Helimagnetic Structure and Heavy-Fermion-Like Behavior in the Vicinity of the Quantum Critical Point in Mn 3 P
H. Kotegawa, M. Matsuda, F. Ye, Y. Tani, K. Uda, Y. Kuwata, H. Tou, E. Matsuoka, H. Sugawara, T. Sakurai, H. Ohta, H. Harima, K. Takeda, J. Hayashi, S. Araki, T. C. Kobayashi
AAPS/123-QED
Helimagnetic Structure and Heavy-Fermion-Like Behaviorin the Vicinity of the Quantum Critical Point in Mn P Hisashi Kotegawa , Masaaki Matsuda , Feng Ye , Yuki Tani , Kohei Uda ,Yoshiki Kuwata , Hideki Tou , Eiichi Matsuoka , Hitoshi Sugawara , Takahiro Sakurai , Hitoshi Ohta , ,Hisatomo Harima , Keiki Takeda , Junichi Hayashi , Shingo Araki , and Tatsuo C. Kobayashi Department of Physics, Kobe University, Kobe 658-8530, Japan Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831 Research Facility Center for Science and Technology, Kobe University, Kobe, Hyogo 657-8501, Japan Molecular Photoscience Research Center, Kobe University, Kobe, Hyogo 657-8501, Japan Muroran Institute of Technology, Muroran, Hokkaido 050-8585, Japan Department of Physics, Okayama University, Okayama 700-8530, Japan (Dated: January 30, 2020)Antiferromagnet Mn P with Neel temperature T N = 30 K is composed of Mn-tetrahedrons andzigzag chains formed by three inequivalent Mn sites. Due to the nearly frustrated lattice with manyshort Mn-Mn bonds, competition of the exchange interactions is expected. We here investigate themagnetic structure and physical properties including pressure effect in single crystals of this mate-rial, and reveal a complex yet well-ordered helimagnetic structure. The itinerant character of thismaterials is strong, and the ordered state with small magnetic moments is easily suppressed underpressure, exhibiting a quantum critical point at ∼ . P, which is most likely effected by the underlyingfrustration effect.
The properties of interacting conduction electrons de-viate from those of free electrons. Such interactions of-ten result in renormalization of the electron mass, whichis the essence in strongly correlated electron systems.They are generally remarkable near an instability, suchas itinerant-localized crossover like the Kondo effect, theproximity of a Mott insulator, and criticality of some de-grees of freedom. Rich physics has been developed insuch backgrounds including their interplay, and quan-tum criticality has been enthusiastically investigated forits ability to induce peculiar behaviors of electrons [1–4].Magnetic frustration is also an important factor to in-duce a wide variety of physical phenomena. It can arisefrom geometrical constraint, or competition of variousshort-range exchange interactions. The effect is gener-ally not well-established in itinerant magnetic systems,because it is thought to be weakened by long-range in-teractions. In itinerant systems, an aspect brought bycompetition of exchange interactions is a stabilization ofhelical magnetic structure, as has long been discussedin systems such as MnP-type materials [5–8]. Addition-ally, in such systems, the Dzyaloshinsky-Moriya (DM)interaction is often effective. It is also short-range ex-change interaction, and contribution to the detailed he-lical structure has been suggested in MnP [9]. The com-peting exchange interactions and the DM interaction arecharacterized by crystal symmetry, general to many ma-terials, and are the key inducers of rich and intriguingphenomena [10–12]. However, their role in the itinerantregime is poorly understood and remains an intriguingissue. Particularly, an interplay of the frustration andquantum phase transition has not been sufficiently ex-plored in d -electron systems.Expecting a remarkable frustration effect and an in- FIG. 1. (color online) (a) Crystal structure of noncentrosym-metric Mn P. Two unit cells are shown. Tetragonal structureof space group I ¯4 is composed of three Mn sites and one Psite. The Mn1 and Mn2 sites form the tetrahedrons, whilethe Mn3 sites form the zigzag chain. (b) The helical mag-netic structure of Mn P was determined at 7 K and at ambi-ent pressure. Three Mn sites are separated into six sites withdifferent modulations of their magnetic moments. Nearly AFcouplings between the magnetic moments almost lying in the ab plane are seen at the Mn1a and Mn1b sites, respectively.The c -axis components are significant at the Mn2 and Mn3sites. duced quantum phase transition, we spotlighted Mn P,whose properties have not been investigated sufficientlyso far. An earlier study of polycrystalline samples hadsuggested an antiferromagnetic (AF) transition at T N =115 K [13], but a recent report has corrected this resultto T N = 30 K [14]. In a latter paper, the M¨ossbauer a r X i v : . [ c ond - m a t . s t r- e l ] J a n spectroscopy and neutron diffraction measurements onslightly Fe-substituted samples suggested an ordered mo-ment below 1 µ B , indicating a remarkable itineracy of theMn 3 d electrons. However, the details of the magneticstructure are unknown [14].Mn P crystallizes in a noncentrosymmetric tetragonalstructure in the I ¯4 space group (No. 82, S ) [15]. Thecrystal structure comprises three inequivalent Mn sitesand one P site (see Fig. 1(a)). The four Mn1 sites, de-noted as α − δ in Fig. 1, are equivalent and form a tetra-hedron. By the symmetrical operation of ¯4, the “ α − γ ”bond is equivalent to the “ γ − β ” bond, leading to anisosceles triangle composed of α, β and γ . The Mn2 sitesare similar in structure to the Mn1 sites, whereas theMn3 sites form a zigzag chain. Another important fea-ture is the short distances between the inequivalent Mnsites: 2.70 ˚A for Mn1-Mn2, 2.54 ˚A for Mn1-Mn3, and2.55 ˚A for Mn2-Mn3. Fifteen different types of Mn-Mnbonds exist within 3.0 ˚A. Competition of the exchangeinteractions is expected among so many Mn-Mn bonds.In this single-crystal study, we revealed that the itin-erant antiferromagnet Mn P possesses unusual physicalproperties, such as a complex helimagnetic structure ac-companied by a double transition and pressure-tunedquantum criticality accompanied by heavy-fermion-likebehavior. The likely cause of these features is the frus-tration effect induced by the underlying structure.Single crystals of Mn P were grown by the self-fluxmethod, as described in the Supplemental Material [16].We performed a range of experiments including resistiv-ity, susceptibility, specific heat, neutron scattering, andNMR measurements. The resistivity and neutron scat-tering measurements were also conducted under pressure.We also calculated the band structure of Mn P. The ex-perimental method, NMR results and the band structurecalculation are described in the Supplemental Material[16].Figures 2 (a-c) plot the temperature dependences of(a) electrical resistivity ( ρ ), (b) T -divided specific heat( C/T ), and (c) magnetic susceptibility ( χ ) of Mn P atambient pressure. The ρ was characteristically convexbelow room temperature and exhibited two anomalies at T N = 30 K and T N = 27 . C/T demonstratedthat both anomalies were second-order-like phase transi-tions. They were also confirmed by successive suppres-sions in χ . As shown in the magnetization curve afterfield cooling to 2 K (Fig. 2(d)), no spontaneous magne-tization occurred in the ground state. Consistent with aprevious report [14], the χ did not follow Curie-Weiss be-havior (Fig. 2(c), inset), indicating the itinerant natureof the magnetism. The intrinsic nature of the doubletransition was further confirmed by NMR measurements[16], and the neutron scattering data shown below.The magnetic structure of Mn P in the ground statewas determined by neutron scattering measurements.The magnetic wave vector was found to be Q =(0 . , . , δ ) with δ ∼ .
16, which differs from the com-mensurate Q suggested in Ref. 14. Fig. 3(a) shows the FIG. 2. (color online) Temperature dependence of (a) elec-trical resistivity, (b) T -divided specific heat ( C/T ), and (c)magnetic susceptibility at 0.1 T. All measurements detectedtwo magnetic transitions; one at T N = 30 K, the other at T N = 27 . ρ above T N is reminiscent of f -electron heavy fermion systems. (d)Magnetization curves at 2 K measured after field cooling. temperature dependence of the scattering intensity at Q = (0 . , . , . T N and exhibited a smallkink at T N , similarly to our NMR results [16]. Themagnetic structure was refined using the 269 magneticreflections measured at ambient pressure. We utilizedtwo programs: the magnetic structure shown in Fig. 1(b)as given by Jana [23], and the modulations of the mag-netic moment along the c axis given by Fullprof [24] andshown in Fig. 4. The resultant magnetic structures areconsistent between the two programs. The crystal and FIG. 3. (color online) (a) Temperature dependence of themagnetic Bragg intensities at (0.5, 2.5, δ ) with δ ∼ . T N and exhibits a smallkink at T N (indicated by arrows). The two transitions arepoorly resolved under pressure, but the scattering is observedat the same wave vector. (b) Temperature dependence of theincommensurability δ of the magnetic wave vector (0.5, 0.5, δ ) at ambient pressure. magnetic-structure analyses are provided in the Supple-mental Material [16]. In the ordered state shown inFig. 1(b), each Mn site is split into two Mn sites withindependently modulated amplitudes and phases. Mn1aand Mn1b (Mn2a and Mn2b) alternate along the c axis,and Mn3a and Mn3b form different zigzag chains. Thecomplex non-collinear structure maintains a two-fold ro-tational symmetry C . At most of the Mn sites, thesize of the total magnetic moment oscillates along the c axis (see Fig. 4), indicating clear ellipticity of the helix.The origin of elliptical helices in d -electron systems isan interesting problem, and has been discussed in FeAs[25, 26]. At all Mn sites, the size of the magnetic momentwas about 1 µ B or less, and was extremely small at theMn2a and Mn3b sites. At the Mn1a and Mn1b sites, themagnetic moments almost presented in the ab plane, andwere approximately coupled in antiparallel between theshortest “ α − β ” (Mn1a) and the “ γ − δ ” (Mn1b) bondsowing to C symmetry, indicating a dominance by AFexchange interactions. In the isosceles triangle composedof α, β and γ , the AF interaction in the “ α − β ” bondfrustrates the equivalent interactions in the “ β − γ ” and“ γ − α ” bonds, similarly to geometrical frustration. Thisyields a significant contribution to disturb the collinearmagnetic ordering. At the Mn2 and Mn3 sites, the mag-netic moments have substantial c -axis components. Thishelical state with the split sites, in which the sizes ofthe moments are significantly different, is conjectured toarise from reduced competition of the exchange inter- FIG. 4. (color online) Variation of the magnetic-moment com-ponents M a , M b , M c , and M total , along the c axis at six Mnsites, measured at 7 K under ambient pressure. M total showsa clear ellipticity of the helix at most Mn sites. M ave de-notes the average value at each Mn site. At the Mn1a andMn1b sites, the magnetic moments almost lie in the ab plane,but the moments are remarkably tilted at other sites. At theMn3 sites, the squares and triangles indicate the two sub-lattices forming the zigzag chain. AF couplings between thesublattices appear in M c ( M a,b ) at the Mn3a (Mn3b) sites.The Fullprof refinement was obtained as R F = 11 . actions among the many Mn-Mn bonds. Contributionof the DM interaction is not clearly seen between theshortest Mn1-Mn1 bonds, which are almost coupled inantiparallel, but it will be important for reproducing theoverall magnetic structure. At T N , the δ abruptly butcontinuously changed by ∼ T N [16], but the weak magnetic intensities in theintermediate phase preclude a detailed analysis of themagnetic structure. We consider that the six Mn sitesundergo partial ordering at T N and that the disorderedMn sites are ordered below T N .The neutron scattering experiment clarified that allthe Mn sites possess static magnetic moments, but theordered states are unusual. At low temperatures, the C/T yielded a large electronic specific heat coefficient γ = 104 mJ/mol K , as shown in Fig. 2(b), correspond-ing to γ V = 3 . K per volume. This is one ofthe larger values in d electron systems [27]. The A co-efficient of the T term in the resistivity was also largevalue ( A = 0 . µ Ωcm/K at ambient pressure). The A/γ ration was 4 . × − , of the same order as theKadowaki-Woods ratio. In the band calculation in thePM state [16], γ band was estimated as 16.3 mJ/mol K for a formula cell. The experimentally obtained γ in theordered state was remarkably enhanced, suggesting thatstrong mass renormalization occurs in Mn P and surviveseven in the magnetically ordered state.
FIG. 5. (color online) (a) Electrical resistivity measured un-der pressure. The two transitions and the residual resistivityare simultaneously suppressed by applying pressure. (b) Re-sistivity behavior in the vicinity of the QCP. The residualresistivity ratio (RRR) is ∼
58. The characteristic convexityappears over a wide temperature range. Inset: The resistivityat low temperature deviates from FL behavior, and obeys a T . dependence. The ρ vs T plots at several pressures areshown in the Supplemental Material [16]. The pressure application drastically changed the elec-tronic state of Mn P, as shown in Fig. 5(a). Both transi-tions (indicated by arrows) were quickly suppressed un-der pressure. The anomaly at T N disappeared above ∼ . T N tended to zero at 1 . − . ∼ µ Ωcm at ∼ . ρ is inherent in the ordered state, although its ori-gin is unclear. The pressure-independent ρ in the PMstate probably originates from imperfectness in the sam-ple, and its small value indicates a high-quality singlecrystal. Figure 6 shows the pressure-temperature phasediagram of Mn P, and the pressure dependences of ρ and A . The continuous suppression of the ordered statesuggests a quantum critical point (QCP) at P c ∼ . P is a rare exam-ple of an easily-inducible QCP; thus far, quantum phasetransitions have been reported in only a few materials[28–30]. Another QCP, where T N reaches 0 K, is alsoexpected at ∼ . δ =0.16 was robust and no pressure de-pendence was observed up to 1 GPa; specifically, δ was0.1608(2) at 0 GPa, 0.160(1) at 0.5 GPa, and 0.162(2)at 1 GPa. The ordered magnetic moment gradually re-duced with increasing pressure, reaching ∼
72% of itsambient-pressure value (on average) at 1 GPa.
FIG. 6. (color online) Pressure dependences of ρ and A coef-ficient, and pressure-temperature phase diagram of Mn P. Inthe vicinity of QCP, the NFL behavior is dominant. The A value is already large in the ordered state and shows a smallpeak at the QCP. Figure 5(b) plots the temperature dependence of re-sistivity at 1.63 GPa, just above P c . The resistivity inthe PM state was convex over a wide temperature range,reminiscent of f -electron heavy fermion systems. Theresistivity remarkably decreased below 50 −
100 K, indi-cating a broad incoherent-coherent crossover in the lowtemperature region. As shown in the inset of Fig. 5(b),the resistivity at 1.63 GPa obeyed a T . dependence atlow temperatures. The vicinity of the QCP was dom-inated by NFL behavior and a distinct peak in the A ,although A was already large in the ordered state. Thisindicates an unusually small entropy release through T N and T N . The A was suppressed at pressures above P c ,and the FL state became stabilized. The observed A in Mn P (0 . − . µ Ωcm/K ) is comparable to those of f -electron heavy fermion systems, and is ∼
30 times largerthan that of MnP, even near the QCP [30] where thehelimagnetic phase terminates [31, 32].Several materials in d -electron systems have been re-ported as heavy fermion systems with strong mass renor-malization. Examples are β -Mn, (Y,Sc)Mn , LiV O ,(Ca,Sr)RuO , Na . Co O , CaCu Ir O , and A Fe As ( A =K, Cs) [33–40]. Such materials commonly exhibit alow characteristic temperature T ∗ , which generally givesa large γ proportional to 1 /T ∗ . Mn P should be in-cluded in this category. In the above oxides and ironpnictides, χ obeys Curie-Weiss behavior at temperaturesabove T ∗ [35, 37–39, 41], suggesting the presence of lo-calized moments at high temperatures. Accordingly, T ∗ is generally interpreted as a localized-itinerant crossover,broadly analogous to that observed in the f -electron sys-tems. The localized moment in d -electron systems pri-marily arises from proximity of a Mott insulator [41–44].However, in Mn P, the deviation of χ from Curie-Weisslaw and the band calculation [16] suggest the obviousitinerant character of the 3 d electrons. It is unlikelythat Mn P is compatible with models based on localized-itinerant crossover.Heavy d -electrons may also arise from the effectsof geometrical frustration, as discussed for β -Mn,Y . Sc . Mn and LiV O . They are divided into mod-els with localized moments [45–50] and itinerant models[51–53]. In the itinerant models, it has been proposedthat frustration enhances the fluctuations of degenerate t g orbitals [52, 53], but Mn P prohibits such degener-acy in principle, because the local symmetry of each Mnsite is low as represented by its notation 1 ( C ). An-other route to heavy mass is degeneracy of the magneticcorrelations inherent in frustration [51]. This route maybe qualitatively adapted to Mn P, which is not a geo-metrically frustrated lattice but exhibits the unusual fea-tures of a strong frustration effect, such as the almost-degenerated magnetic transitions and the splitting intosix Mn sites with quite different magnetic moment. Incontrast to other itinerant heavy-fermion systems, β -Mnand Y . Sc . Mn , Mn P is a clean system, which ex-cludes the possibility that the disorder effect enhances theincoherency. The observed features in Mn P are worthyof further theoretical and experimental elucidations.In conclusion, we performed comprehensive experi-mental characterizations and calculations of antiferro-magnet Mn P. We first identified a complex helimag-netic structure, in which three Mn sites are separatedinto six sites of different sizes and directions of theirmagnetic moments. The helimagnetic structure accom-panied by the double transition suggests competition ofthe exchange interactions, or frustration, in Mn P. Sec-ond, the QCP was easily induced under pressure, a rarebehavior in Mn-based materials. Third, Mn P behavedsimilarly to f -electron-heavy fermion systems, namely, itpresented a convex resistivity curve, a larger electronicspecific heat coefficient than that estimated by the band-structure calculation, and a large A coefficient of the re-sistivity They demonstrate a strong mass renormaliza-tion in Mn P. These noteworthy findings reveal that theitinerant system Mn P comprehends the areas of mag-netism with competing interactions, a quantum critical-ity, and d -electron heavy fermion. It is an excellent ex-ample to merge them and to induce the novel interplay. ACKNOWLEDGEMENTS
We thank Kazuto Akiba for experimental supports.This work was supported by JSPS KAKENHI Grant Number JP15H05882, JP15H05885, JP18H04320, and18H04321 (J-Physics), 15H03689.and 15H05745. This re-search used resources at the High Flux Isotope Reactorand the Spallation Neutron Source, DOE Office of Sci-ence User Facilities operated by the Oak Ridge NationalLaboratory. [1] H. v. L¨ohneysen, A. Rosch, M. Vojta, and P. W¨olfle, Rev.Mod. Phys. , 1015 (2007).[2] P. Gegenwart, Q. Si, and F. Steglich, Nat. Phys. , 186(2008).[3] T. Shibauchi, A. Carrington, and Y. Matsuda, Annu.Rev. C. M. Phys. , 113 (2014).[4] T. Furukawa, K. Miyagawa, H. Taniguchi, R. Kato andK. Kanoda, Nat. Phys. , 221 (2015).[5] S. Takeuchi and K. Motizuki, J. Phys. Soc. Jpn. , 742(1967).[6] A. Kallel, H. Boller, and E. F. Bertaut, J. Phys. Chem.Solids, , 1139 (1974).[7] L. Dobrzynski and A. F. Andresen, J. Magn. Magn.Mater. , 67 (1989).[8] S. Yano, J. Akimitsu, S. Itoh, T. Yokoo, S. Satoh, D.Kawana and Y. Endoh, J. 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Single crystals of Mn P were synthesized using a self-flux method. A mixture of Mn : P = 4 : 1 was putinto Al O crucible and sealed in an evacuated quartzampoule. The ampoule was heated slowly up to 800 o Cand held there for 4 days to promote a homogeneousreaction. After that, the ampoule was heated up to 1100 o C for 2 hours and then cooled to 950 o C for 6 hours.After centrifuging the flux at 980 o C, the bar-shape singlecrystals were obtained. The crystal tends to grow alongthe c axis and the typical size is 1 × × . Thecrystals are robust against air and water, but sensitiveto acid. The remaining flux was removed mechanicallyfrom the crystal.For the single-crystal x-ray diffraction measurement,the data were collected to a maximum 2 θ value of ∼ . ◦ using the angle scans. For all structure analyses, the pro-gram suite SHELX was used for structure solution andleast-squares refinement [1]. Platon was used to checkfor missing symmetry elements in the structures [2]. Theobtained parameters are shown in Tables I and II. It wasfairly consistent with those of the previous polycrystallinesamples [3, 4]. The occupancies estimated at each sitewere more than 99%, ensuring the high quality of thecrystal. TABLE I. Crystallographic data of single-crystal Mn P atroom temperature. Formula Mn PCrystal system tetragonalSpace group I ¯4 (no.82) a (˚A) 9.182 c (˚A) 4.5655 Z R wR P atroom temperature. The l.s. indicates local symmetry.site wyckoff l.s. x y z B eq (˚A ) occup.Mn1 8g 1 0.58059 0.60739 0.7714 0.49 1Mn2 8g 1 0.85655 0.53203 0.0137 0.39 0.99Mn3 8g 1 0.67197 0.71927 0.2468 0.43 1P 8g 1 0.7937 0.54463 0.5109 0.43 1 II. EXPERIMENTAL METHODS
Magnetic susceptibility measurement was performed ina range of 2 −
300 K by utilizing a Magnetic PropertyMeasurement System (Quantum Design). Specific heat measurement was performed in a range of 2 −
50 K by aPhysical Property Measurement System (Quantum De-sign). We measured electrical resistivity using four probeAC method, in which electrical contacts of the wire weremade by a spod-weld method. The high pressure up to3.5 GPa for resistivity measurements was applied usingan indenter-type pressure cell and Daphne 7474 oil as apressure-transmitting medium [5, 6]. The low temper-ature for resistivity measurements was achieved using ahomemade He cryostat, except for the data at 1.38 and1.63 GPa, where a homemade dilution refrigerator wasused. In the He cryostat, the pressure cell was soakedin the liquid, while it was cooled by thermal conductionin the dilution refrigerator. The temperature was mea-suremd from the resistivity of a calibrated RuO ther-mometer.The neutron diffraction measurements at ambientand high pressures were performed on the time-of-flight diffractometer CORELLI at the Spallation Neu-tron Source at Oak Ridge National Laboratory.[7] Highpressure up to 1 GPa was generated with a self-clampedpiston-cylinder cell made of CuBe utilizing Daphne 7373oil. The crystal dimensions were about 1 × × . Thepressure was monitored by measuring the lattice constantof a comounted NaCl crystal. The neutron diffractionmeasurements at ambient pressure were also performedon a triple-axis spectrometer HB-1 at the High Flux Iso-tope Reactor at Oak Ridge National Laboratory. Neu-trons with an energy of 13.5 meV were used, togetherwith a horizontal collimator sequence of 48-80-S-80-240.Contamination from higher-order beams was effectivelyeliminated using pyrolytic graphite filters. The singlecrystal was oriented in the (H, H, L) scattering planeand mounted in a closed-cycle He gas refrigerator.We also performed NMR measurements through aconventional spin-echo method and band calculationsthrough a full-potential LAPW (linear augmented planewave) calculation within the LDA(local density approxi-mation). The results of NMR and the band calculationsare shown in the following sections in the SupplementalMaterial.
III. NUCLEAR MAGNETIC RESONANCE FORPOWDERED SAMPLE
We performed NMR measurements to check that thedouble magnetic transition is intrinsic. Figure 1(a) shows P-NMR spectra measured at several temperatures us-ing the unoriented powdered crystals. In the paramag-netic (PM) state, the sharp spectrum is observed near theposition of zero Knight shift, K . Below T N = 30 K, thespectrum is significantly broadened, and the symmetricbroadening with respect to the K = 0 position indicatesthe AF arrangement of the ordered moments. The sharpsignal, where the internal field is quite weak, remainseven below T N and also T N . A volume fraction of theremaining signal is about 5% at 29 K, which is too small FIG. 1. (a) P-NMR spectra for powdered single crystals ofMn P. The measurements were done with fixing frequency at f = 68 .
35 MHz. The symmetric broadening with the cen-ter at K = 0 position evidences the AF ordered state. Theunexplained sharp signal with small volume fraction remainseven below T N , indicating that it arises from some impurityphase. (b) Temperature dependence of the internal field atthe P site. The continuous and successive development areobserved through T N and T N . to recognize that it is intrinsic. The intensity is gradu-ally weakened as decreasing temperature, and the sharpsignal does not change its shape when crossing T N . Theorigin of this remaining signal is unclear, but it is notrelevant with the second transition at T N . What hap-pened at T N is a change in the shape of the spectrum,which is already broadened below T N . Figure 1(b) dis-plays the temperature dependence of the full-width atthe half maximum, which is estimated with omitting theremaining sharp signals. Obviously, a two-step change isobserved in the spectral width, which is consistent withthe intensity of magnetic scattering shown in Fig. 3(a) inthe main paper, suggesting that the both second-orderlike transitions occur at the identical P sites as the in-trinsic property. IV. BAND STRUCTURE CALCULATION
Band structure calculations were obtained through afull-potential LAPW (linear augmented plane wave) cal-culation within the LDA(local density approximation).Calculated density of states (DOS) for the PM state isshown in Fig. 2, where the partial DOS of the respectiveMn sites and the P site are shown. The DOS at the Fermienergy originates in the 3 d orbitals of three Mn sites, andthe contributions from each Mn site are similar. The 3 d -DOS including all 3 d orbitals roughly forms the rectangleshape with a bandwidth of ∼ . ∼ d electrons. γ band = 16 . is estimated in the PM state for a formulacell. FIG. 2. Density of states (DOS) of Mn P calculated for thePM state. The partial DOS from the respective Mn sites andthe P site are shown by different colors. The 4 s contributionof Mn and 3 s contribution of P are omitted, because they arenegligibly small. The DOS at the Fermi energy is composedof 3 d electrons of three Mn sites. The rectangle shape of the3 d -DOS reflects the itinerant character of Mn P. The γ band =16 . is estimated. V. CRYSTAL AND MAGNETIC-STRUCTUREANALYSES
The neutron diffraction measurements were performedon a time-of-flight neutron diffractometer CORELLI in-stalled at Spallation Neutron Source at Oak Ridge Na-tional Laboratory. The single crystal was mounted with( HK
0) in the horizontal scattering plane in the pressurecell. Thanks to the wide coverage of detectors vertically,Bragg reflections with finite L were observed. The crys-tal and magnetic-structure analyses were performed us-ing the FullProf [8] and Jana [9] packages.The observed Bragg reflection intensity ( I obs ) is pro-portional to the squared structure factor ( F ). I obs = C · N ( λ ) · A ( λ ) · Y ( λ, θ, F cal ) · L ( λ, θ ) · f ( λ, θ ) · F , (1)where C , N , λ , A , Y , L , θ , F cal , and f are constant, inci-dent neutron beam flux, wavelength, sample absorption,extinction correction, Lorentz factor, half the scatteringangle, calculated structure factor, and magnetic form fac-tor, respectively. f is unit for nuclear Bragg peaks. Wefirst performed crystal structure analysis at each pressureto determine C and Y in Eq. (1). As shown in Figs. 3(a),(c), and (e), the analysis of the nuclear Bragg intensitieswas performed reasonably well. The detailed magneticstructure analysis was performed at ambient pressure.The magnetic structure models were evaluated using therepresentation analysis with the program BasIreps inFullprof Suite [10] to determine the symmetry-allowed FIG. 3. (a)-(c) and (e) are plots of F vs. F measuredat 7 K. (a), (c), and (e) display nuclear Bragg reflectionsat 0 ( R F =3.68%), 0.5 ( R F =2.40%), and 1 GPa ( R F =4.24%),respectively. (b) displays magnetic Bragg reflections at 0 GPa( R F =11.3%). The solid lines correspond to F = F . (d)and (f) show the relation between F at ambient and highpressures. The solid lines are the results of fits to a linearcurve. magnetic structure. The magnetic structure, shown inFigs. 1 and 4 in the main paper, was determined. Thefitting was reasonably good, as shown in Fig. 3(b).At 0.5 and 1 GPa, the number of observable magneticBragg peaks is quite limited due to large backgroundsignal and low beam transmission originating from theCuBe pressure cell. Therefore, we did not perform mag-netic structure analysis for the data at the two high pres-sures. Instead, the observed magnetic Bragg intensitieswere compared with those at ambient pressure and theoverall reduction factor for the magnetic moments wasobtained, assuming that the overall magnetic structuredoes not change. Figs. 3(d) and (f) show the relation ofobserved magnetic structure factors at ambient and highpressures. The linearity is reasonably good at 0.5 and 1GPa, suggesting that the above assumption is adequate.The square root of the slope of the linear curve corre-sponds to the reduction factor of the overall magneticmoments. The factor is 1.03(10) and 0.72(7) at 0.5 and 1GPa, respectively. The magnetic moments do not changeat 0.5 GPa but slightly decrease at 1.0 GPa. VI. CHANGE IN THE WAVE VECTOR AT T N Figure 4 shows the magnetic Bragg intensities just be-low T N = 27 . T N without any signatureof broadening or a splitting of the peak. The experimen-tal data suggets that the transition at T N is continuouswithin the experimental resolution. FIG. 4. Magnetic Bragg intensities at (0.5, 0.5, L) for severaltemperatures. The change in the wave vector below T N =27 . VII. MAGNETIC STRUCTURE IN THEINTERMEDIATE PHASE
FIG. 5. Temperature dependence of integrated intensities ofthe magnetic Bragg peaks around (0.5, 0.5, 0.16) and (0.5,0.5, 0.84) measured at ambient pressure. The (0.5, 0.5, 0.16)and (0.5, 0.5, 0.84) intensities are normalized at 5 K.
As described in the main paper, the magnetic structureanalysis in the intermediate phase is challenging, since0the magnetic Bragg intensities are very weak. In orderto understand the magnetic structure in the intermediatephase qualitatively, we measured detailed temperaturedependence of the two magnetic Bragg intensities around(0.5, 0.5, 0.16) and (0.5, 0.5, 0.84). The (0.5, 0.5, 0.16)reflection consists of 83% from the spin component inthe ab plane ( M ab ) and 17% from the spin componentalong the c axis ( M c ). On the other hand, the (0.5, 0.5,0.84) reflection consists of 15% from M ab and 85% from M c . Therefore, the former and latter reflections representmostly M ab and M c , respectively. [The (0.5, 2.5, 0.16)reflection, shown in Fig. 3(a) of the main paper, consistsof 98.4% from M ab and 1.6% from M c .] Figure 5 showstemperature dependence of magnetic Bragg peaks around(0.5, 0.5, 0.16) and (0.5, 0.5, 0.84) normalized at 5 K. Wefound that the (0.5, 0.5, 0.84) intensities are more thantwo times stronger than the (0.5, 0.5, 0.16) intensitiesin the intermediate phase, suggesting that M c is moredominant in the intermediate phase. If the Mn2b site alsocarries large M c in this phase as in the low temperaturephase, the Mn2b moments might order mostly below T N and other moments might order below T N . VIII. SYSTEMATIC RESISTIVITY DATAUNDER PRESSURE
Figure 6 shows the ρ vs T plot for various pressures.At ambient pressure, the ρ obeys a T dependence atlow temperatures, but it is acheived only below ∼ . A coefficient is 0 . µ Ωcm/K . Thetemperature range of the T dependence shrinks nearthe QCP ( P c ∼ . A is maintained upto the pressure just above the QCP, and the A decreasesat higher pressure. FIG. 6. ρ vs T plot for various pressures. The data at lowerpressures are shifted, because ρ is large in the ordered state.The straight line is used for the estimationof the A coefficient.[1] G. M. Sheldrick, Acta Crystallogr., Sect. A 64 (2008)112.[2] A. L. Spek, J. Appl. Crystallogr. 36 (2003) 7.[3] S. Rundqvist, Acta Chem. Scand. , 992 (1962).[4] H. -p Liu, Y. Andersson, P. James, D. Satula, B. Kalska,L. H¨aggstr¨om, O. Eriksson, A. Broddefalk, P. Nordblad,J. Magn. Magn. Mater. , 117 (2003).[5] T. C. Kobayashi, H. Hidaka, H. Kotegawa, K. Fujiwara,and M. I. Eremets, Rev. Sci. Instrum. , 023909 (2007). [6] K. Murata, K. Yokogawa, H. Yoshino, S. Klotz, P. Mun-sch, A. Irizawa, M. Nishiyama, K. Iizuka, T. Nanba, T.Okada, Y. Shiraga, and S. Aoyama, Rev. Sci. Instrum. , 085101 (2008).[7] F. Ye, Y. Liu, R. Whitfield, R. Osborn and S.Rosenkranz, J. Appl. Crystallogr. , 315 (2018).[8] J. Rodriguez-Carvajal, Physica B , 55 (1993).[9] V. Petricek, M. Dusek, and L. Palatinus, Z. Kristallogr. , 345 (2014); V. Petricek et al., Z. Krist.-Cryst.Mater.231