Table Of ContentClustering induced suppression of ferromagnetism in diluted magnets
Avinash Singh,1,2,∗
1Max-Planck-Institut fu¨r Physik Komplexer Systeme, N¨othnitzer str. 38, D-01187 Dresden and
2Department of Physics, Indian Institute of Technology Kanpur - 208016∗
Ferromagnetism in diluted magnets in thecompensated regime p≪x is shown to be suppressed
7
by the formation of impurity spin clusters. The majority bulk spin couplings are shown to be
0
considerably weakened by the preferential accumulation of holes in spin clusters, resulting in low-
0
energy magnon softening and enhanced low-temperature decay of magnetic order. A locally self-
2
consistent magnon renormalization analysis of spin dynamics shows that although strong intra-
n cluster correlations tend to prolong global order, Tc is still reduced compared to the ordered case.
a
J
9 Diluted magnetic semiconductors (DMS) such as In the compensated regime p x, however,the relevant
1 Ga Mn As exhibit a novel carrier-induced ferromag- mechanism is quite different. ≪
1−x x
netism where doped mobile carriers mediate magnetic In this paper we highlight a different aspect of com-
i] interactions between the S = 5/2 Mn++ impurity peting interactions in diluted magnets — a competition
c spins,1,2,3,4,5 with magnetization and Curie temperature between cluster and bulk spin couplings. In the com-
s
- Tc increasing with carrier concentration.6,7,8,9,10 While pensated regime p x, the formation of spin clusters
l ≪
r as-grownsamplesofGa1−xMnxAswithhighMnconcen- (due to impurity positional disorder) and accumulation
t trationexhibitalargedegreeofcompensation,wherethe ofdopedholesintheseimpurity-richregionssignificantly
m
holedensitypisasmallfractionoftheMnimpuritycon- deprivesthe majoritybulk spins ofholes. Becauseofthe
.
t centrationx,recentprogressinfabricationandannealing crucial compensation condition p x, even if cluster
a ≪
has yielded nearly compensation free samples, especially spins constitute a small fraction of the total, they can
m
at low Mn concentrations. accumulate a majority of doped holes. While the cluster
-
d Strong thermal decay of magnetization observed in couplings are therefore dramatically enhanced, yielding
n these diluted magnets has been characterizedrecently in high-energy cluster-localized magnon modes and strong
o termsofspin-waveproperties.11Magnetizationstudiesof intra-cluster ordering, the couplings between bulk spins,
c Ga Mn As samples with different Mn content (thick- whichareessentiallyresponsibleforlong-rangeferromag-
[ 1−x x
ness about 50nm, Mn content ranging from 2% to 6%) netic order, are significantly weakened, resulting in low-
2 usingSQUID(superconductingquantuminterferencede- energymagnonsofteningandenhancedthermaldecayof
v magnetization. These effects of impurity clustering on
vice)measurements,withlinearextrapolationfroma0.3-
4 0.4 T magnetic field to overcomeanisotropy fields which spincouplingsanddynamicshavebeenqualitativelydis-
7
result in spin-reorientation transitions,12 have found the cussed earlier within the impurity-band representation,
84 spontaneous magnetization to be well described by the applicable near the metal-insulator transition.16
0 BlochformM(T)=M0(1 BT3/2),withaspin-wavepa- Ourobjectivehereisto quantitativelyconnect, within
6 rameterB 1 3 10−3 K−−3/2whichisabouttwoorders a more general and unified framework, the different rel-
0 ofmagnitud∼eh−igh×erthanforFeandFeCofilms.11,13 The evant aspects — disorder, clustering, fermion polariza-
t/ calculated values of spin-wave stiffness constant D were tions,spincouplings,magnons,andspindynamics. With
a respecttomagneticeffectsofimpurityclusteringinDMS,
foundtobeofsameorderofmagnitudeasobtainedfrom
m
magneticKerrmeasurementsusingpump-probesetupof as distinguished from comparisonwith the ordered case,
- standing spin waves in Ga Mn As thin films.14 earlierstudieshavefoundsofteningoflow-energymagnon
d 1−x x
modes, weakening (strengthening) of bulk (cluster) cou-
n Recenttheoreticalstudies of magnonpropertiesin the
plings, and formation of high-energy cluster-localized
o dilutedferromagneticKondolattice model(FKLM) pro-
modes,16,17,18 formationofferromagneticdropletsbutno
c vide quantitative understanding of this large magnitude
: significanteffects on T inMonte Carlocalculations,19,20
v of B in terms ofa strongenhancementin density oflow- c
andreductionofT indensityfunctional21andlocalspin-
i energymagneticexcitationsathigherdilution.15 There- c
X density approximation22 studies involving ab-initio eval-
sultingstrongthermaldecayofmagnetizationwasfound
r uationofexchangeparametersandsubsequentstatistical
a tobeingoodagreementwiththeBlochformatlowtem-
analysis.
perature,withB ofsameorderofmagnitudeasobtained
The interplay between itinerant carriers in a partially
in the squid magnetization measurements.
filled band and localized impurity moments is conven-
Generally, presence of competing antiferromagnetic tionally studied within the diluted FKLM
(AF)interactionsisasourceofmagnonsoftening,dueto
the negative-energy contributions from AF bonds when H =t a† a J S .σ (1)
spins are twisted away from the collinear ferromagnetic X i,σ i+δ,σ− 2 X I I
i,δ,σ I
arrangement. Indeed, close to the limit p x of half fill-
∼
ing,competingAFinteractionsbecomestrongenoughto involving itinerant electrons hopping on host sites i and
destabilizetheferromagneticstateinthe dilutedFKLM. alocalexchangecouplingbetweenthelocalizedimpurity
2
0.0008
disordered
0.016
p=1.35% pp==22..8855%% 0.0007 ordered
0.014
0.0006
0.012 J=4 JJJJ====4444
0.0005
x=12.5% xxxx====11112222....5555%%%%
0.01
0.0004
0.008 J(r) 0.0003
J(r) 0.006 0.0002
0.004 0.0001
0.002 0
0 -0.0001
1 1.5 2 2.5 3 3.5 4 4.5 5
-0.002 r
0.5 1 1.5 2 2.5 3 3.5 4 4.5 00 5..55 11 11..55 22 22..55 33 33..55 44 44..55 55
r rr FIG.2: (Coloronline)Substantiallyweakenedbulkspincou-
plings at separations >∼ 2 compared to the ordered case, for
FIG. 1: (Color online) Distribution of spin couplings (single a N = 103 system with J = 4, x = 1/8, and p = 1.35%,
configuration) at two doping concentrations, showing large averaged over20 configurations.
spread in the strongly enhanced cluster couplings and dis-
tinctly non-RKKY nature in the compensated regime (p ≪
x).
couplings can be incorporated by including self-energy
and vertex corrections in the particle-hole bubble within
spinS anditinerantelectronspinσ . Recently,magnon a spin-rotationally-symmetric scheme,23 but are sup-
I I
pressed by the factor 1/S. The above isotropic spin
properties in the mixed spin-fermionmodel (1) were dis-
interaction neglects magnetic anisotropy arising from
cussed extensively with respect to variations in dilution
spin-orbitinteractioninthehostsemiconductor,24,25,26,27
x, exchange coupling J, impurity energy ǫ , and hole
d
and the recently observed strain-induced uniaxial
concentration p within an exact treatment of impurity
anisotropywhichisdependentonholeconcentrationand
positional disorder amd a non-perturbative treatment of
exchange coupling.15 The magnon propagator was stud- temperature.12
For simplicity, we consider (1) on a simple-cubic lat-
iedintherandomphaseapproximation(RPA),wherethe
tice, with periodic boundary conditions. The N mag-
bubble diagrams, representing repeated interactions be- m
netic impurities are placed randomly on a fraction (I)
tween impurity spins mediated by the particle-hole bub-
of the N = L3 host sites (i), with impurity concentra-
ble, provide the lowest-level spin-rotationally-symmetric
tionx=N /N. We considerpositive nearest-neighbour
treatmentoftransversespinfluctuationswheretheGold- m
hopping t, and set t = 1 as the unit of energy scale.
stone mode is explicitly preserved.
For the fully polarized, collinear ferromagnetic ground
Due to spin-rotation symmetry of (1), the impurity
state at T = 0, stability of which is confirmed from
spindynamicsisnaturallydescribedbyanisotropicinter-
action J S .S between impurity spins, and at absence of negative-energy magnon modes, the fermion
− IJ IJ I J eigenvalues E andwavefunctions ψ areobtained
the RPAPlevel, the (weakly-dynamical) carrier-mediated { lσ} { lσ}
by exact diagonalization of the N N fermion Hamil-
spin couplings are given by
×
tonian with effective impurity potentials JS/2 for the
J2 two fermion spins. Furthermore, we con∓sider the sat-
J (ω)= [χ0(ω)] (2)
IJ 4 IJ urated ferromagnetic state with a fully occupied spin-
↑
bandandhole dopingonly inthe pushed-upspin- band
in terms of the particle-hole bubble (p Nholes/2N), so that only the first term in E↓q. (4)
≡ ↓
dω′ contributes to the particle-hole bubble. For p x or
[χ0(ω)]IJ = iZ 2π [G↑(ω′)]IJ[G↓(ω′−ω)]JI (3) lowervaluesofJ,x,wedoobtainnegative-energym∼agnon
modes,indicatingnoncollinearferromagneticordering,as
ψlI↑ψlJ↑ψmI ↓ψmJ↓ found in earlier studies.28 As magnon energies are very
= f (1 f )
l↑ m↓
XEm↓ El↑+ω − low compared to the Stoner gap, the spin couplings are
l,m −
only weakly dynamical, and hence we have set ω = 0 in
ψI ψJψI ψJ
l↑ l↑ m↓ m↓ Eq. (2). Although their nearly static nature is similar
+ (1 f )f (4)
l↑ m↓
XEl↑ Em↓ ω − tothatofRKKYinteraction,theeffectivespincouplings
l,m − −
arestronglynon-perturbativeincharacterduetotheJ,x
evaluated by integrating out the fermions in the broken- dependence ofthe wavefunctionsandeigenvaluesinEqs.
symmetry state. Spin-fluctuation corrections to spin (3) and (4).
3
0.025 45
disordered disordered
ordered 40 ordered
0.02
35
30 J=4
0.015 x=12.5%
25
JI ωN() 20 p=1.35%
0.01
15
10
0.005
5
0 0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
〈σz〉 ω
I
2.5
FIG. 3: (Color online) Correlation between impurity-site dis 1
fermion polarization hσIzi and the local measure JI ≡ 23
PJ6=IJIJ of spin couplings, obtained for 20 configurations 2 4
of a N = 103 system with J = 4, x = 1/8, and p = 1.35%. 5
ordered
Relativetotheorderedcase(bolddot),significantlyenhanced
(reduced) fermion-spin densities and impurity-spincouplings 1.5
correspond to cluster (bulk) spins. 〉 z
S
〈
1
J/t=4
Figure 1 shows the distribution of impurity spin cou-
x=12.5%
plings J obtained from (2,4) as a function of impurity 0.5 p=1%
IJ
separation r r r , for two different hole concen- W=12t=10 eV
I J
≡ | − |
trations p/x 1/10 and 1/4. Relatively very strong
0
≈ ≈
spincouplingsaregeneratedbetweenclusterspinshaving 0 20 40 60 80 100 120 140 160
small separations 1 and √2, which account for the high- T (K)
energy cluster-localized magnon modes.15 A most strik-
ing feature is that for the same impurity separation, the FIG.4: (Coloronline) Thestronglow-T decayofmagnetiza-
coupling magnitudes are spread over a large range. The tion, with a subsequent marginally paramagnetic behaviour
correspondtotheenhancedlow-energymagnonDOSandten-
coupling between two impurity spins is thus not merely
dencyofintra-clustercorrelationstoprolongorder,shownfor
a function of their separation, but actually depends on
5 disorder configurations of a N =123 system in comparison
the whole configuration, suggesting shades of a complex
with theordered case at exactly same dilution x=1/8.
system.29Indeed,forsamex,p,r,thebulkspincouplings
in the disordered case are strongly weakened compared
to the ordered case. This feature is absent in certain
spin-only models where the couplings are fixed functions toasuperlatticearrangementofimpuritieswithimpurity
of the separation, quite independent of disorder. spacing2andconcentrationx=1/8. Thebulkspincou-
Furthermore,whileAFcouplingsaredistinctlypresent plings involving averageseparations >2 are clearly con-
for p/x 1/4 and indeed the collinear ferromagnetic siderably weakened, which is directly∼responsible for the
stateisu≈nstable,inthecompensatedregimep/x 1/10, observed low-energy magnon softening with dilution.15
≈
where the ferromagnetic state is quite stable, the cou- Turning now to include fermions, and the main result
plings are seen to be dominantly ferromagnetic. The of this paper, Figure 3 highlights the essential character
plot of configuration-averaged couplings (Fig. 2) con- ofcarrier-inducedspincouplingsinthepresenceofdisor-
firms that the couplings remain ferromagnetic and do derintermsofadirectcorrelationbetweenlocalfermion
notchangesign. Thisdistinctlynon-RKKYbehaviourof polarization σz and the local measure J J
h Ii I ≡ J6=I IJ
spin couplings in diluted magnets has been highlighted of spins couplings. Fermion polarization valuPes larger
recently in a detailed study.30 than the ordered-case value correspond to cluster sites
In order to examine the effect of impurity clustering which attract a disproportionately higher concentration
on bulk spin couplings we make a comparison with an of holes, yielding proportionately larger spin couplings.
ordered diluted case. Fig. 2 shows the spin couplings Theremainingbulksitesarethusleftwithaholedeficit,
fordisordered(configuration-averagedover20configura- resulting in proportionately weaker spin couplings. The
tions)andorderedcasesforthesameimpurityconcentra- compensation condition p x plays a crucial role here
≪
tion and hole doping. Here the orderedcase corresponds becauseevenifclusterspinsconstituteasmallfractionof
4
thetotal,theycanaccumulatenearlyallthedopedholes. Jx = 0.33eV is close to the corresponding experimental
ThethreeclearlyseparatedclusterpatchesinFig. 3pre- value Jn = J 10% n = 150meV.˚A3 10%
Mn Ga
sumablycorrespondtoone-,two-,andthree-dimensional 2 1022/cm3 = 0.×3eV wi×th J = 150meV.˚A3 tak×en from×
×
clusters, with progressively higher degrees of delocaliza- Ref. [1]. Direct measurements yield J = 1.2 0.2eV
tion and impurity-level band broadening. fromcore-levelphotoemission35andJ =2.4 0.9±eVfrom
The disproportionate accumulation of holes in magneto-transport.1,36 ±
impurity-rich cluster regions is simply due to preferen- Inconclusion,wehavequantitativelystudiedtheinter-
tial filling of cluster impurity states by doped holes. play, within a non-perturbative and unified framework,
Clusterimpuritystatesundergogreaterbandbroadening of the different ingredients of ferromagnetism in diluted
due to stronger effective hopping, and are thus pushed magnets — disorder, clustering, fermion polarizations,
to the valence-band top. Impurity states correspond- spincouplings,magnons,andspindynamics. Due to the
ing to remaining bulk spins lying deeper in the valence competition between impurities for a small fraction of
band are left with hole deficit. With increasing J and doped holes in the compensated regime p x, impurity
≪
morepronouncedimpurity-levelcharacter,thistendency disorderplaysahighlynon-perturbativerole. Thedirect
of differential band broadening is enhanced, resulting in correlationbetweenthe broadlydistributedfermionspin
strongersuppressionof ferromagnetism,highlighting the polarization values with the local measure of spin cou-
non-perturbative character of spin couplings in contrast plingshighlightstheessentialcharacterofcarrier-induced
to the RKKY picture.17,30,31,32 spin couplings in the presence of disorder, clearly distin-
To highlight the macroscopic effects of the disorder guishing between cluster and bulk spins. Cluster cou-
induced changes in bulk and cluster spin couplings, we plings show a largespreadfor same impurity separation,
compare the magnon density of states (DOS) and tem- and bulk couplings show distintly non-RKKY character.
perature dependence of lattice-averaged magnetization Preferentialaccumulationofholes in impurity clusters
with the ordered case at the same dilution. Magnetiza- strengthens (weakens) the cluster (bulk) spin couplings,
tion was obtained using a locally self-consistent magnon resulting in softening of low-energy magnons extended
renormalizationscheme,15 whichincorporatesthespatial over bulk spins and formation of high-energy cluster-
characterofmagnonstatesaswell. WithintheTyablikov localized modes. The locally self-consistent magnon
decoupling formalism, similar self-consistent schemes in- renormalizationschemeisabletocapturethesedisorder-
volving both global33 and local34 self-consistency have induced features in the finite temperature spin dynam-
been employed recently in the context of T calculations ics. While the low-energy magnon softening is respon-
c
for Heisenberg models with realistic spin couplings. sible for the observed strong low-T decay of magnetiza-
Fig. 4(upperpanel)showsthecharacteristicdisorder- tion (B 1.6 10−3 K−3/2 in Fig. 4), strong intra-
≈ ×
inducedfeaturesofsharplyenhancedlow-energymagnon cluster correlationstend to prolongglobalorder through
DOS and high-energy structures due to cluster-localized a marginally paramagnetic temperature regime. Inter-
modes,associatedwiththeweak(bulk)andstrong(clus- estingly,the somewhatslowerdecayduetoclustercorre-
ter) couplings, respectively. The corresponding features lationsbringsthemagnetizationfalloffclosertotheT3/2
inthemagnetizationplot(lowerpanel)arereadilyidenti- formoverabroaderT regime,asobserved.11 Overall,we
fied. The rapid low-T decay follows from the low-energy find that ferromagnetism is considerably weakened com-
magnon softening, and the subsequent marginally para- pared to the ordered case. As increasing p/x enhances
magnetic behaviour (more pronounced in some special AF couplings and destabilizes the ferromagnetic state,
configurations)isduetostrongintra-clustercorrelations; reducing spin clustering appears crucial for enhancing
the overall behaviour is characteristic of two distinct ferromagnetismindilutedmagneticsemiconductors,and
(magnon) energy scales.18 indeed could be a contributing factor in the annealing
InFig. 4(lowerpanel)wehaveconsideredabandwidth process.7,11 Atwo-componentpictureofbulkandcluster
W = 12t = 10eV of the order of that for GaAs. For spins, with a relatively low carrier density in the bulk
J =4t=W/3 3.3eV as considered in our calculations could be useful in interpreting the relatively high resis-
andanominali≈mpurityconcentrationx=10%,thevalue tivity seen even in the most metallic samples.1,7,8
∗ Electronic address: [email protected] 76, 323 (2004).
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