Impurity-Ion pair induced high-temperature ferromagnetism in Co-doped ZnO

0.82 MB·English

by  C. D. Pemmaraju

Impurity-Ion pair induced high-temperature ferromagnetism in Co-doped ZnO
C.D. Pemmarajua, R. Hanafina, T. Archera, H.B. Braunb and S. Sanvitoa
a School of Physics and CRANN, Trinity College, Dublin 2, Ireland and
b School of Physics, University College Dublin, Dublin 4, Ireland
(Dated: February 3, 2008)
Magnetic 3d-ions doped into wide-gap oxides show signatures of room temperature ferromag-
netism, although their concentration is two orders of magnitude smaller than that in conventional
magnets. The prototype of these exceptional materials is Co-doped ZnO, for which an explana-
tion of the room temperature ferromagnetism is still elusive. Here we demonstrate that magnetism
originatesfromCo2+ oxygen-vacancypairswithapartiallyfilledlevelclosetotheZnOconduction
band minimum. The magnetic interaction between these pairs is sufficiently long-ranged to cause
8
percolation at moderate concentrations. However, magnetically correlated clusters large enough to
0
showhysteresisatroomtemperaturealreadyformbelowthepercolationthresholdandexplainthe
0
current experimental findings. Our work demonstrates that the magnetism in ZnO:Co is entirely
2
governed by intrinsic defects and a phase diagram is presented. This suggests a recipe for tailoring

n the magnetic properties of spintronics materials by controlling their intrinsic defects.
a
J
1 ZnO is a piezoelectric conductive oxide, in which free- the opposite effect22,23.
3 carriers coexist with optical transparency1,2. If made v)Theroleplayedbyfreecarriersinestablishingmag-
magnetic, ZnO will become the ultimate multifunctional netism is unclear. Sequential annealing in reducing and
]
i material, with semiconducting, magnetic, optical and oxidizing atmosphere reveals little correlation between
c mechanical properties. This will have a far reaching the electrical conductivity and the magnetic state24.
s
- impact on the emerging field of spintronics3 with ap- Similar conclusions are reached for Al- and H-doping25.
rl plications in optoelectronics4 and quantum computing5. vi) ZnO is often reported to be oxygen deficient. This
mt Moreover it will allow us to go beyond the (Ga,Mn)As has been attributed to either oxygen vacancies (VO)26,27
paradigm6, whose practical use is severely hampered ormulti-centerH28,withZninterstitials(Zn )nowruled
. i
t by the low ferromagnetic critical temperature. This is outbybothexperimental26 andtheoretical27,28 evidence.
a
m why ZnO:Co is perhaps the most studied among all the ThusthepromotionofRTFduetoZnvapourexposure29
diluted magnetic oxides. Room-temperature ferromag- cannotbeattributedtoanincreaseoftheZn concentra-
d- netism(RTF),firstdemonstratedbyUedaetal.7,isnow tion. i
n confirmedbyanumberofgroups8,9,10 (seeTableIofthe vii) Electron paramagnetic resonance30 suggests the
o supplementary material). The experimental situation is presence of two magnetic centers. These are both re-
c howeverstillconfusedandherewelistthemainfindings. lated to Co2+, although they exhibit fine differences in
[
i) Spectroscopy confirms that Co2+ substituting Zn the signal. Interestingly, for a nominal Co concentration

1 is the center responsible for all the different mag- about5%thetwocentersappearwithsimilarabundance.
v neticphasesfoundexperimentally,includingRTF11,12,13,
Existingmechanismsforferromagnetisminthediluted
5 paramagnetism12,14 and spin-glasses15. RTF is usually
limit cannot explain this complex collection of phenom-
94 assigned from magnetometry7,8,9,10. ena. Thep-dZenermodel31 lacksofitsfoundationswhen
4 ii) The saturation magnetization Ms and the rema- appliedtoZn1−xCoxOsincethereislittlecorrelationbe-
. nence are always small and secondary phases are of- tween carriers and magnetism24,25. When present, carri-
1
ten difficult to rule out. However, except for metallic ers are electrons and not holes with small exchange cou-
0
8 Co, most of them are either non-magnetic or antifer- pling to the local spins and therefore the typical critical
0 romagnetic with low N´eel temperatures (CoO, Co2O3, temperatures (TC) are tiny at realistic carrier densities.
v: Co3O4, ZnCo2O4). The coercive field is typically small Similarly super-exchange must be ruled out32. This is
(∼ 100 Oe) and only weakly temperature dependent. short ranged and RTF can be obtained only for x above
i
X iii) M is usually smaller than what is expected the nearest neighbour (NN) percolation threshold. For
s
r for Co2+ with values as low as 0.01 µ /Co12, sug- the wurtzite lattice this is 20 %, much greater than the
B
a
gesting antiferromagnetic interaction among Co2+ and typical experimental concentrations.
frustration13,14,16. Finally a modification of the Zener scheme, called the
iv) Growth conditions and annealing are crucial for “donor impurity band exchange” (DIBE), assumes that
the magnetic state. Chemical methods14,16 and molec- themagneticinteractionismediatedbylargehydrogenic
ular beam epitaxy17 generally lead to paramagnetism, orbitals associated to intrinsic defects and predicts fer-
while pulsed laser deposition produces RTF films7,8,9,10. romagnetism below the donor percolation threshold, i.e.
Typically oxygen deficient growth10 at tuned substrate inabsenceoffreecarriers33. AlthoughthemeanfieldT
C
temperatures18,19,20 promotes RTF. Similarly, annealing obtained with realistic parameters for Zn Co O is ex-
1−x x
in vacuum enhances the magnetic moment and produces tremelysmall(∼10K)33,themodelisfrequentlyusedto
ferromagnetism12,20,21,22, while annealing in oxygen has explain the experimental results. We have investigated
2
such a model with Monte Carlo calculations (see supple- structureisincompatiblewithacarriermediatedpicture
mentary materials) and demonstrated that for Co and of ferromagnetism. In fact, E should be moved by at
F
donor concentrations respectively of 10 % and 1 %, the least 1 eV in order to affect the Co valence and there-
T is only a few degrees K. RTF is obtained only with fore to promote the charge transfer necessary for strong
C
unrealistically high values of the exchange coupling, and exchange coupling33, a task hardly achievable.
therefore the DIBE scheme must also be ruled out. We test this conjecture by calculating the magnetic
In the absence of any simple-scheme for ferromag- coupling between two Co2+ at various distances d .
Co−Co
netism we turn to atomistic density functional theory In table I we show the magnetic coupling energy E
M
(DFT).Forthisproblemthestandardapproximationsto (E > 0 indicates ferromagnetic coupling), obtained
M
the exchange and correlation potential (LDA and GGA) as the total energy difference between the ferromagnetic
arenotappropriatesincetheyover-delocalizeandunder- (FM) and antiferromagnetic (AFM) configuration of a
bindtheCo-dshellresultingintheirincorrectpositioning selection of supercells containing two Co2+ and one in-
with respect to the Fermi level, EF. These failures are trinsic defect far from both the Co2+. Clearly the ex-
only minor in the case of GaAs:Mn34, but they become change interaction between two Co2+ is strong only at
aseriousdrawbackfortheoxideswheretheCoddensity NN position, i.e. when the super-exchange interaction
of states (DOS) has important contributions in the ZnO is effective. In this case the interaction is AFM in
band-gap. We therefore use an approximated version of the a-b plane (d =3.19˚A) and FM along the c axis
Co−Co
the self-interaction correction (ASIC) scheme35, which is (d =3.11˚A). E however drops to zero already at
Co−Co M
free from these problems and produces exchange param- second NN, regardless of the presence of additional in-
eters for transition metal monoxides superior to those trinsic defects. In particular we show data for third NN
obtained with LDA/GGA36. (thedataforsecondNNaresimilar)inpresenceofeither
In Fig. 1 we present the DOS of a Co impurity (at Zn , H or V from which one has to conclude that RTF
i O
theZnsite)ina128atomZnOsupercell(x=0.0156)cal- is not achievable by simply defect doping.
culated with both LDA and ASIC. Although they both
Figure 2 offers an insight on why Zn and V are un-
i O
predict a 2+ valence, the position of the Co d levels is
abletomediateRTF.WepresenttheDOSofa128atom
remarkably different in the two cases. LDA places the supercell containing one Co2+ and one intrinsic defect
(V , Zn and H), and compare the case where the Co2+
O i
40 and the defect are well separated in the cell with that
units) 2300 a) EF TCZonot addl in which they are at NN position. In the case of distant
b. 10 defects the DOS is essentially a superposition of that of
ar 0 Co2+ and the defect. Both Zn and H posses a filled hy-
S (-10 i
O-20 e t drogeniclevelabovetheCBMandratherclosetotheCo
PD-30 2 d-t minoritylevels. IncontrastV displaysadoublyoc-
-40 2 O
40 cupiedimpuritylevel1eVabovetheVBT,almostatthe
nits) 2300 bb)) EF samepositionoftheminorityCo-delevels. Mostimpor-
b. u 10 tantly there is no evidence of interaction between Co2+
ar 0 and the defect levels (the magnetic moment calculated
PDOS (---321000 e t2 fcraosmes,thseimMiluarllliyketnoptohpeuclaatsieonofisC∼o22+.6oµnBl/yC).oTfohriasllmteharnees
-40
-12 -11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 that, despite the energy proximity of the defect levels to
E (eV)
those of the Co 3d shell, the overlap of the hydrogenic
FIG. 1: Density of states of Zn Co O as calculated from wave-function at the Co site is only minimal. For this
1−x x
density functional theory. The simulation cell is a 128 ZnO reason we conclude that the DIBE model as formulated
supercellinwhichoneZnatomisreplacedbyCo(x=0.0156). cannot be sustained by the electronic structure of Co2+
Panels (a) and (b) show LDA and ASIC results respectively. in ZnO.
The right panels of Fig. 2 for NN defects give a dif-
occupied minority e states just below E at the edge of ferent picture. In the case of both H and Zn there is a
F i
theZnOconductionbandminimum(CBM).ASICshifts substantial charge transfer from the defects to the Co2+
thesebyabout2eVdowntothevalencebandtop(VBT) resulting in a partial occupation of the minority t levels
2
in agreement with recent calculations37. In addition the and a reduction of the magnetic moment (1.95 µ /Co
B
ZnO band-gap opens and the Zn d DOS is also down- and 2.05 µ /Co respectively for Zn and H). On-site re-
B i
shifted. This gives us a picture where there is no Co-d pulsion moves upwards in energy all the Co 3d manifold
contribution to the DOS around E , with the first unoc- andnowthemajorityt levelsoccupytheZnObandgap.
F 2
cupied minority states (t ) placed at about 1 eV above In contrast, for V slight charge transfer from the Co
2 O
the CBM. Such a DOS is in excellent agreement with 3d to the vacancy and hybridization move the 3d man-
ultraviolet photoemission (UPS), which places the Co d ifold downwards in energy and the doubly occupied V
O
DOS at a binding energy of about 3 eV with a satellite level spin-splits in the opposite direction to that of Co.
peak at 7 eV and a rather diffuse tail13. This electronic Suchafeaturesuggeststhattheexchangeinteractionbe-
3
TABLE I: Calculated magnetic energy E for various magnetic centers and different dopants configuration. C1 and C2 are
M
the two magnetic centers included in the simulation cell and their relative concentration (concentration of each center), D is
the dopant with its concentration, d in the distance between the two center expressed both in ˚A and in NN shells. For
C1−C2
some of the NN complexes we present the geometrical configuration (after relaxation in the pictures below).
C1 (x) C2 (x) D (y) d (˚A) d (NN) Position D E (meV) Fig.
C1−C2 C1−C2 M
Co (0.015) Co (0.015) – 3.19 1 – -38 1
Co (0.015) Co (0.015) – 3.11 1 – 62 2
Co (0.015) Co (0.015) – 4.54 2 – -1 -
Co (0.010) Co (0.010) Zn (0.010) 8.01 3 Far -1 -
i
Co (0.010) Co (0.010) Zn+ (0.010) 8.01 3 Far 1 -
i
Co (0.010) Co (0.010) H (0.010) 8.01 3 Far 0 -
Co (0.010) Co (0.010) V (0.010) 8.01 3 Far -1 -
O
Co (0.015) Co (0.015) Zn (0.015) 3.180 1 Near 629 3
i
Co (0.015) Co (0.015) Zn (0.015) 2.551 1 Near 3 4
i
Co (0.015) Co (0.015) Zn (0.015) 2.914 1 Near 512 5
i
Co (0.015) Co (0.015) Zn (0.015) 2.557 1 Near 731 3
i
Co (0.015) Co (0.015) V (0.010) 2.585 1 Near 10 6
O
Co (0.015) Co (0.015) V (0.010) 2.795 1 Near -103 -
O
Co (0.015) Co (0.015) V (0.010) & Zn 2.315 1 Near 899 -
O i
Co (0.015) Co (0.015) H (0.010) 3.829 1 Near 12 7
Co (0.015) Co (0.015) H (0.010) 2.713 1 Near 296 8
CoV (0.015) CoV (0.015) – 5.55 2 – -6 -
CoV (0.015) CoV (0.015) H (0.010) 2.30 1 Far 423 -
CoV (0.015) CoV (0.015) H (0.010) 2.27 1 Far 614 -
CoV (0.015) CoV (0.015) H (0.010) 5.55 2 Far 84 -
CoV (0.015) CoV (0.015) H (0.010) 4.51 2 Far 9 -
CoV (0.015) CoV (0.015) H (0.010) 6.94 3 Far 20 -
tween NN Co ions, when mediated by a defect, can be with a substantial increase of the Co-d DOS in the ZnO
extremelylarge. Thisisindeedthecaseasdemonstrated bandgap.
in table I where EM for various Co-Co-defect complexes Since Co2+ alone cannot be responsible for RTF at
is presented. Such complexes effectively behave as small
lowdilutionwehavesearchedforotherpossiblemagnetic
metallicclustersanditisofnosurprisethattheexchange centers and found that Co2+-V pairs (CoV) overcome
O
energy increases quite dramatically for d around
Co−Co the limitations mentioned above. In figures 2d and 2e
and below 2.5 ˚A, which is the NN distance in metallic
wepresenttheDOSassociatedtotheCo-dshellforboth
Co. Co2+ andCoV,ascomparedwithUPSfromreference13.
Clearly both Co2+ and CoV are compatible with UPS,
Is this sufficient for RTF in the diluted phase? Un-
inparticulartheybothshowafiniteDOSatabout-8eV
fortunatelynot. Theseinteractions, althoughstrong, are
from E . This feature is absent in the DOS of both the
still short-range and therefore produce RTF only for x F
Co-Zn and Co-H complexes, which instead present sub-
around the percolation limit. Moreover, the strong FM i
stantial contributions in the ZnO gap and therefore are
interactions require a ratio between Co and the relevant
incompatible with the spectroscopy. Moreover CoV is
donor of 2:1, which at percolation means typical defect
the only complex among the ones studied which main-
concentrations of around 10%. These are impossible to
tains Co in the 2+ valence state.
achieve for any of the defects investigated even under
the most favorable conditions. Finally those complex CoVarealsolikelytobeabundant. OurcalculatedV
O
structures, if abundant, should appear spectroscopically formation energy (see supplementary material) in zinc
4
Finally we have to establish whether CoV couple at
a1) Zn a2)
10 i long-range. Table I shows E for various CoV combina-
M
0 tions. Similarly to the case of Co2+ also the magnetic
-10 Total coupling between two charge neutral CoV is remarkably
Co d
weak already at second NN (-6 meV). However, when
u.) b1) V b2) an additional electron donor is present, the situation
(a. 10 O changes dramatically with second and third NN mag-
S 0 netic coupling reaching up respectively to E ∼80 meV
O-10 M
D and E ∼20 meV. Note that we cannot exclude an even
M
P
longer interaction range, which however is hardly acces-
c1) H c2)
10 sible in our simulations since the size of the supercells
0 becomes prohibitively large.
-10
ZnO CB ZnO CB ZnO CB
-9 -8 -7 -6 -5 -4 -3 -2 -9 -8 -7 -6 -5 -4 -3 -2 t ↓
E (eV) E (eV) Co d› 2 t2 ↓-VO
u.) d E e CUoP Sdfl n-doping region
a. F E
S ( 10 F (a)
O 5
D a↑
P 0 -9-8-7-6-5-4 -3-2 -1 0 -9-8-7-6-5-4 -3-2 -1 0 e↓ 1 a1↓ a1
E (eV) E (eV) e↓
FIG. 2: Density of states for a ZnO 128 atom supercell with
ZnO VB ZnO VB ZnO VB
oneCo2+ andoneadditionaldefect: a)Zn ,b)V ,andc)H.
The left panels are for Co2+ and the defecit well sOeparated in Co2+ CoV VO
the cell, while the right panels are for the NN position. The
arrows indicate the relevant defect position. In d) and e) we 100
(b)
present the Co 3 d density of states as compared with UPS
data from reference13. d) Co2+ substitutional at the Zn site, 80
e)Co2+-VOcomplex. TheUPSsignalhasbeenalignedtothe V) 60 2ndNN-H
calculatedDOSinordertohavethefirstpeakattheminority me 2ndNN
Co e states. (M 40 3rdNN-H
E
20
rich conditions is 0.65 eV, which suggests that the V
O 0
concentration in ZnO can be as large as 1 % at equilib- 0 2.5 5 7.5 10 12.5 15
rium. Since Co does not introduce doping its presence n (1020 cm-3)
will not change considerably the V formation energy.
O
WehavethenonlytoestablishwhethertheVO sitspref- FIG. 3: CoV impurity band. (a) level diagram for Co2+, VO
erentially close to a Co site. By using a 128 atom su- and CoV, (b) E for two 2nd NN CoV as a function of the
M
percell we calculate a reduction in total energy of about donor impurity band electron density. The electron density
340 meV for Co and V moving from third to first NN 6.7·1020 cm −3 corresponds to one electron every two CoV.
O
(pairing energy). This is quite a large gain suggesting
that most of the V are indeed likely to be close to Co WhydoesCoVsustainlongrangecouplingwhileCo2+
O
ions. The large pairing energy also means that the rela- does not? The cartoon of Fig. 3a shows the electronic
tive concentration of CoV (xCoV) with respect to that of structure of Co2+, V and CoV. The main feature is
O
Co2+ (xCo)willincreaseuponoxygenabsorbingprocess- that the Co2+ empty t minority state broadens in CoV
2
ing. Thisisthecaseforbothannealinginanoxygenpoor and forms a t -V hybrid level right at the CBM. This
2 O
atmospheres and long exposure to Zn and Ti vapours26. extendsinspaceoverthethefirstshellofZnionsandan
InthefirstcaseoneexpectsV migrationtotheCosites, impuritybandformsalreadyattinyconcentrations. Such
O
while in the second a preferential V formation close to a band can be easily n-doped. Thus, while for Co2+ the
O
Co. Finally we found that the pairing energy between onlyexchangemechanismsavailableeitherinvolvevirtual
twoCo2+ isalsolarge(∼210meV),whilewedonotfind transitions or weak s-d38 exchange interaction with hy-
any substantial pairing interaction among the CoV in drogenic orbitals, for CoV strong carrier-mediated mag-
any charging configuration. We then expect an inhomo- netic interaction is possible. In figure 3b we show E
M
geneous distribution of the Co2+, which in turn would for two CoV in a 128 atom unit cell as a function of the
lead to an inhomegeneous distribution of CoV and the electron doping. This is introduced by either moving E
F
formation of both high and low concentration regions. in our simulations or by explicitly introducing an H ion
5
cinupthateiosunpoefrctehlel.imClpeuarriltyyEbMandde,pwenitdhsastmroanxgilmyuomntahtehoacl-f 100 x C o / x C o V = 555///012%%%(a) 611%58%% (b) 0.4
bfiallnindg).(n=6.7×1020 cm−3)andvanishingforn=0(empty 10-2 50//37%% 33P06M%% 0.3
)
Thus DFT offers us a mechanism for the ferromag- N m
( 0.2
netismin(Zn,Co)Obasedontwomagneticcenters. Co2+ P10-4
is responsible only for short range coupling, while CoV
0.1
can instead sustain long range interaction via a fraction- 10-6
ally filled impurity band. Can this alone produce RTF?
0
Percolationtheory39setsastrictconditionforamagnetic 100 101 102 103 104 105 0 100 200 300 400
N T (K)
ground state of diluted systems: the concentration of
1.0 1
magnetic impurity should exceed the percolation thresh- (c) (d)
old xc. This depends on the range of the interaction and 0.8 xCo/xCoV= 1/5 % 0.8
forthefcclatticewefind19.8%,13.7%and6.2%forinter- 2/4 %
action extending respectively to 1st, 2nd and 3rd NN40. 0.6 34//32%% 0.6
m m
Therefore our two-center model already produces long
0.4 0.4
range ferromagnetism at xCoV > x ∼ 6%. However, we 0/7 %
c 1/7 %
donotneedsuchalargexforobservinghysteresisinthe 0.2 23//77 %% 0.2
M-H curve at room temperature (i.e. for explaining the 6/6 %
variousexperimentalclaims). Thiscanbeachievedbelow 0.00 100 200 300 4000 100 200 300 4000
x since one just needs a number of percolating clusters T (K) T (K)
c
large enough to be superparamagnetically blocked. Note
that“clusters”heremeanregionswherexislocallylarger FIG. 4: Monte Carlo simulations for the two center model:
thanx . ThepresenceofCoVpushesthislimitfarbelow (a) cluster distribution P(N) as a function of the number N
the 20%c needed by Co2+ and by the recently proposed of magnetic ions in the cluster for different xCo and xCoV
(xCo/xCoV); (b) magnetization curves m(T) at different Co
model where the magnetism originates from uncompen-
concentration for xCo/xCoV = 5; (c) m(T) for a total Co
satedspinsatthesurfaceofantiferromagneticclusters41.
concentration of 6% and different xCo/xCoV; (d) m(T) for
The size of those clusters can be estimated by con- xCoV above percolation (7%) and various xCo (xCo/xCoV).
sidering coherent rotation of the magnetization over an
anisotropy barrier DN S2 (D is the zero-field splitting,
B
N the number of magnetic ions magnetically blocked ulations for a Heisenberg energy (|S |=1)
B i
and S the Co spin). By taking D = 2.76 cm−1 from
EPR measurements17 we obtaine the estimate NB ∼800 H =Co(cid:88),CoVJ S ·S .+Co(cid:88),CoVD(S ·nˆ)2 . (1)
for a blocking temperature T =300 K. This however is eff ij i j i
B
rather conservative. In granular magnets random dipo- (cid:104)i,j(cid:105) i
lar interaction42, random magnetic anisotropy43 or spin-
The exchange parameters are chosen to mimic the short
odaldecomposition44 canpushTB tovaluesconsiderably andlongrangeexchangebetweenCo2+ andCoVrespec-
largerthanthosepredictedforsingleparticlecoherentro- tively. At NN J is AFM for Co2+ pairs (15 meV) and
ij
tation (up to a factor 5). Thus we estimate NB in the FMforCoVpairs(50meV)andbetweenCo2+ andCoV
range of 250 magnetic ions. Therefore, an observation of (50 meV). Moreover it extends to 2nd (15 meV) and 3rd
a hysteresis requires the existence of regions where 250
NN (5 meV) for CoV pairs. The last term accounts for
CoV ions interacting at 3rd NN exist at concentrations anhard-axiseasy-planeanisotropy(|nˆ|=1)17. Notethat
largerthan∼6%. Indeedthisisarathermodestrequire-
we implicitly assume doping in the CoV impurity band
ment.
and neglect the NN FM component of the exchange be-
InFig.4awepresenttypicalclusterdistributionsP(N) tween Co2+ pairs. Given the uncertainty over the pre-
as a function of the cluster size N for various concentra- cisemicroscopicconfigurationandtherelativeabundance
tions of Co2+ and CoV. These have been obtained by of the various complexes (Table I) our numerical values
filling randomly a wurtzite lattice comprising 106 sites. are only representative and certainly conservative. How-
As expected P(N) moves from small to large clusters as ever,evenwiththischoiceT =250Kabovepercolation
C
xCoV is increased with respect to xCo. In particular one (xCoV =7%) suggesting that RTF is indeed possible.
notesthatalreadyforxCoV =2%largeclustersappearin In Fig. 4b we present the reduced magnetization m=
thedistribution,whichbecomesbi-modalatxCoV >6%, M/M asafunctionofT fordifferentCoconcentrations,
s
i.e. above xc. We emphasize that these P(N) have been while keeping xCo/xCoV = 5. The magnetization curves
obtained from a completely random distribution, i.e. ne- show a transition from a concave upwards shape at low
glectingthetendencytoclusteringsuggestedbythepair- concentrations to a convex one for high. As the con-
ing energy. centration increases one encounters the two percolation
Finallyweinvestigatethethermodynamicalproperties thresholds, respectively for CoV and for Co. This pro-
ofourtwocentermodelbyperformingMonteCarlosim- duces the change in shape, which however is complete
6
only above 20%, i.e. when the Co2+’s start to percolate. probes are needed. In particular further insights would
Interestinglymneverreaches1becauseofthestrongNN be provided by a thorough analysis of small angle neu-
AFM interaction among Co. tron scattering data similar to the case of disordered
We also investigate the interplay between Co2+ and ferromagnets43. Alternatives may be muon rotation,
CoV. In figure Fig. 4c m(T) is calculated by keeping the high resolution EPR and energy-dispersive X-ray spec-
total Co concentration to 6% and by varing the ratio troscopy.
between xCo and xCoV, while in Fig. 4d we keep xCoV = Finally we partition the long-range FM region into
7%(abovepercolation)andchangexCo. Inbothpictures two regions separated by the CoV percolation thresh-
xCo is well below the percolation threshold for NN and old xCoV. For xCoV > xCoV percolation among CoV
c c
the high-temperature region of m(T) is almost entirely is achieved and one expects measurable conductivity
dominated by xCoV. For instance one may note that for from the impurity band. Since the exchange is strong
xCoV =5%m(T)approacheszeroroughlyatthesameT an anomalous Hall effect (AHE) should be detected.
regardlessofwhetherxCo is1%(Fig.4c)or25%(Fig.4b This is not expected for ferromagnetism below xCoV
c
foratotalconcentrationof30%). Incontrasttheamount since the conductivity is then dominated by band con-
of Co2+ affects the low temperature region, where the ductivity which is weakly affected by Co2+, given the
strong AFM interaction can drastically alter the total small exchange. Note that this phase diagram says lit-
magnetic moment. For instance m(0) drops from 1 to tle about the overall conductivity of (Zn,Co)O, which
about 0.8 if xCo is increased from 1 to 3%, by keeping in turn can be determined by electrons in the conduc-
xCoV =7%. tion band. Moreover, the precise location of the phase
We are now in a position to propose a phase diagram boundaries depends on details such as the concentration
for(Zn,Co)ObasedontherelativeconcentrationofCo2+ of electron donors. In the extreme case of fully compen-
and CoV (Fig. 5). The most important feature is the satedsamplestheblockedsuperparamagneticphasemay
presence of what we called a blocked superparamagnetic even disappear entirely.
phase. This is below xCoV and xCo, but nevertheless al- In conclusion, by using a combination of DFT and
c c
Monte Carlo simulations, we have demonstrated that
the observed RTF in (Zn,Co)O can be attributed to
blocked superparamagnetism. This develops at concen-
[Co]
trations below those required by long-range ferromag-
Spin-glass or
netism. However,eventhismodelrequiresasecondmag-
Frustrated AFM
neticdopantinadditiontoCo2+substitutingZn,capable
0.2 of mediating magnetic interaction beyond nearest neigh-
=
Co Long-range bours. We have identified the Co-VO pair as the most
xc FM Long-range FM likely candidate and demonstrated that such center can
and AHE indeed promote long range coupling, if additional n dop-
ing is present. These findings draw a new roadmap for
Blocked designingdilutedmagneticoxides,wheretheengineering
PM SPM of intrinsic defects play the leading role. For instance
paramagnetic samples can be turned ferromagnetic by
0.03 xCoV=0.06 [CoV] prolonged exposure to Ti vapours, which produce high
c concentrations of V 26.
O
FIG. 5: Proposed phase diagram for (Zn,Co)O as a func-
tionoftherelativeconcentrationsofCoandCoV.Theyellow Methods
area is the blocked superparamagnetic (SPM) region, where
both magnetic moment and hysteresis can be detected. The
Density functional theory calculations have been per-
dashed line at small x delimits the region where most of the
experiments are conducted. Finally the FM region is parti- formed by using our newly developed approximate ver-
tionintotworegionsdependingonwhetherornotanomalous sion of the self-interaction correction scheme (ASIC)35.
Hall effect can be detected. This is implemented numerically in the localized basis
set code Siesta45. Here the ASIC scaling parameter
lows one the detection of both a net magnetic moment α was set to 1/2, which is the value appropriate for
andhysteresisatroom-temperature. Forlargerxalong- describing the electronic structure of mid- to wide-gap
range FM ground state emerges, which however is lim- semiconductors35. In all the simulations we have consid-
ited by the short range AFM interaction of the Co ions. ered unit cells ranging from 128 to 256 atoms. The basis
Therefore we predict either a frustrated antiferromagnet set used was as follows: Zn: DZ-s, DZ-p, SZ-d, O: DZ-
or a spin-glass for xCo > xCo and xCoV (cid:28) xCo. We s, DZ-p, SZ-d, Co: DZ-s, DZP-p, DZ-d (SZ=single zeta,
c
emphasize that bulk measurements (hysteresis or mag- DZ=double zeta, DZP=double zeta polarized45). The
netization) can hardly distinguish between the FM and grid cutoff (equivalent to plane wave-cutoff) was 650 Ry
the blocked superparamagnetic phase, and more local and we have considered 18 k-points in the full Brillouin
7
zone for the 128 atom cell and appropriate scaling for netic ions and we always use periodic boundary condi-
other cells. Standard conjugate gradient geometrical re- tions. The systems are equilibrated until an initially
laxationwasperformeduntiltheforcesweresmallerthan AFM and FM replica have converged to the same value
0.04 eV/˚A. (typically 10,000 steps) and then the Monte Carlo mea-
Monte Carlo simulations were performed with the surements are taken by sampling 60,000 new steps. Dis-
Metropolis algorithm as implemented in a home-made orderaveragesaretakenover32differentsamplesateach
package. Typical simulation cells for the two center concentration. T forthecaseofxCoV =7%isevaluated
C
Heisenberg model contain between 1,600 to 5,000 mag- from the specific heat (see supplementary materials).
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Acknowledgment
This work is sponsored by Science Foundation of Ireland
under the grants SFI02/IN1/I175 and 05/IN1/I853. We
thankTCHPCandICHECforprovidingcomputationalsup-
port.