Magnetovolume effect, macroscopic hysteresis and moment collapse in the paramagnetic state of cubic MnGe under pressure

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by  N. Martin

Magnetovolume effect, macroscopic hysteresis and moment collapse in the
paramagnetic state of cubic MnGe under pressure
N. Martin,1 I. Mirebeau,1 M. Deutsch,2,3 J.-P. Iti´e,2 J.-P. Rueff,2,4 U.K.
Ro¨ssler,5 K. Koepernik,5 L.N. Fomicheva,6 and A.V. Tsvyashchenko6,7
1Laboratoire L´eon Brillouin, CEA, CNRS, Universit´e Paris-Saclay, CEA Saclay 91191 Gif-sur-Yvette, France
2Synchrotron SOLEIL, L’Orme des Merisiers, Saint-Aubin, 91192 Gif-sur-Yvette, France
3Universit´e de Lorraine, CRM2, UMR UL-CNRS 7036, BP 70239, 54506 Vandoeuvre-l`es-Nancy, France
4Sorbonne Universit´es, UPMC Univ Paris 06, CNRS,
6
1 Laboratoire de Chimie Physique-Mati`ere et Rayonnement, 75005 Paris, France
0 5IFW Dresden, PO Box 270116, 01171 Dresden, Germany
2 6Vereshchagin Institute for High Pressure Physics,
Russian Academy of Science, 142190 Troitsk, Moscow, Russia
n 7Skobeltsyn Institute of Nuclear Physics, MSU, Vorobevy Gory 1/2, 119991 Moscow, Russia
a
(Dated: January 21, 2016)
J

0 Itinerant magnets generally exhibit pressure induced transitions towards non magnetic states.
2 Using synchrotron based X-ray diffraction and emission spectroscopy, the evolution of the lattice
andspinmomentin thechiralmagnet MnGewasinvestigated intheparamagnetic stateandunder

] pressuresupto38GPa. Thecollapseofspin-momenttakesplaceintwosteps. Afirst-ordertransition
i
c withahugehysteresisaround7GPatransformsthesystemfromthehigh-spinatambientpressureto
s alow-spin state. Thecoexistence of spin-statesand observation of history-dependingirreversibility
-
l is explained as effect of long-range elastic strains mediated by magnetovolume coupling. Only in a
r second transition, at about 23GPa, thespin-moment collapses.
t
m
.
t Magnetism in electronic systems is fundamentally spin-orbit coupling resulting in long wavelength helical
a
m unstable with respect to lattice compression. Spin- spin-structures. Among these compounds, MnGe stands
stateinstabilitiesandtransitionsbetweendifferentband- out. Its B20-structure (space group P2 3) is metastable
- 1
d magnetic states cause thermodynamic anomalies under at room temperature and powder samples are obtained
n temperature or pressure changes. Probably, the best byhightemperature(800-2200K)andhighpressure(2-8
o
knownanomalyofthis type isthe invareffect, yielding a GPa) quench during the synthesis. Interestingly, MnGe
c
paused thermal expansion around room-temperature in displays the shortest helical pitch (∼ 30 ˚A) of the B20
[
Fe-Nialloysandvariousmetallicmaterials[1–3]. Thisef- family[19–22],resultinginagianttopologicalHalleffect.
1
fectiswidelyemployedinindustrialapplications. Gener- Toexplainit,acomplexskyrmionlatticewaspostulated,
v
ally, it is believed thatmodifications in the magnetic be- butitsexistencedowntoTandH≃0isdebated[23–25].
2
3 haviorandmagnetovolumecouplingunderlysuchanoma- On the other hand, an inhomogeneous fluctuating chiral
3 lies. An early explanation by Weiss has been based on phase was observed over a very large temperature range
5 existence of a high spin (HS) state with large volume [24].
0
and a metastable low spin (LS) state of reduced volume.
. Following a theoretical prediction [26], pressure-
1 Thermalactivationofthe LSstate counteractsthe usual
inducedcollapseofmagnetisminMnGeshouldtakeplace
0 expansion of the lattice with increasing temperature [4].
6 in two steps between the equilibrium HS-state towards
Band-theory calculations on Fe-based alloys and com-
1 the zero-spin (ZS) state through an intermediate LS-
poundswithinvar-anomalieslatersupportedthebasicas-
: state. Evidence for a HS to LS transition in MnGe was
v
sumptionofadiscontinuoustransitionandthe magneto-
i indeedfoundbyhighpressureneutrondiffraction[27]. At
X volumeeffectassourceofinvaranomalies[5]. Similaref-
lowtemperature,theorderedMnmomentdecreaseswith
fects with intermediate spin-states or reduced magnetic
r increasing pressure in the HS state up to a critical pres-
a moments have been described in other solid-state sys-
sure P ≃ 6 GPa, then remains constant, in excellent
C1
tems, such as certain transition metal oxides [6, 7] or
agreement with calculations on the transition between
molecular complexes [8, 9]. For the metallic invar-like
HS and LS spin-state. The N´eel temperature T was
N
systems, a coherent physical picture of such magneto- seentoreduceatarateof−14K·GPa−1. Atanextrap-
volume effects and the fundamental mechanisms could
olated pressure P ≃ 13 GPa the magnetic long-range
0
notbeachieved. Especiallythe existenceofintermediate
order should vanish, but the pressure-collapse between
spin-states and discontinuous transitions between mag-
the LS and ZS-state was not observed, yet.
netic states is debated [10–12], while being suggested by
several experiments [13–18]. Inthis work,wereportthe observationofa clearfirst-
order transition around 7GPa at room-temperature, far
MnGe belongs to the family of cubic chiral helimag- above the magnetic ordering-temperatureT ≃ 170K in
N
nets,wherethedominantferromagnetismcompeteswith MnGe, and the spin-collapse in the paramagnetic state
2
is found at about 23 GPa. The results demonstrate a In a first run, the applied pressure was increased up
discontinuous evolution and a co-existence of two mi- to ∼ 17 GPa - that is deeply inside the LS state - and
croscopic spin-states in an ordered metallic compound. then progressivelyreleased. The compressioncurve does
This remarkable invar-like effect highlights the impor- not display drastic change of behavior (see Fig. 2a). We
tance of long-range lattice strains in the spin-transition attribute the EoS corresponding to this process to the
taking place in a chemically clean system. Such elastic initial HS-state which progressively transforms into the
strainsmediated by magnetovolumiceffect arecrucialto LS-state. Parameters from Murnaghan EoS fit to the
explain the anomalous properties of MnGe. data are gathered in Tab. I. However, upon decompres-
In order to understand the spin-state-transitions in sion, a remarkable structural hysteresis occurs, signaling
MnGe, we performed experiments to detect the collapse the occurrence of a phase transition, across which a siz-
of the local Mn moment, and not only the ordered one. able LS proportion remains stabilized until pressure is
Moreover, by monitoring the evolution of the lattice pa- fully released.
rameter under pressure, we could detect magnetovolume
effectsinducedbythedifferentvolumesandcompressibil-
ities of the HS and LS-state. Synchrotron-based X-ray 110
T = 300 K, run 1
tseucrhesniwqeulelsabaoreveidPeallycasnuibteedrefaocrhtehdebseyutwsiongtamskems.bPrarnees-- 3V (Å) 108 CDoemcopmrepsrseisosnion
C,2 me 106
ttvyhopelueemaderilasi.merFodnigad.taa1nwviisitlhtchereellss(uP(lt,DsTAp)C-rpe)shewansitetehddviiaengrrytahmsims lcaeoltlmtsebari.mnipnlge Unit cell volu 110042
100 a) PC1
0 5 10 15 20
300 2.5
High Spin Low Spin Zero spin Pressure (GPa)
Temperature (K) 22115505000000 HPaerliammaaggnnePettC1 P0 NoPn-Cm2agnetic 2110....0505 BOrdered moment (/Mn)µ 3Unit cell volume V (Å) 1111119910000086086420 b) PC1 PC2T = 300 K, run 2
0 0.0
0 5 10 15 20 25 30
0 5 10 15 20 25 30
Pressure (GPa)
Pressure (GPa) 110 T = 300 K 200 HS+LS
108 LS 2000
T1Fdlme.iIht8ffeGtenreµ.atrvBc)1aa.t:,rliueo(iTennPcwl,oe[rT2foae7)drt-]lhw)pyeaehasnlaenled-xspdsepeaspeldryoraaniwiatrmcegahdr(teramenobdmttear∼lrlooolenywfg0-Mi.dbot7eahnntsµesGeeBrdTwem,Nxiiit(inn-nhPrfeea)dbrhy(rliougegtrdreehde)cefeh(rnor0noer.misdd7qoeulmrn≤iedeesod)umm(ltmtirenhono≤eints-. 3Unit cell volume V (Å) 11110000964208 HS LS ZS ∆E (meV / f.u.)11150250050 ∆E (P = 0) 51100400500E (K / f.u.)∆
(black squares) along with the DFT rescaled results for the 96 c) d)
0 0
HS (red circles) and LS (bluecircles) are also added. 0 5 10 15 20 25 30 100 104 108 112
Pressure (GPa) Unit cell volume (Å3)
FIG.2: Experimental(P,V)EoSofMnGededucedfromour
X-ray powder diffraction (XRD) discriminates differ- high pressure diffraction experiment. Results from the first
entspinstatesbymeasuringhigh-resolution(P,V)equa- a)andsecondb)runareoverlayedinc). SolidlinesareMur-
naghan EoS fits to the data. d) Volume dependence of the
tions of state (EoS). XRD was performed at room tem-
perature on the PSICHE´ beamline of the synchrotron energyfortheHS+LSandLSstatesofMnGecalculatedfrom
theexperimental parameters of theEoS.
SOLEIL. No indication of a structural phase transition
wasfounduptothehighestpressureof30GPa,implying
theabsenceofsymmetrychangeoratomicdisplacements
withintheunit-cell([28]). Wethereforefocusontheunit In order to study the metastability of the HS-LS mi-
cell volume V =a3 where a is the cubic lattice constant crostructure, also observed in invar alloys [30], we have
deduced from Rietveld refinements. In order to describe prepared a second sample that was loaded in a DAC at
itspressure-dependence,weusetheso-calledMurnaghan an initial pressure of ≃ 7 GPa, maintained for about a
equation of state [29], V(P)/V = [1+P B′/B ]−1/B0′ weekpriortothemeasurement. Wehavethenquicklyre-
0 0 0
′
where B is the isothermal bulk modulus, B its first leasedtheappliedpressureanddeterminedtheEoSupon
0 0
pressure derivative and V =V(P→0). compression in the 0-30 GPa range. As seen on Fig. 2b,
0
3
aclearchangeintheEoSslopenowoccursatca. 7GPa. follow minor hysteresis loops.
A Murnaghan fit to the whole dataset gives unphysical In order to address the magnetic collapse in MnGe on
′
values, B0 = 90(5) GPa and a very large B0 = 13.5(6) a local scale, we performed hard X-ray emission spec-
We conclude that the low pressure range concerns a HS- troscopy (XES) measurements under pressure at 300 K.
LS composition that depends on the thermal and pres- Hard X-ray emission spectroscopy (XES) is sensitive to
sure prehistory. Considering the pressure range above 7 the local moment and earlier detected the pressure in-
GPa, we obtain the parameters for the EoS that can be ducedcollapseofmagnetismininvaralloys[33].Theemis-
attributed to the LS-state (Tab. I). sionspectrawererecordedupto38 GPaontheGALAX-
Our results show that a first-order transition takes IESbeamlineofthesynchrotronSOLEIL(see[34]). The
place where specific volume and compressibility are dis- element-specific photonemissionatthe Kβ line ofMnis
continuouslychanged. Amaximalpressureof7.2(5)GPa bound to spin-sensitive selection rules. While the sys-
is estimated where the two states can coexist. Following temis excitedby the incomingphotons,the finalstateis
the observation on the evolution of the magnetic mo- characterized by a core hole (3p) that interacts with the
ments around the same pressure in the earlier magnetic 3dn electrons via intra-atomic exchange. This results in
neutron diffraction [27], we identify the coexisting two the energy splitting of the emission line, yielding a main
states as HS and LS spin-states and the estimated max- peak Kβ at a photon energy of 6485 eV paired with
1,3
imum pressure compares rather well with the previous a low-energy shoulder Kβ′ located around 6475 eV (see
determination of PC1. Fig. 3a). AdecreaseinthelocalMnmomentshouldyield
′
a decrease in the intensity ofthe Kβ line relative to the
main peak. In an itinerant magnet such as MnGe, the
TABLEI:ComparisonofMurnaghanEoSparametersderived
variationoftheXESsignalishowevermuchsmallerthan
from DFT and determined experimentally.
inoxydes[35]. Phenomenologically,awaytomonitorthe
B0 (GPa) B0′ V0 (˚A3) evolution of the local moment is to consider the integral
HS+LS(run 1, Fig. 2a) 154(3) 2.6(4) 110.26(4) of the difference between a spectrum measured at a cer-
HS+LS(run 2, Fig. 2b) 119(7) 3.4(9) 109.48(8) tain applied pressure P with reference spectrum (in our
LS (run 2, Fig. 2b) 237(3) 4.3(2) 106.6(4) case measured at P = 38 GPa), both being normalized
tounityafterappropriatebackgroundsubtraction([28]).
HS(DFT) 148 2.5 107.9
The integralis solely performed around the satellite fea-
LS (DFT) 165 3.7 103.2
tureinordertogetridofpressure-dependentbroadening
ZS (DFT) 177 4.7 102.4
of the main peak ([36]).
In Tab. I, density functional theory (DFT) results on
the (P,V) equation of state (EoS) demonstrate the ex-
pectedmagnetovolumeeffects inthe threedifferentspin- The result is displayed in Fig. 3. The differential in-
states. This determination of the EoS calculation uses tensity δ decreases as pressure increases, up to about
the full potential local orbital approach [31], and has ≃25GPawhereitsaturatesto0withinerrorbars. This
an improved accuracy by using an extended set of basis is indicative of another spin transition towards a state
states corresponding to the state-of-the-art [32]. These with a moment value that is lower than that of the LS
calculations yield qualitatively similar changes for the state. Based on the good correspondence with DFT re-
EoS between HS and LS state (see Tab. I). The lower sults [27], we identify this transition as the local LS-ZS
equilibriumvolumesV canbeexplainedbythefactthat transition expected in this pressure range as a complete
0
the DFT-results can reproduce only the homogeneous collapse of spin-polarization. To estimate the associated
T = 0 ground-state (namely, they do not include ther- critical pressure, we have fitted the data by the power
mal lattice expansion and also neglect certain effects of law, δ(P) = δ ·(1−P/P )β for P ≤ P and 0 oth-
0 C2 C2
magnetic fluctuations). There are notable differences for erwise, yielding a critical pressure P = 22.7(1.8) GPa
C2
the bulk moduli B , but both findings agree in that the with δ = 1.1(1)·10−3 and β = 0.38(15). Such a scal-
0 0
LS state possesses a smaller V than the HS state, while ingisexpectedifthereisafluctuation-dominatedtransi-
0
being much less compressible. tionfromparamagnetictonon-magneticstateandshould
There is a remarkable history dependence of the effec- obey3D-Isingcriticality,butwithpressureascontrolpa-
tive EoSand hence the spin state composition in MnGe. rameter that drives the transition because of the differ-
The results of lattice parameter vs pressure from both ent volumes of the LS and ZS-state. On the other hand,
experiments runs are overlayed in Fig. 2c. The differ- the HS-LS transitionis hardly observable using the XES
encebetweenthecompressioninthefirstandsecondrun techniqueinthismetalliccompound,asthefinemultiplet
proves that the internal mixed state, starting at ambi- structure is not well established in comparisonto the lo-
ent pressure, must have been different. The two cycles calizedspin-states ofaninsulator andmay be influenced
probedhere,bythemixednatureofinitialstates,clearly by temperature.
4
10x10-3 cleationofmetastableLSspinstatesinthedominantHS
a) T = 300 K state. Such scenario would explain the large variability
sity 8 Kβ' 53.82 GGPPaa of magnetic properties reported in literature depending
n
d inte 6 Kβ1,3 ownitthhiemspyunrtihteiessis, cooffn-sdtiotieocnhsioomfeMtrnyGoer,rwahnidcohmisdniosotrldinekr.ed
ze 4
ali Onthe otherhand, the suppressionofLS-statesinthe
m
or 2 ambient pressure HS-matrix, and their metastable co-
N
0 existence implied by the pressure-hysteresis requires a
coupling that prevents a simple pressure-driven transi-
6470 6480 6490 6500
tion in a jump-like process. In the paramagnetic state,
Outgoing photon energy (eV)
where long-rangemagnetic orderis absent, the elasticity
-1V) T = 300 K of the lattice remains the sole explanation for the real-
δe (e 1.0x10-3 PC2 itzeantdioednopfretswsoureenerargnegtei.caAllysidzieffaebrleenmtsapginne-tsotavtoelsumineaenffeexct-
c
en 0.5 implies that the lattice is strained when locally a spin-
er
diff statetransitiontakesplace. Thesestrainseffectivelyme-
ctral 0.0 diate long-range couplings between the sites that slowly
pe b) decaywithdistanceasr−3,actingasanenergeticbarrier
S
-0.5
0 10 20 30 40 againstasizablenucleationofLS-states[37–40]. Namely
the local strains prevent the sites from permanently oc-
Pressure (GPa)
cupying the minority spin state. The spin state could
bechangedbetweenHSandLSthroughthermalfluctua-
FIG. 3: a) Typical response measured at room temperature
at5.2GPa(HS)and38GPa(ZS),illustratingtheweakinten- tions, realizing the conditions of an ”open” system. The
sitydecreaseatthelowenergysatellite. Grayedzonedenotes elastic energy does not depend on the spatial arrange-
the energy range used for the integration. b) Pressure evo- ment of the LS-sites (a fact known as Crum-Bitter the-
lution of the integrated difference between spectra measured orem for isotropic elastic two-phase bodies [37]). In the
at each applied pressurevalueand thehighest pressurespec-
idealcaseofahomogeneoussystem,thisbarrierprevents
trum. Solid line is a power-law fit (see text).
the transformationuntil the stability limit of the matrix
phase is reached. In the real case, the coexistence of
thetwospinstates,consideredasthermodynamicphases,
We alsomeasuredthe evolutionofthe XESsignalver- occurs at a microscopic level, yielding hysteresis in the
sus temperature in the range 5≤T ≤300 K at ambient physical observables. This ‘thermodynamics of an open
pressure ([28]). Essentially, the data are not indicative two-phase system’ in a coherent elastic solids has been
of a thermally driven HS-LS transition. Rather, a slight analyzed in another context by Schwarz and Khachatu-
increasewithtemperatureisseenthatmaybeassociated ryan [39, 40], but it exactly applies to the case of spin-
with a thermal re-population in the multiplet structure. state transitions because spin-states can be changed by
At ambient pressure the Mn local moment at 300K is spin-lattice relaxation ([28]). Improvements to this sim-
essentiallyinthesameHSstateasatlowtemperaturein ple thermodynamic picture may introduce certain corre-
the ordered phase. This justifies a posteriori the X-ray lations betweensites of the nucleating phase, e.g. by the
experimentsdoneatroom-temperatureandthecompari- elastic anisotropy of the cubic lattice, but cannot funda-
sonwithDFTdataat0K,butitraisesthequestionwhy mentally change this physical picture.
theHS-LStransitiondoesnotoccurwithtemperatureas
In conclusion, the magnetic collapse in MnGe occurs
in other spin crossovercompounds.
intwo-steps,intheparamagneticregimeaswellasinthe
In order to answer this question, we have calculated magneticallyorderedstate. Thedirectobservationofthe
the energy curves of the LS and ambient pressure HS ultimate collapse ascertains the nature of the intermedi-
state from the relation P =−∂E/∂V by using the Mur- ate phase, which at low temperatures is a weak itiner-
naghanEoS(seeFig. 2d)withtheXRDresultsfromrun ant band ferromagnetic state. This somewhat contrasts
1fortheHS-LSinitialmixtureandrun2fortheLSstate with high pressure studies in other B20-helimagnets like
(Tab. I). One gets an energy gap ∆E ≃125 meV/f.u.≃ MnSi and FeGe, where quantum phase transitions to-
1450 K/f.u. between the two states at ambient pressure. wards a non magnetic state have been found with in-
Thisenergygapislargerthanthetemperatureswhereall termediate regimes characterized by non Fermi liquid
reported magnetic measurements were performed (up to character and/or partial magnetic order [41, 42]. The
300Ktypically). ItexplainswhynoHS-LStransitionoc- huge pressure-hysteresis at the transition between the
curs versustemperature. At the same time, one can also ambient paramagnetic and the pressure-induced inter-
speculatethatthesynthesisconditions(upto2200Kand mediate phase proves the co-existence of different spin-
8 GPa)followedbya thermalquenchcouldyieldthe nu- states. The thermodynamic anomalies, in particular the
5
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zen, F. Sette, and R. Verbeni, Phys. Rev. B 63, 132409 [41] C. Pfleiderer, D. Reznik, L. Pintschovius, H. v. Lohney-
(2001). sen, M. Garst, and A.Rosch, Nature 427, 227 (2004).
6
[42] A. Barla, H. Wilhelm, M. K. Forthaus, C. Strohm, constant,themaineffectofpressureisaLorentzianpeak
R. Ru¨ffer, M. Schmidt, K. Koepernik, U. K. Ro¨ßler, broadening by a factor <∼2, as seen in Fig. 4.
andM.M.Abd-Elmeguid,Phys.Rev.Lett.114,016803 As explained inthe maintext, no trace ofa structural
(2015).
transition could be found throughout the data analysis.
[43] A. Tsvyashchenko, Journal of the Less Common Metals
The positional x parameters for Mn and Ge stay fairly
99, L9 (1984).
constant and vary by less than 1 % within the pressure
[44] J. Rodr`ıguez-Carvajal, Physica B: Condensed Matter
192, 55 (1993). range we explored (see Fig. 5). Such a variation most
[45] J. P. Rueff,M. Krisch, Y. Q.Cai, A.Kaprolat, M. Han- is most likely a bias of the refinement procedure, linked
fland, M. Lorenzen, C. Masciovecchio, R. Verbeni, and with anisotropic peak broadening occurring due to the
F. Sette,Phys. Rev.B 60, 14510 (1999). geometry of the pressure cell.
[46] C.-S. Zha, H.-k. Mao, and R. J. Hemley, Proceedings of
theNational Academyof Sciences 97, 13494 (2000).
[47] E. Altynbaev, S.-A. Siegfried, V. Dyadkin, E. Moskvin,
a)
D. Menzel, A. Heinemann, C. Dewhurst, L. Fomicheva, P = 0 GPa
A. Tsvyashchenko, and S. Grigoriev, Phys. Rev. B 90, 60x103 MCaelacsuularetedd
174420 (2014). a.u.) 40 Difference
y (
sit
n 20
e
nt
SUPPLEMENTARY MATERIAL I
0
SAMPLES
2 3 4 5 6
Q (Å-1)
Polycrystalline MnGe was synthesized at 8 GPa in 60x103 b)
P = 30 GPa
a toroidal high-pressure apparatus by melting reaction 50 Measured
Calculated
with Mn and Ge. The purity of the constituents was u.) 40 Difference
99.9% and 99.999% for Mn and Ge respectively. The y (a. 30
pellets of well-mixed powdered constituents were placed nsit 20
e
in rock-salt pipe ampoules and then directly electrically nt 10
I
◦
heated to T ≃ 1600 C. The samples were subsequently 0
quenched to room temperature before releasing the ap-
plied pressure as described by Tsvyashchenko[43]. The 2 3 4 5 6
samplequalitywascheckedbyX-rayandneutrondiffrac- Q (Å-1)
tion, yielding an amount of impurity less than 2%. The
FIG. 4: Examples of measured diffractograms at a) 0 and b)
samples used in the experiments described in this letter
30GPa, showing thehigh dataquality. Redlineistheresult
were withdrawn from the very same synthesis that was
of a Rietveld refinement of themeasured data.
used in previous studies[20, 24, 27].
X-RAY POWDER DIFFRACTION
Highresolutionx-raypowderdiffraction(XRD)exper- X-RAY EMISSION SPECTROSCOPY AT THE
iment was performed at the wiggler beamline PSICHE´ Kβ-LINE OF MN
(Synchrotron SOLEIL). The incident X-ray wavelength
was 0.3738 ˚A. Pressure was applied on the sample with Ourx-rayemissionspectroscopy(XES)measurements
the help of a diamond anvil cell (DAC). Ne was used as havebeenperformedinabackscatteringgeometry(2θ =
◦
pressure transmitter as it offers excellent homogeneity. 135 ) at the RIXS spectrometer of the undulator beam-
The X-ray patterns were averaged over Debye-Sherrer line GALAXIES[34] (Synchrotron SOLEIL). Emitted
cones, after supressing few non isotropic contributions photon energies were determined by reflection on a
from the pressure cell (e.g. Bragg spots from the di- Si(440) analyzer at 84◦ Bragg angle. Detection was en-
amond anvils). The X-ray patterns were refined using suredbyanavalanchephotodiode. WehaveusedaDAC
the Fullprof routine [44]. The R values situate between and a 4:1 methanol/ethanol mixture as pressure trans-
F
1.3 and 3.0 %. The very good refinements confirm that mitter. The incoming photon energy was selected to be
the sample remains in the B20 structure with negligible 7 keV, as a trade off between transmission through the
texture effect up to the highest pressure. Examples of pressure cell’s diamond and emitted photon flux, while
refined diffractograms taken at low and high pressures maximizing the signal-to-backgroundratio. An example
are displayedin Fig. 4. Besides the change in the lattice of a full spectrum is displayed in Fig. 6.
7
1.0
0.8 δ(P)= ωmax(σ(ω,P)−σ(ω,P ))
x (r.l.u.) 00..64 <<xxMGen>> == 00..185695((57)) σ(ω,P)=PIω((Imω(i,ωnP,P)−)−IbIgb(gω()ω)) ref (1)
whereωistheemPittedphotonenergy,ω =6467eV,
min
0.2 ωmax = 6480 eV and the subscript bg refers to the lin-
a) early ω-dependent background mainly originating from
0.0
the high energy tail of the Kα emission line. We have
0 5 10 15
checked that changing the reference spectrum simply
Pressure (GPa)
1.0 shifts the values of δ by a constant offset without alter-
ingitsrelativeevolution. Moreover,extendingtheenergy
0.8
boundary to the whole measured range, we verify -as it
x (r.l.u.) 00..64 <<xxMGen>> == 00..184309((56)) vpmaruoliscdtea-dttuehsraett.htehbeaδcktugrronusnodutsutbotbraec0tioant aalnldprneosrsmuraelsiz.aTtihoins
0.2
b)
0.0 10x10-3
Ambient pressure
0 5 10 15 20 25 30 5 K
Pressure (GPa) sity 8 Kβ' 300 K
n
FIG. 5: Pressure dependence of the positional parameters of nte 6 Kβ1,3
Mn and Ge atoms -xMn and xGe respectively - as a function zed i 4
of the applied pressure for the first (a) and second (b) run ali
m
(see main text). or 2
N
0
6470 6475 6480 6485 6490 6495 6500
50x10-3 Kβ1,3 Room temperature Outgoing photon energy (eV)
5.2 GPa
sity 40 38 GPa FIG. 7: Typical response measured at ambient pressure at 5
en and 300 K, showing a small increase in the emitted intensity
alized int 3200 Kβ' iastgtrheeatKlyβ′imsaptreolvlietedpcoosmitpioanre.dOtnoetchaenhniogtheptrheastsuthreessttuatdiys,tiacss
m in the latter case the incoming and emitted photon flux was
Nor 10 drastically reduced bythe diamond anvils.
0
6465 6470 6475 6480 6485 6490 6495 6500
Outgoing photon energy (eV)
Asmentionedinthemaintext,wehaverecordedsome
FIG. 6: Example of XES spectra measured at room temper- ambient pressure spectra between 5 and 300 K (see Fig.
atureforP=5and38GPa. Theverticalbarsdelimitatethe 7). The spectraldifferenceisca. 5timesweakerbetween
integration boundaries. base and room temperature as compared with the value
obtained under pressure between 0 and 38 GPa. Thus,
′
this weak spectral change at the Kβ position doesn’t
supportaHS-LStransitiontriggeredbyheating. Rather,
it suggests a thermal population of the multiplet struc-
ture.
As explained in the main text, we have evaluated the
evolution of the local Mn moment value as a function of
HIGH-PRESSURE GAUGE
pressure by means of a qualitative analysis. The latter
is based on the comparison of the integral intensity of
′
the Kβ -low energy shoulder measured at a certain ap- In both the XRD and XES experiments, pressure was
pliedpressureP withthatofareference. Inourcase,we measured in situ before and after each single measure-
assumethespectrumtakenatP =38GPatoberepre- ment via recording laser-stimulated ruby fluorescence
ref
sentative of the non-magnetic state of MnGe. Thus, the lines. Wehavevisuallyinspectedthepositionoftheruby
integral difference that is linked with the local moment within the cell andcheckedthat it was centeredandem-
value [33, 45] is calculated as follows: bedded in the same medium as the sample (see Fig. 8).
8
A double Lorentzian function was fitted to the obtained ELECTRONIC STRUCTURE CALCULATIONS
two-peaks spectra (see Fig. 9). The wavelength position BY DENSITY-FUNCTIONAL THEORY
λ of the main line is then inputed into the following
m
function Theoretical calculations of the equation of state
EoS have been done within the density-functional the-
ory approach using the full-potential local-orbital ap-
proach, as implemented in the FPLO code [31]. The
A λ [nm] B generalized-gradientapproximation (GGA) was used for
m
P [GPa]= · −1 (2)
B λ the exchange-correlation potential, and the lattice cell
"(cid:18) 0 (cid:19) #
was optimized for each calculated volume and the dif-
ferent spin-states. The values of the Murnaghan EoS in
table I of the main text have been determined from the
where A = 1904 GPa, B = 7.715[][46] and λ =
0
theoretical(total) energycurves, E (V) in the rangeof
694.36 nm, the latter being calibrated by measuring the tot
volumes 98 < V < 111 ˚A3. For the LS-state, we apply
response of a ruby at room pressure.
theGGA*XcorrectionswithreductionfactorfortheXC-
potential (ξ = 0.720, as in Ref. 27) in order to consider
some of the long-range spin-fluctuations affecting the
spin-state. Atequilibriumvolumethenetspinmomentis
0.8µ /f.u. inthis spin-statewhichismetastablewithin
B
u.) 1400 λrubis = 694.76 nm this calculation. For the HS-state, the bare GGA results
ensity (a. 11200000 Pmeas= 1.09 GPa anreeglreecptosratelldlwonhgic-hracnogreressppionn-fldutcotuaantieolencstr(oξni=cs1ta).teTthhaist
e int 800 ideal spin-state has a net spin-moment of 2.0 µB/ f.u.
orescenc 640000 SporompeerqtiueasnbtiettawteiveendperveisaetnitonesvailnuatthieonEaonSdaenadrlimeragcanlectuic-
u
by fl 200 lations [26, 27] are explained by a refinement in the one-
u
R 0 electron local-orbital basis. For the present evaluation,
690 692 694 696 698 700 wehaveusedthedoubledbasissetforvalenceband-states
Wavelength (nm) thatwasalsousedinarecentdeterminationoftheEoSof
thecrystallinestatesoftheelementstobereportedinthe
FIG. 8: Picture of the sample within the DAC taken at the
bench-markingcomparingvariousDFT-codes performed
end of theXES experiment.
and organized by LeJaeghere, Cottenier et al. [32],[? ].
Using this improved basis set of the FPLO-code yields
maximum deviations of about 1 meV / ion for the en-
ergy curves near equilibrium states compared to other
high-precision full-potential codes. In this respect, the
solution of the EoS as given by the DFT Kohn-Sham
equations are well converged.
1500
u.) λruby = 694.76 nm
ensity (a. 1000 Pobs. = 1.09 GPa SPIN-STATEELTARSATNICSILTAIOTNTIICNESCOHERENT
nt
e i
nc
e
esc 500 In this section, we discuss in more details the elemen-
or
u tary thermodynamic picture of the transition between
uby fl different spin-states that we have used to identify long-
R
0 range elastic stresses as the crucial factor for the large
690 692 694 696 698 700
and history-depending hysteresis in the high pressure
Wavelength (nm)
XRD experiment.
FIG. 9: Example of ruby fluorescence spectrum as used for SchwarzandKhachaturyandisscussedtheco-existence
pressure determination. of two solid phases α and β under hydrogen-loading
when only an external partial hydrogen pressure con-
trols the amount of interstitially disolved H-ions in the
two lattices[39, 40]. The thermodynamics of a paramag-
net with a HS to LS transition with a magnetovolume-
induced strains can be mapped onto the elementary
9
model developed by them to explain the macroscopic coupling. To simplify the discussion of the basic mecha-
hysteresis in such open two-phase systems. The ele- nism, the elastic and materials coefficients are assumed
mentary model used by them to explain the macro- to be the same in the two phases (A = A = A). It
HS LS
scopic hysteresis in such open two-phase systems can be isthenpossibletoderivethe expressionforthe totalfree
mapped onto the case of a paramagnet with HS and LS energy of a two phases closed system via Eqs. 4-5:
transition with magnetovolume-induced strains. Here,
F (V,T,µ¯,µ ,µ ,ω) = ωf (V,T,µ )
the spin-polarization of the electronic structure can be ν HS LS LS LS
changedbyspin-latticerelaxation,whichmeansthatthe + (1−ω)f (V,T,µ )(6)
HS HS
system can change the local spin-polarization and the + Aµ¯(1−µ¯)
associated volume strain freely. In this sense we are
discussing an open thermodynamic system where two In an open system, the average moment µ¯ is not fixed
different phases can coexist, but the nucleation causes anymore and can be tuned by an appropriate external
long-range strains to appear in the system. For sim- potential (in our case, the applied pressure). The corre-
plicity only two spin-states ν = {HS,LS} are consid- sponding Gibbs free energy reads:
ered that can exist in a certain range of values for the
G (V,T,P,µ¯,µ ,µ ,ω) = F (V,T,µ¯,µ ,µ ,ω)
squared normalized spin-polarization given by the ratio ν HS LS ν HS LS
µ =(M /M0)2(0<µ <1),whereM istheactuallo- − κP µ¯ (7)
ν ν ν ν ν
calmagnetizationorspin-densitypervolumeandM0the
ν
where P is the applied pressure and κ a magneto-elastic
fullspin-polarizationforthegivenspin-state. Identifying
coupling constant (in units of volume per moment) such
the reducedµ withthe H-concentrationsc andc , the
ν α β
as the average unit cell volume V = κµ¯. By virtue of
elementarymodelcanbeformulatedbysimplyrewriting
Eq. 4, Eq. 7 can be rewritten in the form of a second
the basic equations for free energy contributions of the
order polynomial function:
different phases and the elastic strain. In the following,
we reproduce the equations and basic arguments from
G (V,T,p,µ ,µ ,ω) = φ (µ )
for such a paramagnet with magnetoelastic coupling us- ν HS LS 0 HS
+ φ (µ ,µ ) ω (8)
ing the notation from Ref. 39, 40 to mark the essential 1 HS LS
equivalenceofthismodelwiththecaseofhydrogenload- − φ (µ ,µ ) ω2 ,
2 HS LS
ing in a two-phase material.
with
Thestraincausedbythemagnetovolumeeffectsforthe
nucleationof sites with different spin-state in the matrix
φ =f (µ )+Aµ (1−µ )−κPµ
0 HS HS HS HS HS
increases the elastic energy. The situation corresponds φ =(µ −µ ) fLS(µLS)−fHS(µHS) +A(1−2µ )−κP
to the inclusion of misfitting spheres in the holes of a 1 LS HS µLS−µHS HS
recipient elastic matrix. This energy cost reads: φ =A(µ −µ h)2 i
2 LS HS
(9)
1+σ
E =NAµ¯(1−µ¯) with A=v G ǫ2 , (3) Depending on the relative values of the φi (i={0,1,2})
el 0 s 1−σ 0 terms,the Gibbsfreeenergymaymonotonicallyincrease
ordecreaseasafunctionofthephasefractionω,yielding
where N is the number of lattice sites involved in the
a minimum atω =0 (macroscopicHS-state) or atω =1
transformation, v the volume of a single site, G the
0 s (macroscopic LS-state). An interesting third possibility
shear modulus, σ the Poisson ratio, ǫ = da/adµ¯ the
0 arises when a maximum of G occurs at an intermedi-
volume dependence of the average spin moment and the ∗
ate value ω = φ /(2φ ) (see Fig. 1 of Ref. 39). In
average moment µ¯ is defined as: 1 2
this case, the elastic strains create a macroscopicbarrier
against nucleation of a stable minority phase. For the
µ¯ =ωµ +(1−ω)µ , (4)
LS HS
spin transition to be triggered, the Gibbs free energy at
ω =0 must cease to be a local minimum and hence, the
where ω is the volume fraction of the LS phase. Impor-
linear term φ in Eq. 9 must be cancelled since φ is
tantly, the form of Eq. 3 is independent of the arrange- 1 2
always > 0:
ment of the sites with deviating spin-moment. In the
extreme case, we may even have single lattice sites un-
f (µ )−f (µ )
LS LS HS HS
dergoingHStoLStransitionsinaHS-matrix. UsingEq. +A(1−2µHS)−κP =0 (10)
µ −µ
LS HS
3,wecanwritethe Helmholtz freeenergyperlattice site
for a given phase: The pressure at which the HS phase is stabilized can be
calculated by differentiating Eq. 7 with respect to µ:
F (V,T,µ )=f (V,T,µ )+A µ (1−µ ) , (5)
ν ν ν ν ν ν ν
∂F ∂f
HS HS
κP(µ )= = +A(1−2µ )
HS HS
where the first term corresponds to the magnetic contri- ∂µ ∂µ
(cid:12)µ=µHS (cid:12)µ=µHS
butionandthesecondtermexpressesthemagneto-elastic (cid:12) (cid:12) (11)
(cid:12) (cid:12)
(cid:12) (cid:12)
10
Inserting Eq. 11 into Eq. 10, one eventually gets: text). However,internalstresses like defects (originating
from the high pressure synthesis of MnGe powder), but
f (µ )−f (µ ) ∂f (µ )
LS LS HS HS HS HS alsocrystallitesizesandtheirshapes,willmassivelyinflu-
− =0 (12)
µLS−µHS ∂µHS encethetransformationprocessesandmayfavorthesta-
bleinclusionofminorityLSsitesintheambientmajority
In this sense, the stability limit for the HS phase will be
HS matrix. Thus, the HS-LS transition will not display
reachedwhentheenergycurvesfortheHSandLSstates
a marked ”jump-like” behavior at the critical pressure
will have a common tangent for the first time, i.e. when
and will be replaced by a smooth crossover. An analo-
the energybarrierforthe stabilizationofthe LSphase is
gouscaseofmacroscopichysteresisphenomenaandtheir
overcomeunderpressure. Thisreasoningmaybeapplied
dependence on microstructure are classicalferromagnets
to the backwardLS-HS transformation. Since for the LS
with dipolar stray-fields,as realizedin the huge variabil-
state:
ity of the hysteresis in permanent magnetic materials,
∂F ∂f whereintrinsicmagnetic propertiesarenotchanged,but
LS LS
κP(µ )= = +A(1−2µ )
LS ∂µ ∂µ LS magnetizationprocessesmayyielddifferencesinthecoer-
(cid:12)(cid:12)µ=µLS (cid:12)(cid:12)µ=µLS (13) civefields by ordersofmagnitude. In the presentdiscus-
(cid:12) (cid:12)
we anticipate the o(cid:12)bservation of a(cid:12)finite square-like hys- sion,forthesakeofsimplicity,wehaveneglectedtherole
of magnetic correlations which are still sizable at room
teresisofwidthκ |P(µ )−P (µ )|inthe(P,V)equa-
HS LS
temperature, as seen e.g. by neutron diffraction[27] and
tion of state of the material since V depends on the av-
small-angle scattering[47]. Also, unavoidable deviation
erage spin state.
from perfect hydrostaticity in the pressure medium was
Transposing these considerations to MnGe, it means
neglected. Anyway, the model presented above explains
that transformation between the HS and LS state will
the essential features of the V(P) equation of state de-
be marked by a large hysteresis of the unit cell volume
termined experimentally.
through macroscopic magneto-elastic coupling, as ob-
servedinthefirstrunofourXRDmeasurement(seemain