Signature effects of spin clustering and distribution of spin couplings on magnetization behaviour in Ni-Fe-Mo and Ni-Fe-W alloys

0.51 MB·English

by  Mitali Banerjee

Signature effects of spin clustering and distribution of spin couplings
on magnetization behaviour in Ni-Fe-Mo and Ni-Fe-W alloys
Mitali Banerjee,1 Avinash Singh,2 A. K. Majumdar,1,∗ and A. K. Nigam3
1Department of Materials Science, S.N. Bose National Centre for Basic Sciences,
JD Block, Sector III, Salt Lake City, Kolkata 700098, India
1
2Department of Physics, Indian Institute of Technology, Kanpur 208016, India
1
0 3Department of Condensed Matter Physics and Materials Science,
2 Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400005, India

n
(Dated: January 14, 2011)
a
J The spontaneous magnetization as a function of temperature is investigated for a number of

3 disordered Ni-Fe-Mo and Ni-Fe-W alloys using superconducting quantum interference device mag-
1 netometry, with a focus on the low-T behavior as well as the critical exponents associated with
the magnetic phase transition. While the low-T magnetization is found to be well described by

] Blochs T3/2 law, an extraordinary enhancement of the spin-wave parameter B and the reduced
ci coefficient B3/2=BTC3/2 are observed with increasing Fe dilution as compared to conventional 3d
s ferromagnets, whereas the critical amplitudes are found to decrease systematically. Recent locally
-
l self-consistent calculations offinite-temperaturespindynamicsin ageneric diluted magnet provide
r
an understanding in terms of two distinct energy scales associated with weakly coupled bulk spins
t
m in the FM matrix and strongly coupled cluster spins. In view of similar behaviour observed in di-
luted magnetic semiconductors and otherferromagnetic alloys, it is proposed that thesedistinctive
.
t features corresponding to thethree important temperature regimes provide macroscopic indicators
a
m of signature effects of spin clustering on magnetization behaviourin disordered ferromagnets.
- PACSnumbers: 75.30.Ds,75.40.Cx
d
n
o
I. INTRODUCTION scopic magnetization behaviour? A similar question
c
[ was recently addressed in the context of squid mag-
netization studies7 of diluted magnetic semiconductors
2 Magnetic phases of alloy systems with both ferromag-
Ga Mn As, which also exhibit strong disorder effects
v 1−x x
netic and antiferromagnetic exchange interactions have
duetoMnpositionaldisorder. Recenttheoreticalstudies
3
been studied in details by many groups including ours.
4 offinite-temperaturelocalspindynamicswithinageneric
Ni-rich Ni-Fe-Cr and Ni-Fe-V alloys1,2 have shown ex-
3 model for diluted magnets have provided fundamental
0 otic magnetic phases and hence have been probed ex- understandingofmacroscopicmagnetizationcharacteris-
1. tensively. Compositions near Ni80% - Fe20% have been ticsintermsofmicroscopicspindisorder,spinclustering,
0 widely investigated in bulk form not only for their rich and distribution of spin couplings.8–10
magnetic phases but also for the unique quality of low
1
1 magnetostriction and high initial permeability3. These There are three importanttemperature regimes in the
: are ideal properties for making magnetic cores for light magnetization behaviour of a generic disordered ferro-
v
electrical equipment as well as magnetic shielding mate- magnet. The low-temperature regime characterized by
i
X rials. the spin-wave parameter, the intermediate-temperature
regimenearlyuptoT characterizedbytheoverallshape
r C
a The observation of several signatures of a spin-glass- of the magnetization decay, and the critical regime very
like phase in recentdc magnetizationand ac susceptibil-
close to T involving critical fluctuations, divergent spin
C
ity studies of Ni-Fe-Mo and Ni-Fe-W alloys4–6 indicates
correlationlength,andcharacterizedbythecriticalexpo-
presence of competing ferromagnetic and antiferromag-
nents and critical amplitudes. In this paper we propose
netic spin interactions, resulting in frustration-induced
that the magnetization behaviours in all three tempera-
frozen spin configurations in spin clusters. Generally, a
ture regimes actually provide macroscopic indicators of
broad distribution of magnetic spin couplings would be
signatureeffectsassociatedwithmicroscopicspindynam-
expectedcorrespondingto differentseparationsandcon-
ics of weakly coupled bulk spins and strongly coupled
figurationsresultingfromthepositionaldisorderofmag-
cluster spins. For that purpose we have made a detailed
netically active Fe atoms in these alloys.
study of the temperature dependence of magnetization
Are there any distinctive finite-temperature spin dy- in Ni-Fe-Mo and Ni-Fe-W alloys and the critical expo-
namics effects associated with this disorder-induced dis- nents and critical amplitudes associated with the phase
tribution of magnetic interactions in the Ni-Fe-Mo and transition.
Ni-Fe-W alloys which are observable in their macro-
2
II. EXPERIMENTAL DETAILS % Cr in Ni. The split-band model of Berger12, however,
gives the same critical concentration of ∼ 12 % for Cr,
Mo, and W since they fall in the same group in the pe-
Alloys of both ternary series (Mo and W) were pre-
riodic table. Detailed magnetic measurements, both dc
pared by arc melting of the required amount of con-
and ac, have been done on Mo13.5 to establish that the
stituents of Johnson-Mathey, 99.999 % pure Ni, Fe, Mo,
second phase is indeed a mixed ferro-spin-glass phase.
and W in pure argon atmosphere. Then the shiny but-
The M(H) measurements show a ferromagnetic nature
tons were sealedin quartz tubes, flushed with ultra pure
withafeeblenon-saturatingcharacterat5Kbutgrossly
argon gas, and homogenized at 1300oC for 48 hours.
itisferromagnetic. Thepeakinacsusceptibilityshiftsto
Then they were cold-rolled and cut to size. A final an-
higher temperatures with increasing frequency and also
nealing was done in argon atmosphere at 1100oC to re-
the field cooled (FC) and zero-field cooled (ZFC) curves
ducestrainduetocoldrolling. Wekepttheconcentration bifurcate until an applied field ∼ 100 Oe. These charac-
ofNiroughlyaround80-83at. %andvariedFefrom17
teristic features resemble those of spin glasses. All the
to3at. %andMofrom2to14at. %. Foralloyscontain- above will be presented in detail elsewhere6.
ingW,Niconcentrationremainsroughlyaround79to86
at. %whilethatofFe steadilydecreasesfrom18to3at. The alloys of both the series show very thin M-H
% and that of W increases from 3 to 10 at. %. All mag- loops just like their parent permalloy. However,the per-
netic measurements were done using Quantum Design meability found is not comparable to those of permal-
SQUIDmagnetometer(QDMPMS).TheCurietempera- loys. Magnetic annealing of the alloys could give higher
tures (T ) were determined from the rate of fall of mag- permeability3. The saturation magnetization falls from
C
netizationasafunctionoftemperatureinappliedfieldsof 0.85 µB to 0.08 µB on increment of Mo in the alloys4
20 - 50 Oe. The compositions of the samples have been while it decreases from 0.85 µB to 0.11 µB for the W
confirmedbyenergydispersivespectrum(EDS)measure- series5. The ac-susceptibility measurements show the
ments using a scanning electron microscope (SEM). The same TC as found from dc measurements.
details of measurements of other quantities will be dis-
cussedalongwith the presentationofdata in sectionIII.
B. Low-temperature high-field magnetization of the
alloys
III. RESULTS AND DISCUSSION
In conventionalferromagnets, the temperature depen-
dence of the spontaneous magnetization M (T) far be-
s
A. Magnetic phase of the alloys low T is dominated by long-wavelength spin-wave ex-
C
citations. The excitation energy E(k) of spin waves in
the limit of small wave vectors (k≪a−1, a is the lattice
TheX-raydiffractionpatternrevealsthatallthealloys
spacing) is given by
are of single FCC phase with no additional phase. The
lattice parameter varies by 1.5-2 % from that of pure Ni E(k)=¯hω(k)=gµ H +Dk2+Ek4+..., (1)
B int
and increases with the increase of Mo/W. The alloys of
both the Mo and W series show paramagnetic to ferro- where the first term is an energy gap due to the pres-
magnetictransitionatTC,whichvarieswidelydepending ence of an effective internal field Hint, arising from the
ontheamountofFepresentinthealloy. Themagnetiza- applied field, the anisotropy field, and the spin-wave de-
tion of the alloys with low Fe content are measuredin ∼ magnetizing field. D is the spin-wave stiffness constant,
20Oeappliedfieldandwithlowtemperatureattachment and E is a proportionality constant. Even in disordered
ofQDMPMS andthatfor the alloyswith higherconcen- ferromagnets, there is ample experimental evidence that
trationofFeweremeasuredin∼50Oeappliedfieldwith long-wavelengthspin-wavemodesareausefulwaytorep-
high temperature attachment of QDMPMS. The TC for resent the low-energy magnetic excitations.13
all the compositions are already reported by Banerjee et
al.4,5 and are given in Table I. It is clear that increas- Inthe low-temperaturelimit, accordingto the Heisen-
bergmodel,thechangeinthespontaneousmagnetization
ing Mo and W content and accompanying decreasing Fe
due to the excitation of spin waves can be written as:14
content in the alloys lowers the T rapidly.
C
In the Mo series, the sample Mo13.5 shows a sec- 3 T
M(T) = M(0)[1−Bz , g T3/2
ond phase transition apart from the paramagnetic- (cid:18)2 T (cid:19)
ferromagnetictransition found in all the alloys of the se-
5 T
ries. No such additional phase has been found in the W − Cz , g T5/2+...], (2)
(cid:18)2 T (cid:19)
series which we anticipate is due to lack of high W sam-
ples around the critical concentration. We have found
whereM(0)isthemagnetizationat0K,T =gµ H /k
that the T extrapolates to 0 K for ∼14.2 at. % Mo g B int B
whereastheCSlater-Paulingcurve11 givesTC ∼0Kfor18 is the gap temperature, and z 23,TTg and z 52,TTg are
(cid:16) (cid:17) (cid:16) (cid:17)
3
thecorrectiontermswhichreducetounityiftheeffective recently using a systematic non-perturbative expansion
internal magnetic field vanishes. The two temperature scheme.17 The presence of thermally excited long wave-
terms above come from the harmonic (k2) and anhar- length spin wavesin the ferromagnetic state at low tem-
monic(k4)termsinthespin-wavedispersionrelation[Eq. perature allows virtual excitation of majority-spin elec-
(1)]. For simplicity, disregarding anharmonicity and the trons to the minority-spin band due to electron-magnon
gap corrections, Eq. (2) reduces to the so-called Blochs coupling,resultinginspectral-weighttransferandconse-
T3/2 law: quent reduction of the magnetization hn i−hn i.
↑ ↓
M(T)=M(0)(1−BT3/2). (3) The low-temperature magnetization was measured
from 2 K to 0.1 T or till 15 K at ∼ 6000 Oe for all the
C
In conventional spin-wave theory the spin wave parame- samples which is above their saturation fields of 1000-
terB andthespinwavestiffnesscoefficientD arerelated 2000 Oe, although sample Mo13.5 did not actually satu-
through rate even at 6000 Oe due to its low-temperature mixed
ferro-spin-glass phase. Figure 1 is a plot of M vs. T for
k 2.612gµ 2/3 T ≪T . The solid lines are the best-fitted curves as in-
B B C
D = . (4)
4π (cid:18) M(0)B (cid:19) dicated in the legends of the figures. The gap correction
[Eq. (2)]wastriedbutitwasfoundtobenegligiblesince
the gap temperature came out to be less than 1 K.
Thethermaldemagnetizationprocessofferromagnetic
metalsatlowtemperatures(T≪TC)canbeexplainedby From Fig. 1 as well as Table I we find that the fits
both localized15 and itinerant16 models. In the localized forsome ofthe 12samples thatwe havestudied improve
model, with static interactions between spins associated significantly if we include the T5/2 term in Eq. (2). Not
with localized electrons on atomic sites, spin waves cor- only the values of χ2 are smaller by a factor of 2-3 (∼8
respond to coherent superposition of local spin devia- for W7.2), the deviation of the best-fitted curve from
tions in the ferromagnetic state, and their equilibrium the data vs. T is random as against systematic when
number density at finite temperature according to the we consider only the T3/2 term. For the other samples
Bose-Einstein distribution function yields the magneti- the improvements are insignificant. We must note that
zation reduction according to Eq. (2). In the itiner- for higher T samples (low Mo and W), ∆M/M is only
C
ant model, electrons move in the average field of other 1% and only very high resolution SQUID measurements
electrons/ions, and effective inter-site spin couplings are are able to isolate the anharmonic term in the magnon
generated by exchange of the particle-hole propagator, dispersion relation.
strongcorrelationeffectsinwhichhavebeeninvestigated
Thefirstsignificantfeatureemergingfromthepresent
investigation is the order of magnitude of the spin-wave
parameter B. These values are found to be strongly en-
hanced with increasing Fe dilution in both the series of
0.270 alloys, and are about 10 to 100 times larger than those
3.330
found for conventional3d ferromagnets (for bulk Fe it is
3.325 0.255 3.4×10−6K−3/2). ThissharpincreaseinthevaluesofB
withincreasingFedilutionreflectsanenhancementinthe
3.320 0.240
u) 3.315 Ni81.0Fe16.7Mo2.3 Ni83.2Fe3.3Mo13.5 density of low-energy magnetic excitations due to weak-
m 3/2 0.225 3/2 eningoftheferromagneticcouplingsbetweenthebulkFe
e 3.310 T3/2 5/2 T3/2 5/2 spinsformingthe percolatingferromagneticmatrix. The
( T +T 0.210 T +T spin-wavestiffness constantsarealsocorrespondinglyre-

nt 15 30 45 60 75 0.5672 4 6 8 10 12 14 duced, as seen in Table I.
e
m1.100 0.566
o
M1.095 0.565
0.564
1.090 C. Intermediate-temperature behaviour
Ni81.0Fe11.8W7.2 0.563
1.085 3/2 0.562 Ni83.5Fe7.6W8.9
T
1.080 T3/2+T5/2 0.561 T3/2 Inordertoallowaqualitativecomparisonbetweenthe
magnetization behaviour of different ferromagnetic sys-
10 20 30 40 50 60 70 5 10 15 20 25
tems over a much broader temperature scale extending
Temperature (K)
nearly up to T , it is convenient to use the normalized
C
coefficient B defined through the relation:18,19
3/2
FIG. 1: (Color Online) Magnetisation vs. Temperature plot
forfoursamplesaregiven,amongthemthreesamplesfitbet-
ter using Eq. (2) including anharmonic term another one M (0)−M (T) T 3/2
s s
shown fitted to Blochs T3/2 law. M (0) =B3/2(cid:18)T (cid:19) (5)
s C
4
TABLE I: Magnetization vs. temperature data are fitted to Eq. (2) and the coefficients B, C along with normalized χ2, R2,
TC, and D are tabulated below.
Sample M(0) B C B3/2 TC normalized χ2 R2 D
Composition (emu) (10−4 K−3/2) (10−6 K−5/2) (K) (10−7) meVA˚2
Ni81.0Fe16.7Mo2.3 3.33323±0.00005 0.1189±0.0004 - 0.23 720 0.099 0.99843 219
3.33286±0.00004 0.094±0.001 0.023±0.001 0.036 0.99941
Ni80.4Fe12.9Mo6.7 1.4606±0.0002 0.294±0.005 - 0.32 495 2.91 0.97166 144
1.4601±0.0003 0.24±0.03 0.1±0.06 0.036 0.99941
Ni83.4Fe10.7Mo5.9 0.79083±0.00006 0.345±0.003 - 0.35 470 2.08 0.99064 155
0.79038±0.00007 0.24±0.01 0.19±0.02 1.28 0.99439
Ni83.5Fe7.6Mo8.9 3.13870±0.00002 0.864±0.007 - 0.49 320 3.31 0.99981 110
Ni83.1Fe6.0Mo10.9 1.20957±0.00005 3.68±0.01 - 0.90 182 0.75 0.99916 45
Ni83.2Fe3.3Mo13.5 0.2770±0.0001 42.7±0.1 - 1.29 45 63.9 0.99871 20
0.2755±0.0001 36.2±0.5 0.44±0.03 27.7 0.99944
Ni78.9Fe18.1W3.0 4.0269±0.0001 0.0982±0.0007 - 0.21 775 0.52 0.99332 255
Ni79.4Fe14.1W6.5 1.25119±0.00002 0.302±0.001 - 0.37 530 0.15 0.99945 133
1.25098±0.00002 0.275±0.002 0.040±0.002 0.07 0.99976
Ni81.0Fe11.8W7.2 1.10312±0.00002 0.359±0.001 - 0.42 515 0.31 0.99921 137
1.10280±0.00001 0.311±0.002 0.070±0.002 0.04 0.9999
Ni83.5Fe7.6W8.9 0.56687±0.00001 0.796±0.001 - 0.41 300 0.47 0.99956 120
Ni82.6Fe6.9W10.5 1.66087±0.00003 2.906±0.004 - 0.77 191 0.22 0.99963 53
Ni86.6Fe3.1W10.3 0.54141±0.00001 19.6±0.1 - 0.8 55 18.08 0.99825 28
in terms of the low-temperature magnetization (Eq. 3), two impurity spins at lattice sites i and j were calcu-
whichyieldsB3/2=BTC3/2. Significantly,thisreducedco- lated from Jij=J2(2S)φij in terms of the particle-hole
efficientB3/2 providesaneffective measureofthe overall propagatorφij evaluated in the ferromagnetic state. For
shape of the magnetization decay. For crystalline ferro- the same impurity separation, the calculated magnetic
magnets, where the magnetizationfalls sharply near T , couplingswerefoundtoexhibitabroaddistribution,im-
C
oneobtainsB ≈0.2,irrespectiveoftheCurietemper- plying that the coupling between two impurity spins is
3/2
ature. Ontheotherhand,forreference,ifthemagnetiza- not simply a function of their separation, but actually
tion were to fall off as M(0)(1−BT3/2) nearly upto T , depends on the whole disorder configuration, suggesting
C
thenclearlyB ≈1. TableIshowsthatwithincreasing shades of a complex system.
3/2
Fe dilution, the B values for both series of alloys sys-
3/2 Thebroaddistributioninthecalculatedferromagnetic
tematically increase from ∼ 0.2 to ∼ 1, indicating that
spin couplings was ascribed to the formation of small
the magnetization decay near T becomes significantly
C impurity spin clusters due to positional magnetic disor-
slowerintheFe-pooralloys. Thisisthesecondsignificant
der. Theclusterspincouplingswerefoundtobestrongly
result which emergesfrom the intermediate-temperature
enhanced due to preferential accumulation of carriers in
regime.
impurity spin clusters,whereasthe consequentdepletion
Both these distinctive features of macroscopic magne- of carriers from the bulk resulted in weakened bulk spin
tization behaviour observed in Ni-Fe-Mo and Ni-Fe-W couplings between the bulk spins forming the ferromag-
alloys — strong enhancement of B with increasing Fe netic matrix.8–10
dilution and B approaching unity — have also been
3/2 The effects of such a broad distribution of ferromag-
observed in squid magnetization studies of diluted mag-
netic spin couplingsonfinite temperaturespin dynamics
neticsemiconductorsGa Mn As,7wheretheywereas-
1−x x was recently investigated in diluted magnets using a lo-
cribed to two distinct energy scales involved in the local callyself-consistentmagnonrenormalizationscheme,8–10
spindynamicsassociatedwiththeformationofspinclus-
inanalogywithrenormalizedspin-wavetheoryinordered
ters. A brief review of the theoretical analysis of finite- ferromagnets.20 The localmagnetizationhSziofeachin-
i
temperature spin dynamics within a minimal model for
dividualimpurityspinatsiteiinaquantumspin-Sdisor-
diluted magnets will be helpful in understanding the ob-
deredferromagnetwasobtainedbyself-consistentlysolv-
served magnetization behaviour.
ing the three coupled equations:
Theoretical investigations in diluted magnets do
yield competing ferromagnetic and antiferromagnetic (S−Φ )(1+Φ )2S+1+(S+1+Φ )Φ2S+1
hSzi= i i i i ,
interactions.19 Within a minimal model for diluted mag- i (1+Φ )2S+1−Φ2S+1
i i
nets involving spin-S localized impurity spins and host (6)
band fermions (carriers), magnetic interactions between where the local site-dependent boson occupation num-
5
bers:
5
J, ε = 4, 0
d
|φi|2 4 6, 0
Φi =Xl eβωll−1 (7) 3/2) 3 2, 2
-K
3
in terms of the renormalized magnon energy eigenvalues -10 2
ωl andeigenfunctions φl obtainedfromthe renormalized B (
magnon Hamiltonian:
1
0
H = 2hSzi J2[χ0] 2hSzi. (8)
ij i ij j 5 10 15 20 25 30 35 40 45 50
q q
(cid:0) (cid:1)
x (%)
The locally self-consistent magnetization hSizi thus ob- 3 dis 1
tainedclearlyshowedrapidthermaldemagnetizationand 2
2.5 3
nearly paramagnetic behaviour of the weakly coupled 4
5
bulk (FM matrix) spins, whereas the strongly coupled
ordered
2
cluster spins resist thermal demagnetization and thus
prolong the magnetic order near TC. The overall pic- 〉S z 1.5
turefromtheaveragedmagnetizationwasthatwhilethe 〈
ferromagnetic T was suppressed by positional-disorder-
C 1
induced spin cluster formation, the magnetization decay J/t=4
was distinctly stretched near TC. 0.5 xp==112%.5%
W=12t=10 eV
These distinctive features of microscopic spin dynam-
0
ics behaviour in a generic diluted magnet were shown 0 20 40 60 80 100 120 140 160
to quantitatively affect the two readily accessible char- T (K)
acteristics B and B of macroscopicmagnetizationbe-
3/2
haviour. While the calculated spin-wave parameter B FIG. 2: (Color Online) Rapid enhancement of the calcu-
was found to be sharply enhanced with increasing dilu- latedspin-waveparameterBwithdilution(upperpanel),and
tion [Fig. 2 (upper panel)] due to weakened bulk spin (lower panel) nearly T3/2 fall-off of site-averaged magnetiza-
couplings and softening of low-energy spin excitations, tionandstretchingofmagneticordernearTC (from ref. [7]).
the stretching ofmagnetic ordernear T due to strongly
C
coupled cluster spins resulted in enhanced B ∼1, in-
3/2
dicating slower thermal demagnetization approximately
D. Magnetic phase transition and the associated
as T3/2 (dashed line) over a much broader temperature
critical exponents
range, as shown for five different disorder configurations
(lowerpanel). Insharpcontrast,theorderedferromagnet
yielded, for the same dilution and carrier concentration, In the present work, we have also studied the criti-
a conventionalmagnetizationdecay with B3/2 ≈0.2 and calexponentsassociatedwiththe magneticphasetransi-
T3/2behaviour(dashedline)onlyinthelow-temperature tionofthe alloyscontainingmorethan8 at.%ofMo/W.
regime. Here the given notation refers to impurity con- The specific reason to study the critical exponents and
centration(x),carrierconcentration(p),bandwidth(W), the amplitudes is to see the consistency of our findings
and host-impurity energy offset (ǫ ). with respect to the above magnetization data analysis.
d
In the last section we have seen evidence of spin clusters
As shown in Table I, a very similar behaviour is ob-
with relatively strongly coupled spins in the alloys with
tainedfor the twomagnetization(spin-wave)coefficients
dearth of Fe content. How does this spin clustering af-
B and B from our finite-temperature magnetization
3/2 fect the critical exponents and critical amplitudes? As
investigationofNi-Fe-MoandNi-Fe-Walloys. Whilethe
a first step, ac susceptibility has been measured in all
BvaluesaresharplyenhancedwithincreasingFedilution
alloys with a temperature increment of 0.2 K or less so
(thespinstiffnessiscorrespondinglysharplysuppressed),
that their T could be found accurately within ± 0.1 K.
theB valuesapproach1withincreasingFedilutionin C
3/2 The magnetization was measured in fields ranging from
both alloys, as compared to about 0.2 for ordered ferro-
0to20kOeatvarioustemperaturesaround2%ofT on
magnets. Similar enhancements were reported recently C
either side.
fromSQUIDmagnetizationmeasurementsonthediluted
magnetic semiconductors Ga Mn As,7 and in earlier The criticalexponent β associatedwith M , the spon-
1−x x s
studies on amorphous ferromagnetic alloys in compari- taneousmagnetization,isgivenbyM =B|ε|β forT<T ,
s C
son with crystalline ferromagnets.14,15 where ε=(T −T )/T and B is the corresponding crit-
C C
6
ical amplitude. The critical exponent γ is related to 4
2.0x10
χ , the zero-field dc susceptibility, through the equation
0
χ−1=Γ−1|ε|γ, where Γ−1 is the critical amplitude. The Ni Fe Mo
0 4 83.1 6.0 10.9
magnetic field dependence of M at TC gives us the third 1.5x10
critical exponent δ through the relation M=DH1/δ, D 1/ )
g
being the critical amplitude. All these three exponents u/ 4
m1.0x10
follow a static scaling relationship, δ=1+γ/β. In order
e
to find these exponents one has to know the transition M
3
temperatures very accurately. First the isothermal mag- (5.0x10
= 0.275
netizationM wasplottedasin asimple Arrottplot(M2
vs. H/M) which uses the mean-field exponents. How- = 1.03
0.0
ever, when they did not give linear isotherms, the modi-
0 400 800 1/1200
fiedArrott-Noakes(AN)plotwastried,i.e.,M1/β versus [(H/M)Oe/(emu/g)]
(H/M)1/γ.
40
10
ToplottheANisothermsonehastoknowthevaluesof Ni Fe Mo
83.1 6.0 10.9
β andγ beforehand. Insteadofguessingtheinitialvalues 30 8
we have calculated β and γ using a simple FORTRAN
Y
code where we applied the method of least squares fit- 6 X
ting. We have calculated the averagechi-square (χ2) for 20
4
all possible combinations of β and γ, where β was var-
ied between 0.2 and 0.6 in steps of 0.001 and γ from 10
= 0.31 = 1.05 2
0.9 to 1.5 in similar steps. Then the combination of
β and γ, for which the average χ2 is a minimum and 0 0
the slopes of all the straight lines are almost equal, i.e., 172 176 180 184 188 192
the set of isotherms are truly parallel to each other, are T(K)
taken. Amazingly, the extrapolated isotherm at TC in- FIG.3: (ColorOnline)ThemodifiedArrott-Noakes(AN)plot
deedpassesthroughtheoriginmakingusquiteconfident and X (T) and Y (T) vs. temperatureplots for Mo10.9 alloy
ofthe plots. Thenusing those values of β and γ we have withvaluesofβ andγ forwhichthebest-fittedisothermsare
plotted in Fig. 3 the modified Arrott-Noakes (AN) plot, obtained.
a set of isotherms for 10 % Mo alloy, the y-intercepts
of these straight lines give M and the x-intercepts χ−1.
s 0
Figure 3 (upper panel), a typical AN plot, also gives the panel) also shows the typical X(T) and Y(T) plots for
valuesofβ andγ forthe best-fittedisotherms. Although 10 % Mo alloy. Both of them intersect the T-axis at
γ is close to the mean-field value of 1, β is much lower T=T =180.5 K.
C
thanthemean-fieldvalueof0.5. Deviationsofthevalues
We have used both the methods for all our samples to
of β and γ from the mean-field values have been found
check the consistency of the results. Except for Mo13.5
both in many magnetic glasses as well as in crystalline
alloy, all the other samples gave quite reasonable val-
ferromagnets.
ues of the critical exponents. However, it is to be noted
There is another way of calculating β and γ, that is thatunlikeotheralloys,Mo13.5hasareentrantspin-glass
using the Kouvel-Fisher method. The set of equations phase(RSG)phasebelowT =10K(T is∼44K)which
g C
used in this method are: mightpossibly affectthe criticalbehaviorsincethe tran-
sitionmaynotbeapureferro-paraonealthoughweused
−1
dlnMs 1 M(T,H)dataonlyfromT=43-45K,farawayfrom10K.
Y(T)= = (T −T ) (9)
C
(cid:18) dT (cid:19) (cid:18)β(cid:19) Nevertheless, our calculated values of the critical expo-
nents from the two methods did not match at all. More
and importantly, the value of β was above and that of γ was
below the mean-field values. This is quite unphysical as
dlnχ−1 1
X(T)= 0 = (T −T ). (10) themean-fieldcaseisthelimitingone,sinceitisarather
C
(cid:18) dT (cid:19) (cid:18)γ(cid:19)
crude theory. Presence of more than one phase due to
improper homogenization and/or final annealing might
Theseequationsarevalidonlyinthecriticalregionwhere
cause such deviations in the critical exponents. Table
temperatures are very near T . Here we have used the
C II below gives the values of T , β, γ, and δ. The criti-
M and χ−1 values from the intercepts of the AN plots. C
s 0 calexponentvaluessuggestthatMo8.9andW8.9almost
InthecriticalregionY(T)vs. T andX(T)vs. T areboth
follow the mean-field model whereas the others, except
straightlineswithslopesof(1/β)and(1/γ),respectively.
Mo13.5 are close to 3D-Ising rather than that of pure
InthismethodaprioriknowledgeofT isnotneededand
C nickel, which follows the 3D-Heisenberg model.
if the results are consistent, then Y(T) and X(T) will
intersect the T axis at the same point. Figure 3 (lower We have also calculated the critical amplitudes since
7
TABLE II: Alloy compositions, values of TC and critical ex-
paloonnegnwtsi,tohbtthaoinseedfobroptuhreexNpierainmdenthtaolsleyfaronmdferxoimstiKngFtahneaolryiessis. -4 Ni86.6Fe3.1W10.3 ) 7.2 Ni82.6Fe6.9W10.5
Sample TC β γ δ n-5 M ( 7.0
CNoi8m3.p5Foesi7t.6ioMno8.9 E31x6p.t0. 3K16F.4 E0x.4p8t. 0K.4F2 E1x.2p1t. 1K.3F1 E3x.5p4t. 13+.50βγ l-6 = 1.3 ln 6.8 = 4.67
Ni83.1Fe6.0Mo10.9 180.5 180.4 0.275 0.31 1.03 1.05 4.68 4.74 = 4.63 kOe g/emu 6.6 D0 = 5.13
Ni83.2Fe3.3Mo13.5 44.55 46.2 0.59 0.54 0.92 0.88 2.22 1.64 -5.6 -4.9 -4.2 -3.5 7.2 7.8 8.4 9.0 9.6
Ni83.5Fe7.6W8.9 305.0 300.0 0.43 0.45 1.30 1.37 4.22 4.05 ln ln H
NNii8826..66FFee63..91WW1100..53 15981.1.05 15980.1.05 00..3343 00..3395 11..32034 11..1299 44..6973 44..6544 s 2.0 Ni83.1Fe6.0Mo10.9 Ni86.6Fe3.1W10.3
Nia 627.4 0.378 1.34 4.58 4.54 M 1.8 1|00
Mean-field 0.50 1.00 3.00 ln M/|
1.6
3D-Ising 0.312 1.25 5.00 = 0.289 T<Tc
3aDR-eHfeeriseenncbee2rg1. 0.378 1.405 4.76 1.4 -5.6 -4.B90 = -149.2.95 e-m3.5u/g 10105 T1>0T7c 109 1011 1013
ln H/||
they are important for a complete knowledge of the FIG. 4: (Color Online) . χ−1 vs. lnε plot for W10.3 alloy,
0
critical behavior near ferro-para transition. The mag- lnM vs. lnH plot for W10.5 alloy, lnM vs. lnε plot for
s s
netization above and below T satisfies a single scal- Mo10.9 alloy and M/|ε|β vs. H/|ε|βδ plot for W10.3 alloy.
C
ing equation given by m=f (h) where m=|ε|−βM(ε,H)
±
and h=|ε|−βδH are called scaled magnetization and the
scaled magnetic field. The above relations show that m haviour.
as a function of h falls on two different universal curves
The locally self-consistent spin dynamics analysis
f (h) for T<T and f (h) for T>T . If the values of
− C + C
showsthatasthetemperaturefallsbelowT ,thecluster
thecriticalexponentsfoundherearecorrect,thenallthe C
spins rapidly get magnetized.8–10 A similar behaviour is
data will fallon twodistinct curves confirmingtheir cor-
rectchoice. ThecriticalamplitudesB =m ,Γ−1=h /m expected above TC in an external magnetic field. Thus,
0 0 0 0
the clusterspins contributedominantlyto the highmag-
andD areobtainedfromthe interceptsofthe ln-lnplot
0
of M vs. |ε|, χ−1 vs. |ε| and M vs. H(ε = 0), re- neticsusceptibilityofthealloysnearTC,andhenceresult
spectsively. Figure0 4 shows, respectively lnχ−1 vs. lnε in a suppression of χ−01 and of the critical amplitudes.
0
plot for W10.3, ln Ms vs. lnH for W10.5, lnMs vs. lnε Particularlyfor the Mo13.5sample, the extremely low
for Mo10.9, and M/|ε|β vs. H/|ε|βδ for W10.3 including measured values of critical amplitudes along with the
the calculated values of the corresponding critical am- unusual critical exponent values (see Table II) indicates
plitudes. Figure 4 (lower right panel) shows clearly the dominant finite-cluster contribution just below the per-
correctness of our procedure. Table III gives the criti- colation threshold, with no diverging spin correlation
cal amplitudes of some of the samples. We observe that length and no true critical behaviour. This is consistent
with the increasing Fe dilution, the critical amplitudes with the emergence of spin-glass behaviour at this com-
Γ−1, B0, and D0 decrease systematically and all three position due to frustrated cluster spins locked in frozen
have values lower than those for Ni. This is the third non-collinear orientations which also results in smaller
observableconsequenceofstronglycoupledclusterspins, spontaneous magnetization along the z-direction.
indicatingsignificantlyreducedparticipationbythebulk
spinsformingtheferromagneticmatrixinthecriticalbe-
IV. CONCLUSION
TABLEIII:Thecriticalamplitudesofthealloysandthoseof
Ni for comparison. In conclusion, we find that all three macroscopicmag-
Sample Γ−1 B0 D0 netization characteristics— the spin-waveparameter B,
composition (kOe-g/emu) (emu/g) the reduced coefficient B , and the critical amplitudes
3/2
Ni83.5Fe7.6Mo8.9 8.4 21.4 7.36 Γ−1 — corresponding to the low, intermediate, and the
Ni83.1Fe6.0Mo10.9 4.2 20.0 5.14 critical temperature regimes respectively, yield distinc-
Ni83.2Fe3.3Mo13.5 1.7 2.5 1.9 tive spin dynamics signatures associated with strongly
Ni83.5Fe7.6W8.9 8.7 27.2 7.27 coupled cluster spins in these alloys.
Ni82.6Fe6.9W10.5 6.1 25.9 5.13
Ni86.6Fe3.1W10.3 4.6 8.8 4.79 The dilution behaviour of the two magnetization co-
Nia 19 83 30 efficients extracted from the macroscopic magnetization
a Reference 21. behaviouroffinite-temperaturespindynamicsinthetwo
8
alloysystemsisindicativeofabroaddistributionofmag- consistentwith this picture. Similar suppressionof criti-
netic spin interactions between the magnetically active calamplitudeswasobservedinFe Ni P B alloys,22
x 80−x 14 6
Featoms. Thesharpenhancementinthemeasuredspin- and was ascribed to the growth of spin clusters as T ap-
waveparameterB accompanyingthespinstiffnessreduc- proachesT andthe consequentreducedparticipationof
C
tionwithFedilutionclearlyindicatesweakenedbulkspin remaining bulk (FM matrix) spins in the FM-PM phase
couplings and softening of low-energy spin-wave modes. transition.
Furthermore, the magnitude of the reduced coefficient
Except for the Mo13.5 alloy, which has a spin-glass
B rapidly approaches 1 with increasing Fe dilution,
3/2 phase below T =10 K, all the other samples gave
andtheslowermagnetizationdecaywithtemperaturein- g
quite reasonable values of the critical exponents. Non-
dicates presence of strongly coupled cluster spins which
universalvalues of critical exponents with β ≈0.55 have
resistthermaldemagnetizationandstretchthe magnetic
also been found in recent experimental studies23 of the
order near T .
C ferromagnetictransition in spin-glass re-entrantmetallic
As also observed in diluted magnetic semiconductors alloys such as Au Fe .
0.81 0.19
and metallic glasses, this sharp enhancement with dilu-
tion is thus suggestive of the two magnetization coeffi-
cientsasuniversalmacroscopicindicatorsofspincluster-
ing and disorder-induced distribution of magnetic inter-
Acknowledgements
actionsindisorderedferromagnets. The reductionofthe
measuredcriticalamplitudes withFe dilution due tosig-
nificantly reduced participation by the bulk spins form- MB acknowledges financial support from UGC, Govt.
ing the ferromagnetic matrix in the critical behaviour is of India.
∗
Electronic address: [email protected] Instituteof Technology Kanpur(2008).
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