvils in the DAC, whereas thepolycrystalline rings are from the silver foil and the stainless steel gasket, as exemplarily indicated by the white arrows in thetop right panel.
MAGNETISM UNDER PRESSURE FROMNEUTRON AND µ SR STUDIESNeutron scattering under pressure
In Fig. S14, we show the integrated intensity data asa function of temperature of the (1 0 0) Bragg peak (a)and the (0 0 2) Bragg peak (b), which were obtained inneutron diffraction experiments at HB1. For the (1 0 0)Bragg peak, we find a clear increase of the intensityupon cooling through the ferromagnetic transition tem-perature T FM ( p ) (see blue arrows). The positions of thearrows correspond to the transition temperatures deter-mined from our thermodynamic measurements and ap-proximately coincide with the points where the behaviorin the integrated intensities deviates from the extrapo-3 I (a.u.) p = 1 . 5 G P a ( a ) I (a.u.) p = 1 . 9 G P a ( b ) I (a.u.) [ H H p = 4 G P a ( c ) FIG. S12. Cuts along the (
H H
0) direction of the (2 2 0) peakfrom high-energy x-ray diffraction data on a LaCrGe singlecrystal under pressure. The data shows no peak splitting orbroadening, indicating the absence of a symmetry-loweringstructural phase transition. Cuts of the (2 2 0) peak are shownat several temperatures for p = 1.5 GPa (a), 1.9 GPa (b), and4 GPa (c). Data are offset for clarity. lated high-temperature behavior. The increase of the(1 0 0) Bragg peak is fully consistent with a ferromagneticorder with moments aligned along the crystallographic c axis and corresponds to 1.5(3) µ B at p = 0 GPa and T = 5 K. The value of magnetic moment was determinedfrom the (1 0 0) Bragg peak intensity, I , relative to aset of nuclear Bragg peaks and compared to calculatedintensities . For low pressures, the temperature depen-dence of the (1 0 0) Bragg peak shows a typical order-parameter behavior and is thus fully consistent with thesecond-order nature of the FM transition. Upon increas-ing pressure, T FM is shifted to lower temperatures. For1.9 GPa, the ordered moment is 1.4(3) µ B at T = 9 K.We also observe an increase of the (0 0 2) Bragg peakintensity for almost all pressures at T FM , as shown inFig. S14 (b). This effect, however, is likely not relatedto the magnetic order itself, but rather due to extinc-tion effects, i.e., a change of the mosaicity of the crys-tal upon cooling through a transition can lead to a sud-den increase of the neutron intensity as a consequence ofthe suppression of multiple scattering . For example, amagnetic transition can change the mosaicity of a crys-
02 0 04 0 06 0 08 0 01 0 0 01 2 0 0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 002 0 04 0 06 0 08 0 01 0 0 01 2 0 0 ( a ) I (a.u.) p = 1 . 9 G P a( b ) I (a.u.) Q ((cid:176) ) p = 2 . 6 G P a FIG. S13. Powder x-ray diffraction intensities of LaCrGe under pressure. The data shows no significant change of peakintensities which would imply the absence of a symmetry-lowering structural component to all salient phase transitionsat p = 1.9 GPa (a), and 2.6 GPa (b). Data is offset for clarity. tal through magnetoelastic effects and the formation ofmagnetic domains. With x-ray single crystal diffraction,we verified the presence of strong extinction effects byRenninger scans of Bragg peaks by rotating the crystalabout the axis of the scattering vector. We found a vari-ation of Bragg peak intensities as the azimuthal angle ischanged at ambient pressure, characteristic for strong ex-tinction effects. Extinction effects can be expected to belarge for strong nuclear Bragg peaks, such as the (0 0 2)Bragg peak, whereas they are negligible for weak nuclearBragg peaks, such as (1 0 0). In addition, if the changein intensity of the (0 0 2) Bragg peak was magnetic innature, then other Bragg peaks, e.g., (1 0 2) and (2 0 2),corresponding to the same magnetic order, should showa similar increase of magnetic intensity, which was notobserved in our experiment. Since extinction effects aredependent on factors such as the size and shape of aspecific sample, the strain applied to a specific sample,and the scattering configuration, the effect is expectedto be strongly sample-dependent. For our measurementson LaCrGe , extinction release coincides with T FM for p ≤ . p > ∼ . T and T . In Figs. S15 and4
02 0 04 0 06 0 08 0 01 0 0 01 2 0 01 4 0 0 0 2 0 4 0 6 0 8 0 1 0 07 0 0 08 0 0 09 0 0 01 0 0 0 01 1 0 0 01 2 0 0 01 3 0 0 0 T ( b )
0 G P a 0 . 6 G P a 1 . 4 G P a 1 . 9 G P a I (100) (a.u.) ( a ) T F M I (002) (a.u.) T ( K ) FIG. S14. Integrated intensity of the (1 0 0) Bragg peak (a)and the (0 0 2) Bragg peak (b) in elastic neutron scatteringexperiments at HB1 on a single crystal of LaCrGe underpressures up to 1.9 GPa (a) and 3.5 GPa (b). Blue arrowsindicate the position of the ferromagnetic transition at T FM .Data in (b) was offset for clarity. Data at ambient pressurewere taken in experiment N0, data up to 1.9 GPa were taken inN1, data at 2.5 GPa and 3.5 GPa in N2. The positions of thearrows correspond to the transition temperatures determinedfrom our thermodynamic measurements and approximatelycoincide with the points where the behavior in the integratedintensities deviates from the extrapolated high-temperaturebehavior. S16 we show scans along the high-symmetry H and L directions over wide regions of reciprocal space at basetemperature for pressures of 1.9 GPa (a-c), 2.5 GPa (d-f), and 3.5 GPa (g-i). Based on our T - p phase diagram,LaCrGe is ordered ferromagnetically at base tempera-ture for 1.9 GPa and 2.5 GPa, and is in the new mag-netic phase for 3.5 GPa, for which a modulated AFM q was expected . We do not observe a significant inten-sity of the (1 0 0) Bragg peak for 2.5 GPa and 3.5 GPa.The nuclear contribution to the Bragg peak is likely re-duced due to pressure-induced shifts of atomic positionsand changes of lattice parameters. At the same time, theweak (1 0 0) Bragg peak at 2.5 GPa implies at maximumonly a weak FM contribution at base temperature [see in-set in Fig. S15 (d)]. This increase in the intensity is withinthe significance level of our experiment, but nonethelessconsistent with the formation of FM order with smallcorrelation length at base temperature (see also the large λ T in µ SR experiments at a similar pressure, discussedin the main text). At 3.5 GPa at low temperatures, amagnetic contribution to the (1 0 0) Bragg peak can beruled out within our experimental limits, discussed be- low. In addition, we do not observe magnetic superstruc-ture peaks along the high symmetry directions at anypressure, indicating the absence of AFM order, in partic-ular for 3.5 GPa, for which LaCrGe was proposed to bein the AFM Q region down to lowest temperature . Theonly peaks observed during the experiment are structuralBragg peaks from LaCrGe , the pressure cell, and thepressure medium, as indicated in Fig. S15 and Fig. S16.Given the absence of magnetic Bragg peaks, we cancalculate a lower boundary for the observable magneticmoment ( µ ) in our experiment for particular cases oflong-range magnetic order. We discuss three cases ofmagnetic order which were suggested in [3]: (i) long-range FM with µ k c , (ii) an AFM structure consistingof FM- ab planes with µ k c , which are stacked along the c axis in a ++++++++++ −−−−−−−−−− sequence,and (iii) the intermediate case with FM- ab planes stackedin a 100 × + and 100 × − sequence along the c axis. TheFM order would yield a Bragg peak at (1 0 0) with aminimum observable moment of 0.3 µ B at 3.5 GPa. Thesecond structure would yield satellite peaks at positions(1.1 0 0) and (0.9 0 0), which are clearly separate from the(1 0 0) Bragg peak. For those satellite peaks we are sen-sitive to an ordered moment of 0.5 µ B at 3.5 GPa. Wewould observe the small- q peaks for the third structureas a broadened peak at position (1 0 0), but the sensitiv-ity for observing a peak at this position is the same asfor the FM structure at 3.5 GPa, µ = 0.3 µ B .Since we found no evidence for AFM order along high-symmetry directions in the crystal, we then searchedlarger sections of reciprocal space using CORELLI.CORELLI allows for the simultaneous measurement oflarge sections of three-dimensional reciprocal space byutilizing a white-beam Laue technique with energy dis-crimination by modulating the incident beam with a sta-tistical chopper . This allows CORELLI to efficientlyseparate the elastic and diffuse scattering from the sam-ple, and is useful for identifying short- and long-rangeorder. By applying pressure in a DAC at CORELLI wewere able to reach pressures from 0.8 GPa to 3.2 GPa atbase temperature. In Fig. S17 we show a clear increase ofintensity of the (1 0 0) Bragg peak when cooling throughthe FM transition temperature at 0.8 GPa. This observa-tion shows clearly that we are sensitive to the FM transi-tion at CORELLI, and that we can expect to detect AFMorder or short-range FM order with a minimum estimatedcorrelation length of 15 nm with a similar magnetic mo-ment for higher pressures. Note that the estimate of cor-relation length for our CORELLI experiment is 15 nmvs. 12 nm for the HB1 experiment. To search for super-structure peaks we increased the pressure up to 3.2 GPa,at which point we no longer observed the (1 0 0) Braggpeak increasing upon cooling and for which the phasediagram in the main text as well as in Ref. [3] indicatesthe presence of the new magnetic phase below ≈
50 K. InFig. S18 we show 2D images of the (
H K
H H
0) and5( H L ) reciprocal planes at T = 5 K (a, c, e) and 30 K (b,d, f), for which LaCrGe is below T at those pressures.Bragg peaks from LaCrGe are clearly seen alongsiderings from the polycrystalline steel gasket. After an ex-haustive search of peaks within the 3D reciprocal space atCORELLI we found no evidence of superstructure peaksindicative of AFM from LaCrGe . For the FM and AFMphases, introduced above in the discussion of the HB1 re-sults, our sensitivity at CORELLI amounts to µ = 0.4 µ B for the FM phase with µ k c and µ = 0.7 µ B for the AFM(++++++++++ −−−−−−−−−− ) phase with µ k c .In conclusion, we found no indications of long-rangemagnetic order within the high-pressure phase in carefulmeasurements along the high-symmetry directions on thetriple-axis HB1, and within the full 3D reciprocal spacemeasurements done at CORELLI. Summarizing all re-sults from HB1 and CORELLI, our sensitivity for mag-netic order would be a lower limit for correlation lengthof 15 nm for an ordered moment of 1.5 µ B like in the FMphase, or a lower limit of 0.7 µ B for any long-range AFMorder. µ SR data under pressure
General introduction -
During a µ SR experiment, al-most 100% spin-polarized muons are implanted into thesample of interest, where they thermalize at interstitiallattice sites. Once stopped, the muon precesses aroundthe direction of the local magnetic field B at the stop-ping site with the Lamor frequency ω m = γ m µ B with γ m / (2 π ) = 135 . µ sand decays into a positron and two neutrinos. The time-and the direction-dependence of the positron emission ismonitored during a µ SR experiment. From this informa-tion on the emitted positron, the muon precession andrelaxation can be inferred, and thus, directly the localmagnetic field in the sample. The muon therefore is amagnetic micro-probe that allows for tracing of the inter-nal magnetic fields at a local level, and for investigationsof the static and dynamic magnetism.When µ SR experiments are performed on a magneticsample with simple magnetic order, which implies a well-defined magnetic field B at any of the n inequivalentmuon stopping sites ( n ≥
1, depending on the sam-ple), then the superposition of the signals from all ofthe muon stopping sites is observed experimentally. The
P CP CP CP MP M ( i )( h )( g ) ( c )( b )( a ) p = 1 . 9 G P a T = 5 K I (100 cts/min) [ H
0 0 ] ( e )( d ) ( f ) [ H
0 1 ] p = 1 . 9 G P a T = 4 K p = 1 . 9 G P a T = 5 K [ H
0 1 ] p = 2 . 5 G P a T = 3 K I (100 cts/min) [ H
0 0 ] p = 2 . 5 G P a T = 3 K [ H
0 2 ] p = 2 . 5 G P a T = 3 K [ H
0 1 ] p = 3 . 5 G P a T = 3 K I (100 cts/min) [ H
0 0 ] p = 3 . 5 G P a T = 3 K [ H
0 2 ] p = 3 . 5 G P a T = 3 K [ H
0 1 ]
FIG. S15. Selected scans from neutron diffraction experi-ments on a single crystal of LaCrGe at HB1 along the highsymmetry directions [ H H H . Data in (a-c) weretaken in experiment N1, (d-i) in N2. Inset in (d) shows the[ H measured asymmetry (i.e., the normalized difference be-tween positron counts on the detectors in forward andbackward direction) in zero magnetic field for a powdersample, or an aggregate of crystals with random orienta-tion, is given by A (0) P ZF ( t )= n X i =1 A i (0) (cid:20)
13 exp( − λ L, i t ) + 23 exp( − λ T, i t ) cos( γ m B int, i t ) (cid:21) , (S4)with A (0) ( A i (0)) the initial asymmetry of the muon en-semble (of the muon at the i -th stopping site) and P ZF ( t )the time-dependent polarization function of the muon en-semble. The spatial averaging due to the random orien-tation leads to a non-oscillating component with a weightof 1/3 for muons, whose spins are parallel to the inter-nal field vector at the stopping site, B int, i , and thereforeshow an exponential relaxation with rate λ L, i , as wellas an oscillating component with weight 2/3, for which6 - 0 . 5 0 . 0 0 . 502 04 06 00 . 5 1 . 0 1 . 502 04 06 0 0 . 5 1 . 0 1 . 502 04 06 01 . 5 2 . 0 2 . 502 04 06 0 - 0 . 5 0 . 0 0 . 502 04 06 0 0 . 5 1 . 0 1 . 502 04 06 01 . 0 1 . 5 2 . 002 04 06 0 - 0 . 5 0 . 0 0 . 502 04 06 0 0 . 5 1 . 0 1 . 502 04 06 0 P MP M A lP CP C P CP CP C P CP CP M ( i )( h )( g ) ( c )( b )( a ) p = 1 . 9 G P a T = 5 K [ 1 0 L ] ( e )( d ) ( f ) P M [ 0 0 L ] p = 1 . 9 G P a T = 5 K p = 1 . 9 G P a T = 4 K [ 1 0 L ] I (100 cts/min) p = 2 . 5 G P a T = 3 K I (100 cts/min) [ 0 0 L ] p = 2 . 5 G P a T = 3 K [ 1 0 L ] p = 2 . 5 G P a T = 3 K [ 1 0 L ] p = 3 . 5 G P a T = 3 K I (100 cts/min) [ 0 0 L ] p = 3 . 5 G P a T = 3 K [ 1 0 L ] p = 3 . 5 G P a T = 3 K [ 1 0 L ] FIG. S16. Selected scans from neutron diffraction experi-ments on a single crystal of LaCrGe at HB1 along the high-symmetry directions [0 0 L ] (a, d, g), and [1 0 L ] (b, c, e, f, h,i) at base temperature for pressures of 1.9 GPa (a-c), 2.5 GPa(d-f), and 3.5 GPa (g-i). Different regions of reciprocal spacewere measured in the data due to the supports in the PalmCubic Cell blocking incoming and outgoing neutrons for someconfigurations. The labels Al, PC and PM indicate that theobserved peaks are associated with Aluminum (Al), the pres-sure cell (PC), and the pressure medium (PM). Unlabeledpeaks correspond to Bragg peaks associated with the crystalstructure of LaCrGe . Data in (a-c) were taken in experimentN1, (d-i) in N2. the muons precess around the internal field vector. Therelaxation rate, λ T, i , which is associated with the oscil-lating component, is a measure of the width of the staticfield distribution ∆ B int, i /γ m , whereas λ L, i is solely re-lated to dynamical magnetic fluctuations. Note that forLaCrGe , an earlier, ambient-pressure, study showedthat the muon time-spectra was best fitted by consid-ering three inequivalent muon stopping sites. However,the three internal field values B int ,i were found to be soclose to each other, that, for simplicity, we will consideronly one muon stopping site for fitting the data inside thepressure cell, given that the higher background contribu-tion in pressure-cell experiments does not allow for tak-ing high-enough statistics to reliably distinguish differentmuon stopping sites with very similar internal fields.In addition to zero-field experiments, µ SR measure-ments can also be performed in external fields. Here,weak-transverse field (wTF) measurements, in which asmall external field, µ H ext , is applied perpendicularto the initial muon-spin direction, is a commonly usedmethod to determine the onset magnetic transition tem-perature and the magnetic volume fraction. When muons [ H
0 0 ] T (K) p = 0 . 8 G P a T F M I (a.u.) FIG. S17. Neutron diffraction intensity of a LaCrGe sin-gle crystal as a function of [ H p = 0.8 GPa. The arrow, labeled with T FM ,indicates the position of the ferromagnetic transition, as de-termined from our thermodynamic measurements under pres-sure. Data were taken in experiment N3. stop in a non-magnetic sample, the external magneticfield causes a steady precession of the muon spin aroundits direction, giving rise to long-lived oscillations in themeasured µ SR asymmetry. In contrast, when muons stopin a magnetically-ordered sample, then the µ SR signalbecomes more complex and reflects the precession aroundthe vector combination of B int and µ H ext , which dueto the random orientation of the crystallites leads to abroad distribution of precession frequencies. Therefore,the contribution to the muon asymmetry from muons,which do not experience a finite internal fields, can beaccurately determined as a function of temperature. Forthe case of a weak transverse field, i.e., µ H ex (cid:28) B int ,the fitting function of the µ SR asymmetry becomes sim-plified such that A (0) P wTF ( t )= A nmag (0) cos( γ m µ H ext t + φ ) exp( − σ t A mag (0) P ZF ( t ) , (S5)with A nmag (0) [ A mag (0)] the initial non-magnetic [mag-netic] asymmetry, φ a phase factor, σ nm the relaxationrate caused by nuclear moments, and P ZF ( t ) the functiondefined in Eq. S4.Overall, in pressure-cell experiments, a large fractionof the muons stop in the pressure cell ( ≈
50 %). Thisadditional contribution has to be included in the data7
FIG. S18. Several slices of neutron diffraction data taken at CORELLI of a LaCrGe single crystal in a DAC with p = 3.2 GPa.For each panel crystallographic Bragg peaks are indicated in black or white text, and the reciprocal space direction of thecrystal is indicated in blue with arrows. Polycrystalline rings are from the steel gasket with strong intensity modulation arisingfrom the texture and strain in the gasket. Cuts of the neutron data are shown for the ( H K
0) plane (a, b), (
H H L ) plane (c,d), and ( H L ) plane (e, f) at two temperatures, T = 5 K (a, c, e) and 30 K (b, d, f). Data were taken in experiment N3. analysis, so that the measured asymmetry A ( t ) reads as A ( t ) = A s (0) P s ( t ) + A pc (0) P pc ( t ) , (S6)with A s (0) [ A pc (0)] being the initial sample [pressure-cell]asymmetry and P s ( t ) [ P pc ( t )] the sample [pressure-cell]polarization function. The sample polarization functioneither corresponds to P ZF ( t ) for the case of zero-field ex-periments, as defined in Eq. S4, or to P wTF ( t ) for the caseof weak-transverse field experiments (see Eq. S5).The background of the pressure cell is typically de-termined in an independent set of experiments and canthen be described by two depolarization channels (one from nuclear moments and one from electronic moments),using a damped Kubo-Toyabe depolarization function, A ZFpc ( t ) = A ZFpc (0) (cid:18)
13 + 23 (1 − σ t ) exp( − σ t ) (cid:19) exp( − λ PC t ) , (S7)with λ PC the relaxation rate, which is related to elec-tronic moments, and σ PC the relaxation rate, related tothe nuclear moments. For the case of LaCrGe underpressure, it also needs to be taken into account that sam-ples, which do exhibit a strong macroscopic magnetiza-8tion, will induce a magnetic field in their surrounding,which can be felt by muons that stop in the pressurecell. Typically, this is the case for superconducting orferro- and ferrimagnetic samples. As a result, the muonsstopping regions of the pressure cell closest to the sampleundergo a precession around the magnetic field, which isthe vector sum of the applied field and the field inducedby the sample with strong magnetization (the sum is de-noted as B PC ). This leads to an additional depolarizationof the muon spin polarization, the size of which dependson the external field, the field created by the sample aswell as the stopping site distribution of the muons in thepressure cell with respect to the spatial distribution of B PC . In these cases, the pressure cell contribution can-not be determined in an independent set of experimentsor described by the Eq. S7 above, and instead follows inwTF experiments A wTFPC ( t ) = A (0) exp( − λ PC t ) exp( − σ t /
2) cos( γ m B PC t + φ ) , (S8)with λ PC the relaxation rate, the size of which is de-termined by the influence of a sample with macroscopicmagnetization on the pressure cell as well as the elec-tronic relaxation rate, and σ PC the relaxation rate, re-lated to the nuclear moments. The electronic relaxationrate is found to be temperature-independent and was de-termined to be ≈ . µ s − in the non-magnetic stateof LaCrGe , i.e., for T > T FM at ambient pressure, forthe used pressure cell. Therefore, if the muons stoppingin the pressure cell do not experience any field, that iscreated by the sample, then λ PC ’ . µ s − , and thepressure cell asymmetry shows long-lived oscillations. In-stead for λ PC > ∼ . µ s − , the signal is damped, re-flecting the additional depolarization of the precession ofmuons, that stop in the pressure cell, as a result of thefield created by the sample.Following the same line of arguments, any sample thatexhibits a large, remanent magnetization, will distort thepressure-cell µ SR signal. Thus, µ SR measurements insidethe pressure cell also allow for the estimation of whetherthe sample exhibits a remanent magnetization or not.A remanent magnetization is typical for ferromagnets,however, the remanency can be very small, potentiallyeven beyond the resolution of µ SR experiments.Experimentally, the test for a remanent magnetizationis performed in the following way. First, the sample iscooled below the transition temperature in zero field, andan initial µ SR spectrum is recorded. In a next step, theexternal field is ramped isothermally to a specific, finitevalue, held constant for a short period of time, and thenremoved again. Then, at zero field, the µ SR spectrum isrecorded again. The recorded pressure-cell response afterthe application and subsequent removal of the magneticfield can be described by following function A ( t ) = A (0)[(1 − ζ ) G KT ( t ) exp( − λ PC t ) + ζ ] , (S9)with 1 − ζ being the spectral weight of the relaxing com-ponent, G KT ( t ) = 1 / / − σ t ) exp( − σ t /
2) beingthe Gaussian Kubo-Toyabe depolarization function re-flecting the field distribution at the muon site created bynuclear moments and λ PC is the exponential relaxationdescribing the influence of the sample on the pressurecell. Again, λ PC ’ . µ s − implies no remanent mag-netization, whereas λ PC > ∼ . µ s − implies a remanentmagnetization and the exact value of λ PC is expected tobe dependent on the external field that was applied priorto the collection of the spectrum (as long as the appliedfield is smaller than the saturation field). In the presentexperiment, we performed a set of these experiments athigh pressures, p = 2 .
55 GPa, at three distinct temper-atures. At each temperature, in total 5 different fieldswere applied and a spectrum was recorded each time af-ter decreasing the respective field back to zero. µ SR measurements in zero field (ZF) at p = 0 . GPa-
Figure S19 shows selected zero-field µ SR spectra ofLaCrGe at p = 0 . ≤ T ≤ T FM (0 . ’
82 K.The T = 10 K data is shown again separately below inFig. S25. For T < T FM , a well-defined muon spin preces-sion is observed, which confirms the presence of a finiteinternal field B int . For temperatures just below T FM ,i.e., for T = 80.3 K, weak and highly damped oscillationsare observed. For T > T FM (see T = 89.5 K data), noprecession of the muon spins is discernible, indicatingthat B int = 0. The solid lines in Fig. S19 correspondto fits to the experimental data to Eqs. S4 and S6. Thetemperature-dependence of the fit parameters for all in-vestigated temperatures will be discussed below. µ SR measurements in weak-transverse field (wTF) at p = 0 . GPa -
Next, we show selected µ SR time-spectraof LaCrGe at p = 0 . T FM in Fig. S20 (a) as well as on en-larged scales around t ≈ µ s (b). For T > T FM , largeand only weakly-damped oscillations with maximum am-plitude close to 0.25, i.e., the maximum for the used spec-trometer, are observed. This observation corresponds tothe expected precession of the spins in the non-magneticsample and the non-magnetic pressure cell, induced bythe wTF. In contrast, for T < T FM , the oscillations aredamped, since the sample exhibits a strong internal field,but also the pressure cell are exposed to a strong mag-netic field, which is created by the ferromagnetic sample9 ( b ) Asymmetry A t ( m s )Z F , p = 0 . 2 0 G P a ( a ) Asymmetry A t ( m s ) FIG. S19. Short-time µ SR spectra (symbols) of LaCrGe ,taken in a zero field (ZF) and at a pressure p = 0 .
20 GPa fordifferent temperatures. Lines correspond to fits of the data toEqs. S4 and S6. Data in (b) are the same as in (a), but offsetfor clarity. ( b )
Asymmetry A t ( m s ) w T F m H = 3 0 O e p = 0 . 2 0 G P a ( a ) Asymmetry A t ( m s ) FIG. S20. Muon-time spectra (symbols) of LaCrGe , takenin a weak-transverse field (wTF) of 30 Oe and at a pressure p = 0 .
20 GPa for different temperatures (a-b). (b) shows thedata, presented in (a), on enlarged scales around the localmaximum close to t ≈ µ s. Lines correspond to fits of thedata to Eqs. S5 and S6. inside the pressure cell. The ordering therefore leads toan additional depolarization of the muons, which explainsthe strongly reduced amplitude of the oscillations. Themaximum size of the oscillation amplitude for T < T FM is fully consistent with full-volume fraction, which will beelucidated below in more detail, when discussing the de-tailed evolution of the fit parameters with temperature,which are extracted from the fits to Eqs. S5 and S6 (solidlines in Fig. S20), will be discussed in the following. Temperature evolution of µ SR fitting parameters at p = 0 . GPa and comparison with thermodynamic mea-surements -
In Fig. S21, we show the temperature ( T )evolution of the µ SR fitting parameters (a-d) at a pres-sure of 0.2 GPa. This includes the evolution of the in-ternal field, B int , (a) and the transverse relaxation rate, λ T , (b) which were both extracted from fitting the ZF µ SR data, as well as the magnetic asymmetry, A mag ,(c) and the relaxation rate of the pressure cell, λ PC , (d)which were extracted from fitting the wTF data. Wecompare this data with data of the specific heat, ∆ C/T ,(e), the relative length change along the c axis, (∆ L/L ) c and the thermal expansion coefficient, α c , (f) and thetemperature-derivative of the c -axis resistance, d R c /d T ,(g), all taken at similar pressure values.We find that B int ’ µ SR data . B int decreases upon increasing T and ex-trapolates to zero close to T FM ≈
82 K. At lowest tem-peratures, λ T ’ µ s − , somewhat larger, but still con-sistent with previous reports , and increases with increas-ing temperature, until it reaches a peak at T FM , abovewhich λ T decreases rapidly [see Fig. S21 (b)]. This be-havior of λ T , which quantifies the width of the staticfield distribution at the muon stopping site, correspondsto the typical behavior for a sample which undergoes amagnetic transition. Only very close to the phase tran-sition, the field distribution becomes wide as the fieldstarts to occur in the sample when magnetic order devel-ops, whereas well below the ordering temperature, λ T issmall, reflecting the well-ordered magnetism in LaCrGe at 0.2 GPa (see below for a comparison of λ T at 0.2 GPaand 2.55 GPa). A mag is almost constant for T ≤ T FM ata value of ≈ A mag strongly suggests that themagnetic volume fraction reaches 100% at T FM , giventhat the maximum asymmetry of the setup is close to0.25, which was determined in a separate experiment.Above T FM , A mag decreases rapidly to zero, as the sam-ple becomes non-magnetic. λ PC ’ . µ s − at lowesttemperatures [see Fig. S21 (d)], the finite size of which re-flects the influence of the magnetic field, created by theferromagnetic sample inside the pressure cell, on the pres-sure cell. Upon increasing T , λ PC initially stays roughly0
01 0 0 02 0 0 03 0 0 001 0 02 0 00 . 00 . 10 . 00 . 10 . 20 . 30 . 40 . 5 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0 1 1 0 1 2 0 B int (Oe) T F M l T ( m s-1) A mag l PC ( m s-1) ( d ) p = 0 . 2 0 G P a( c ) p = 0 . 2 0 G P a( b ) p = 0 . 2 0 G P a( a ) p = 0 . 2 0 G P a D C / T (a.u.) ( e ) p = 0 G P a ( D L / L ) c (a.u.) ( f ) p = 0 . 2 0 G P a Da c (a.u.)d Rc /d T (a.u.) T ( K ) ( g ) p = 0 . 2 4 G P a FIG. S21. Comparison of several low-pressure microscopicand thermodynamic data sets close to a pressure, p , of 0.2 GPaas a function of temperature, T . (a,b,c,d) Internal field, B int (a), transverse relaxation rate λ T (b), magnetic asymmetry, A mag (c) and relaxation rate of the pressure cell, λ PC , fromzero-field (a,b) and weak-transverse field (wTF) (c,d) µ SRmeasurements at p = 0 . C/T at p = 0 GPa; (f) Relative lengthchange and thermal expansion coefficient along the crystal-lographic c axis, (∆ L/L ) c (left axis) and α c (right axis), re-spectively, at p = 0 . c axis, d R c /d T , at p = 0 .
24 GPa. In (a)-(d), blue dashed lines indicates the position of the anomalies,associated with the ferromagnetic transition at T FM . constant and then starts to decrease as T is approach-ing T FM . However, instead of λ PC just approaching avalue close to zero, the behavior of λ PC is more com-plex. In more detail, λ PC first goes through a minimumat T ≈
76 K, followed by a maximum at ≈
82 K andthen decreases and saturates at a value close to zero for
T > T FM . This complex behavior of λ PC was not dis-covered in the previous study , likely due to the largedata point spacing in temperature. Whereas the maxi-mum in λ PC is highly likely related to the FM ordering Assymmetry A t ( m s ) FIG. S22. Muon-time spectra (symbols) of LaCrGe , takenin zero field (ZF) and at a pressure p = 2 .
55 GPa for differenttemperatures. Lines correspond to fits of the data to Eqs. S4and S6. Data were offset for clarity. at T FM , we speculate that the minimum is rather relatedto the proposed crossover from FM to FM2 in LaCrGe upon cooling at low pressures. However, as we will showbelow, we do not find any corresponding feature in ourthermal expansion measurements. The observed featuresin the µ SR fitting parameters at T FM are well consistentwith the positions of the FM anomalies in the thermo-dynamic and transport studies of the present work (Notethat the shown specific heat data set was taken at ambi-ent pressure, and thus at a slightly smaller pressure thanthe other data sets, which were taken at ≈ µ SR measurements in zero field (ZF) at p = 2 . GPa-
Fig. S22 shows a larger set of zero-field µ SR spectra for10 K ≤ T ≤
61 K. A subset of this data was shown inFig. 3 of the main text, and the evolution of B int and λ T was shown in Fig. 4 of the main text and discussed there. µ SR measurements in weak-transverse field (wTF) at p = 2 . GPa -
Figure S23 shows selected µ SR time-spectra of LaCrGe at p = 2 .
55 GPa, which were takenin a weak-transverse field of 30 Oe after zero-field cool-ing. The temperature evolution of the fit parameters,extracted from fitting Eqs. S5 and S6 to this data, werealready shown and discussed in the main text in Fig. 4 (c)and (d). At this point, we would only like to describethe behavior of these selected data sets, in analogy tothe discussion of the wTF data at 0.2 GPa, across thecharacteristic temperatures T ( p = 2 .
55 GPa) ’
56 Kand T ( p = 2 .
55 GPa) ’
49 K. Figure S23 (a) shows thewTF muon-time spectra for temperatures
T < T and T > T . For T > T , large and only weakly-dampedoscillations with amplitude close to 0.25, i.e., the maxi-mum for the used spectrometer are observed, which cor-respond to the expected precession of the spins in the1 ( d )( c ) ( b ) Asymmetry A t ( m s ) ( a ) Asymmetry A t ( m s )
5 K 2 5 K 4 0 . 3 K 4 5 . 5 K 5 0 . 6 K 5 1 . 7 K 5 3 K 5 4 . 5 K 5 5 . 7 K 5 8 . 3 K 6 0 . 8 K 9 0 K
Asymmetry A t ( m s ) w T F m H = 3 0 O e p = 2 . 5 5 G P a Asymmetry A t ( m s ) FIG. S23. Muon-time spectra (symbols) of LaCrGe , taken in a weak transverse field (wTF) of 30 Oe and at a pressure p = 2 .
55 GPa for different temperatures (a-d). (d) shows the data, presented in (c), on enlarged scales around the localmaxmimum close to t ≈ . µ s. Lines correspond to fits of the data to Eqs. S5 and S6. non-magnetic sample and the non-magnetic pressure cell,induced by the wTF. In contrast, for T < T , the muonoscillations are strongly damped, since the sample ex-hibits a strong internal field, but also the pressure cellis exposed to a strong magnetic field, which is createdby a sample with macroscopic magnetization inside thepressure cell. The ordering therefore results in an addi-tional depolarization of the muons, which explains thestrongly reduced amplitude of the oscillations. FigureS23 (b) shows wTF spectra for temperatures, which arecloser in temperature to T and T . At T = 60.8 Kand 55.7 K, we find oscillations with large amplitude,whereas for T ≤ T and T is shown inFig. S23 (c) and on enlarged scales in (d). These plotsshow that the oscillations become maximally reduced inamplitude at T , whereas their amplitude is still large at T . Note that the magnetic asymmetry, inferred at thispressure at low temperatures, is identical, within the er-ror bars, to the magnetic asymmetry at low pressuresand low temperatures [see Fig. 4(c) in the main text andFig. S21 (c)], consistent with the notion of full magneticvolume fraction. Altogether, this leads to the conclusionfull magnetic volume fraction can only be observed for T ≤ T . Experimental test for a remanent magnetization ofthe sample in µ SR measurements at p = 2 . GPa-
In this section, we present our experimental test of
Asymmetry A ( a ) T = 5 K z e r o f ie ld a f t e r 6 0 0 0 O e Asymmetry A ( b ) T = 3 5 K ( d ) a f t e r 0 O e a f t e r 3 0 0 O e a f t e r 1 0 0 0 O e a f t e r 2 5 0 0 O e a f t e r 6 0 0 0 O e l PC ( m s-1) T ( K ) p = 2 . 5 5 G P a Asymmetry A t ( m s ) ( c ) T = 6 0 K FIG. S24. (a,b,c) Initial zero-field µ SR time-spectra and zero-field spectra after increasing and decreasing the magnetic fieldto 6000 Oe (symbols) at T = 5 K (a), 35 K (b) and 60 K (c)at p = 2 .
55 GPa. Solid lines are fits to the experimental databy Eq. S9; (d) Pressure-cell relaxation rate, λ PC , as a functionof temperature, T , at p = 2 .
55 GPa. λ PC was extracted fromfitting the experimental data. whether LaCrGe exhibits a remanent magnetization at p = 2 .
55 GPa by µ SR measurements, by searching fora change of the pressure-cell µ SR spectrum after mag-netizing the sample. We note that similar experimentswere already conducted in Ref. [3] at a similar pressure of2.3 GPa, i.e., clearly in the region of the phase diagram,in which the new phases exist. In this previous report,2it was argued that no significant response of the pres-sure cell was found at 2.3 GPa, which was interpreted asan absence of an ordering with ferromagnetic componentat this pressure. In the present study, we revisited thisquestion by performing measurements with higher statis-tics. As we will argue below, this new set of data stronglysuggests that LaCrGe does exhibit a remanent magne-tization below T . As outlined in detail in the main text,this observation together with (i) the finite λ PC below T in wTF experiments (see main text) and (ii) the ab-sence of a clear magnetic intensity in the vast majorityof reciprocal space in neutron scattering experiments (seeSI), has led us to redefine the nature of the new magneticphase that is associated with the avoidance of ferromag-netic criticality.The detailed experimental procedure, which was usedto check for a remanent magnetization in µ SR measure-ments, was described in the introduction above. Fig-ure S24 shows the initial ZF muon-time spectrum afterzero-field cooling, together with the ZF time-spectra af-ter increasing the field to 6000 Oe and subsequently goingback to ZF at T = 5 K (a), 35 K (b) and 60 K (c). Ac-cording to our thermodynamic phase diagram [see Fig. 2of the main text or Fig. S4 (b)], LaCrGe is in the FMphase at 5 K (a), in the T phase at 35 K (b) and inthe high-temperature paramagnetic phase at 60 K (c).Solid lines are fits to the experimental data by Eq. S9.The raw data already indicate that for T = 5 K and35 K the muon-time spectra are different from the ini-tial time-spectra after the application of any finite field.In contrast, at T = 60 K, the application of any fieldleaves the ZF time-spectra almost unmodified from theinitial ZF time spectrum. This already strongly suggeststhe presence of a remanent magnetization not only for T = 5 K, but also for 35 K. A quantitative analysis ofthis behavior is obtained from considering the evolutionof the fit parameter λ PC with temperature and magneticfield, which is shown in Fig. S24 (d). We note that the λ PC values obtained here from ZF experiments are usu-ally larger by a factor of ≈ . λ PC values ob-tained in wTF experiments (shown in the main text inFig. 4). Whereas the ZF λ PC is small for 60 K and almostindependent of field, λ PC is clearly larger and strongerfield-dependent for 35 K and 5 K. This all suggests thatfor 5 K and 35 K there exists a remanent magnetization,which creates a field that the muons, which stop in thepressure cell, are exposed to. Thus, this result speaks instrong favor of the fact that the magnetic phase below T also has a remanent magnetization. In this regard, wewould like to note that the finite λ PC below T , whichwas inferred from wTF measurements and is shown inthe main text in Fig. 4(d), is also fully consistent withthat notion. Direct comparison of low- and high-pressure µ SR data
Z F p = 0 . 2 0 G P a T = 1 0 K Asymmetry A Z F p = 2 . 5 5 G P a T = 1 0 K Asymmetry A t ( m s )( b )( a ) ( d ) w T F p = 0 . 2 0 G P a T = 2 0 K Asymmetry A ( c ) w T F p = 2 . 5 5 G P a T = 2 0 K Asymmetry A t ( m s ) FIG. S25. Comparison of zero-field [ZF; (a-b)] and weak-transverse field [wTF; (c-d)] muon-time spectra (symbols) forlow pressure, p = 0 .
20 GPa (a,c) and high pressure, p =2 .
55 GPa (b,d). ZF data were taken at T = 10 K, wTF datawere taken at T = 20 K. Lines are fits to the experimentaldata by Eq. S4 for the ZF data and Eq. S5 for the wTF data. - Finally, we want to explicitly compare the ZF muon-time spectra at low temperatures ( T = 10 K) for 0.2 GPa(a) and 2.55 GPa (b), as well as the wTF time spectraat T = 20 K for the same pressures (c-d). The com-parison of the ZF spectra shows that the precession ismuch stronger damped for high pressures, as also quan-tified by the respective λ T values, which are depicted inFig. S21 and Fig. 4 in the main text, respectively. Thisimplies that the static field distribution (i.e., the disor-der in field the muon experiences) for 2.55 GPa is threeto four times larger than the one at 0.2 GPa, whereasthe size of the internal field B int remains similar. Similarobservations were also made in Ref. [3]. In addition, thecomparison of the respective wTF data shows that thedamping of the muon precession is larger for low pres-sures of 0.2 GPa than for high pressures of 2.55 GPa, i.e.,that λ PC (0 . > λ PC (2 .
55 GPa). This result impliesthat the macroscopic magnetization of the sample assem-bly is smaller for 2.55 GPa than for 0.2 GPa despite thevery similar values of the internal field. Overall, in themain text, these observations have led us to a reinterpre-tation of the magnetism below T for 2.55 GPa in termsof a short-range magnetically ordered state with ferro-magnetic component. ESTIMATION OF THE POSITION OF THETRICRITICAL POINT
In the following, we discuss our estimation of the posi-tion of the pressure-induced tricritical point at ( p tr , T tr ),at which the character of the ferromagnetic (FM) tran-sition changes from second order to first order. To thisend, we will focus on an analysis of the anomalies in thethermal expansion coefficient, given the presence of pro-3 Anomaly asymmetry p ( G P a ) Anomaly width
FIG. S26. Asymmetry (left axis) and width (right axis) of thethermal expansion anomalies in LaCrGe along the crystallo-graphic ab axis, which was shown in Fig. S6. The asymmetrywas determined from ( T r − T m ) / ( T m − T l ), with T m being thetemperatures, at which the peak of the thermal expansionanomaly occurs, and T r and T l being the temperatures, atwhich the thermal expansion reaches 50% of the peak value,respectively. Correspondingly, the width was calculated as( T r − T l ) /T m . Dashed lines are to guide of the eyes to high-light the change of the behavior of the asymmetry close to1.5 GPa. nounced features in this quantity over a wide range ofthe phase diagram. Analysis of the shape of the anomaly -
In thermody-namic quantities, a second-order phase transition oftenmanifests itself in a strongly asymmetric, mean-field typeanomaly, whereas a first-order phase transition usuallyshows up as a symmetric, somewhat broadened peak. Asdiscussed in the main text as well as the SI above, we ob-serve a change from an almost mean-field-type jump inthe thermal expansion coefficient ∆ α i for low pressures toan almost symmetric, sharp peak for high pressures. Thissignals a pressure-induced change of the character of thetransition from second-order to first-order. To quantifythis change and to determine the position of the asso-ciated tricritical point, we evaluated the asymmetry ofthe expansion anomaly ∆ α a by using the following ex-pression ( T r − T m ) / ( T m − T l ) with T m being the temper-ature, at which ∆ α ab reaches its maximum value, and T r ( T l ) being the higher (lower) temperature, at which∆ α ab exhibits 50% of the maximum value of ∆ α ab . Theevolution of the so-determined asymmetry is shown inFig. S26 (left axis). For low pressures, the asymmetryparameter is less than 0.5, signaling a very asymmet-ric anomaly. With increasing pressure, the asymmetryparameter increases rapidly to a value close to 1 (cor-responding to a perfectly symmetric peak), and flattensoff (see dashed line) at a value of ≈ .
2. This behavior therefore meets the expectation for the above-describedchange from second order to first order. Thus, we usethe pressure, at which the asymmetry parameter lev-els off, to determine the position of the tricritical point.This results in p tr = 1 . T cr = 53(3) K was inferred from the thermodynamicphase diagram in Fig. 2 of the main text or Fig. S4 (b)in the SI above. We can also consider the width of the∆ α feature, which we determine via ( T r − T l ) /T m and isdisplayed on the right axis of Fig. S26. The width clearlyshows a strong decrease right around p tr , consistent withan increase in sharpness of the transition feature, oncethe transition becomes first-order. We assign the subse-quent increase of the width with pressure for higher pres-sures, which is on the first glance not consistent with thenotion of a sharp-first order transition, to an increasedslope d T FM /d p , which naturally accounts for an increasein broadening, the higher the pressure is. Measurements of thermal hysteresis -
As a complemen-tary approach, we can also consider the evolution of thethermal hysteresis at the ferromagnetic transition withpressure. In Figs. S27 (a) and (b), we show two exam-ple data sets of the anomalous contribution to the ther-mal expansion coefficient, ∆ α c , around the ferromagnetictransition upon warming and cooling at p = 0 .
35 GPa(a) and 1.9 GPa (b). Whereas we find only a tiny thermalhysteresis for low pressure, which is probably related tothe intrinsic hysteresis of our experimental setup, we ob-serve a clear hysteresis for larger pressures. This confirmsclearly that the transition becomes first order for higherpressures. A quantitative analysis of the evolution of thethermal hysteresis, ∆ T , defined as the difference betweentransition temperatures upon warming and cooling, withpressure is shown in Fig. S27. ∆ T starts to increase at ≈ . PROBING THE PROPOSED FM2 TRANSITION
A previous study on LaCrGe demonstrated, based onresistance measurements at ambient and finite pressures p < ∼ . undergoes a crossover from the well-establishedferromagnetic (FM) state to another ferromagnetic state,which was correspondingly labeled with FM2. However,no clear feature associated with this crossover was de-tected in previous specific heat measurements at am-bient pressure. In this section, we want to discuss towhat extent our present set of thermodynamic, µ SR andneutron scattering measurements under pressure providefurther insight into the presence of this crossover.4 ( c )( b ) p = 0 . 3 5 G P a Da c (10-6/K) ( a ) D T (K) p ( G P a ) Da c (10-6/K) T ( K ) p = 1 . 9 4 G P a FIG. S27. Thermal hysteresis at the ferromagnetic transitionin LaCrGe ; (a) Anomalous contribution to the thermal ex-pansion coefficient along the c axis, ∆ α c , at p = 0 .
35 GPaupon warming (red) and cooling (blue). The rate of temper-ature change was ± α c at p = 1 .
94 GPaupon warming (red) and cooling (blue); (c) Thermal hystere-sis, ∆ T , defined as the difference between transition temper-atures upon warming and cooling, as a function of pressure.Red dotted lines indicate the onset of a measurable thermalhysteresis beyond the experimental hysteresis of the setup. a ab (10-6/K) a c (10-6/K) T ( K ) FIG. S28. Thermal expansion coefficients along the crystallo-graphic ab direction, α ab (top), and along the crystallographic c direction, α c (bottom), vs. T for LaCrGe for p = 0.21 GPa.The orange stars indicate position of the minimum in λ PC ,inferred from the present µ SR measurements at a pressure of0.2 GPa.
Figure S28 shows a plot of the thermal expansionanomalies α i ( i = ab, c ) at p = 0.21 GPa. The orangestars indicate the position of minimum in λ PC , whichwas observed in our µ SR data at 0.2 GPa (see Fig. S21)and which might be potentially related to the FM-FM2crossover. However, our data of the thermal expan-sion coefficients do not show any discernible feature at ( b ) Rc (m W ) ( a ) T T d Rc /d T ( mW /K) T ( K ) T F M
FIG. S29. (a) Resistance of LaCrGe along the crystal-lographic c direction, R c , as a function of temperature, T , for different pressures 0.24GPa ≤ p ≤ R c /d T ,vs. T for the same pressures as in (a). Blue, black and redarrows indicate the position of the anomalies, that are asso-ciated with the phase transitions at T FM , T and T , respec-tively. this temperature nor at any other temperature (also theambient-pressure thermal expansion data does not revealany signature of the crossover). Thus, we cannot provideany thermodynamic evidence for this crossover from ourdata. Similarly, we did not find any anomaly in our neu-tron data of the intensity of the (1 0 0) Bragg peak. RESISTANCE DATA UNDER PRESSURE
In this section, we want to provide more details of ourresistance data set of LaCrGe under pressure. We notethat, in contrast to the previously-published resistanceunder pressure data , which were performed with cur-rent in the ab plane ( R ab ), we performed the presentresistance data set with the current along the crystal-lographic c direction to infer R c . In this way, we explorethe directional anisotropy of the resistance in order todemonstrate that the herein-reported phase transition at T leaves a clear fingerprint in R c ( T ) for high pressures.Figure S29 shows selected data of R c as a func-tion of temperature, T , for different pressures in therange 0.24 GPa ≤ p ≤ R c upon coolingthrough the ferromagnetic (FM) transition at T FM , asso-ciated with the loss of spin-disorder scattering. For veryhigh pressures, e.g., for 2.41 GPa, we find first a smallincrease of R c ( T ) upon cooling through T , before the re-sistance drops quickly below T . This behavior becomesmore apparent when considering the T -derivative of the R c data. For low pressures, d R c /d T , shows a step-likefeature at T FM , which is followed by a broad maximumat lower temperatures. The broad maximum was alsoobserved in previous work and was associated with acrossover to another ferromagnetic state at T FM2 . Wehave discussed the ambiguity of the thermodynamic evi-dence for this additional crossover in the previous section.Irrespective of this discussion, the mid-point of the step-like increase of d R c /d T can be used to infer the transi-tion temperature T FM for low pressures. Upon increasingpressure, the step-like feature in d R c /d T evolves into aclear peak. At the same time, above a finite pressure closeto 1.5 GPa, the broad maximum associated with the po-tential T FM2 becomes indiscernible. Whenever d R c /d T shows a clear peak rather than a step-like feature, see,e.g., the data sets for p ≥ .
66 GPa in Fig. S29 (b), weused the peak position in d R c /d T to infer T FM . Note thatwe find a signature of the FM transition for T ≥ R c ( T ) up to 2.55 GPa, the highest pressure measured inthis experiment (the corresponding data is shown in themain text). In addition to the features that are associ-ated with T FM , we also find clear anomalies at T and T in d R c /d T , see all data sets for p ≥ .
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