Exchange Interaction and $T_c$ in Alkaline-earth-metal-oxide-based DMS without Magnetic Impurities: First Principle Pseudo-SIC and Monte Carlo Calculation

0.18 MB·English

by  Van An Dinh

Exchange Interaction and T in Alkaline-earth-metal-oxide-based DMS without
c
Magnetic Impurities: First Principle Pseudo-SIC and Monte Carlo Calculation
∗
Van An Dinh, Masayuki Toyoda, Kazunori Sato, and Hiroshi Katayama-Yoshida
The Institute of Scientific and Industrial Research,
Osaka University, 8-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan.
The prospects of half-metallic ferromagnetism being induced by the incorporation of C atoms
into alkaline-earth-metal-oxides are investigated by the first principle calculation. The origin of
the ferromagnetism is discussed through the calculation of the electronic structure and exchange
8
coupling constant by using the pseudo-potential-like self-interaction-corrected local spin density
0
method. TheCurietemperature(Tc)isalsopredictedbyemployingtheMonteCarlosimulation. It
0
is shown that by taking theelectron self-interaction into account, thehalf-metallic ferromagnetism
2
induced by C in the host materials is more stabilized in comparison with the standard LDA case,

n andtheC’s2pelectron statesinthebandgapbecomemorelocalized resultinginthepredominance
a oftheshort-rangedexchangeinteraction. Whiletheferromagnetism inMgO1−xCx isstabilizeddue
J to the exchange interaction of the 1st-nearest neighbor pairs and might be suppressed by the anti-

5 ferromagneticsuper-exchangeinteractionathigherx,theferromagnetisminCaO1−xCx,SrO1−xCx,
1 andBaO1−xCx isstabilized byboththe1st-and2nd-nearest neighborpairs,andTc monotonously
increases with theC concentration.

]
i
c
s Besidestheattemptstodiscoverdilute magneticsemi- In this Letter, we will discuss the origin of the ferro-
- conductors (DMSs) by incorporating transition metals magnetismthatisinducedbysubstitutingCforOinfour
l
r into various materials to realize ferromagnetic DMSs for alkaline-earth-metal-oxides−MgO,CaO,BaO,andSrO
t
m spintronic devices, recently the ferromagnetism induced −throughthecalculationofnotonlytheelectronicstruc-
. bynonmagneticimpuritieshasalsoattractedgreatatten- ture but also the exchange coupling constant and Curie
t
a tion among both theoreticaland experimental scientists. temperature(Tc). Itshouldbe notedthatthe prediction
m A novelclassofmagnetic materialscanbe formedbyin- of ferromagnetism and T in ref. 3 and ref. 11 is based
c
corporatingnonmagneticimpuritiesorbylatticedefects. on the electronic structure calculation using the density
-
d With regardto this issue,the magnetisminduced by the functionaltheorywithin the localdensityapproximation
n cationvacancy in MgO [1], Ca vacancy in CaO [2, 3], Hf (LDA) and the mean field approximation (MFA) which
o
c vacancyinHfO2 with Tc exceeding500K[4, 5, 6,7], and cannot be successful for describing many materials, es-
by hydrogen in graphite [8, 9] and in carbon nanotubes pecially for strongly correlated systems. The LDA often
[
[10] etc. has been reported. It has also been predicted overestimates the hybridization between electron states

1
that magnetism can be induced by nonmagnetic impu- due to the underestimation of the bandgap energies of
v
rities such as C and N by substituting the O atoms in semiconductors. Moreover, the calculation of T within
9 c
various oxides. For example, the ferromagnetism caused theMFAoftenpredictsT withanexcessivelyhighvalue
4 c
2 by N and C in alkaline-earth-metal-oxides [3, 11] and evenif the substitutionalconcentrationislowerthanthe
2 thehalf-metallicferromagnetisminducedbyNinquartz- percolationlimit[15]. Therefore,inordertodescribeand
1. SiO2 [12]. predict the properties of these systems, we need a more
accurate method. There have been several attempts to
0 It is suggested that the magnetism might arise in the
8 improve the LDA to overcome these inaccuracies. One
host materials if impurities have a finite local magnetic
0 of the most popular methods is LDA+U which improves
moment that interacts with each other to form a mag-
: the LDA by taking the strong correlationeffects into ac-
v netic moment net. In the materials in which the mag-
i netism is induced by incorporatingnonmagnetic impuri- count through the screened Coulomb parameter U [16].
X
Theotherpopularapproachesaretheself-interactioncor-
ties,thesubstitutionalionsmayhaveanonzeromagnetic
r rection (SIC) [17] and the recent pseudo-SIC [18] meth-
a moment and the 2p-electrons of these ions, rather than
ods. SIC methods differ from LDA+U in that LDA+U
the 3d-electrons, play an essential role in inducing the
uses the additionalparameterU, while the SIC methods
magnetisminthehostmaterials. Theyformanimpurity
do not require any additional parameter at all. Con-
band in the deep bandgap, and ferromagnetism can be
sequently, in order to investigate the origin of ferromag-
induced if the Fermi level lies in these impurity bands.
netisminalkaline-earth-metal-oxide-basedDMSs,weap-
Furthermore, recent research also shows the role of the
ply the pseudo-SIC method based on the improvement
2p-like impurity band formed by C and N in the stabi-
of the MACHIKANEYAMA2000 packagecoded by Akai
lization of the ferromagnetism in Ga1−xMnxAs [13] and
[19](LDA+SIC)inthecalculationoftheelectronicstruc-
In1−xMnxN [14]. tures, and then we use the formula by Liechtenstein et
al.[20] to calculate the exchange interaction J between
ij
two impurities at the ith- and jth-sites in the ferromag-
∗Electronicaddress: [email protected] neticcoherentpotentialapproximationmedium. Finally,
2
FIG. 1: Total DOS (solid line) and C’s 2p - PDOS (dashed
line)forCaO0.9C0.1. Theleft-handfigure(a)showsthestan-
dardLDAcalculationandtheother(b)showsthepseudo-SIC
results. Theupperplanecorrespondstothemajorityspinand
lower plane tothe minority spin.
we employ the Monte Carlo simulation to estimate T .
c
FIG. 2: Total DOS (solid line) and C’s 2p - PDOS (dashed
The substitution of O with C in alkaline-earth-metal-
line) obtained bypseudo-SICcalculation for MgO0.9C0.1 (a),
oxides is treated randomly. For convenience, the lattice
CaO0.9C0.1 (b), SrO0.9C0.1 (c), and BaO0.9C0.1 (d).
constants of alkaline-earth-metal-oxide-based DMSs are
fixedtothevaluesofundopedcrystals[21](a=4.123˚A,
4.909 ˚A, 5.160˚A, and 5.520˚A for MgO, CaO, BaO, and
SrO, respectively) and no distortion in lattice structure
from Fig. 2, all materials in question are half-metallic.
is assumed. Throughout the electronic structure calcu-
However,theexchangemechanismcausingtheferromag-
lations, 624 independent k-sampling points in the first
netism is somewhat different. Figure 2(a) illustrates the
Brillouin zone are used. The potential form is restricted
DOS of MgO0.9C0.1. With the smallest lattice constant,
to the muffin-tin type, and muffin-tin radiiare chosenin
MgO0.9C0.1 has the largest bandgap energy. The C’s 2p
such a way that the ions at lattice sites can touch each
states are located near the top of the valence band orig-
other. The relativistic effect is also taken into account
inated from anion p-states, resulting in the strong hy-
by using scalar relativistic approximation.
bridization of 2p electron wave functions. The majority
In order to compare the density of states (DOS) spin states hybridize with O’s 2p states, leading to the
calculated within the standard LDA with that within appearanceoftheimpuritybandthatconnectstothetop
LDA+SICweplottheDOSofthetypicalcaseofalkaline- of the valence band and causes a narrowermajority spin
earth-metal-oxides, CaO1−xCx, at x = 0.10 in Fig. 1. bandgap. The minority spins create an impurity band
The figure on the left-hand side correspondsto the DOS in the bandgap, which includes the Fermi level. This
calculated within the standardLDA (Fig. 1(a)), and the impurity band is broadened with a half-width of about
figure on the right-hand side illustrates the DOS within 1.5 eV. The exchange splitting in this material approxi-
LDA+SIC (Fig. 1(b)). It can be seen that, taking the mates to 2.14 eV and is the smallest compared with the
self-interaction of electrons into account, the LDA+SIC three remaining materials. These impurity bands can
calculation gives a wider bandgap than the standard be broadened more strongly with increasing C concen-
LDA. While the position of the minority spin states of tration, leading to the antiferromagneticsuper-exchange
C’s 2p electrons remains unchanged, the majority spin interaction being easy to occur and the compensation
states calculated within LDA+SIC shifts about 1.1 eV of majority and minority spins. These mean that the
in comparison with the standard LDA one, leading to a ferromagnetismmightbesuppressedathigherCconcen-
higherlocalizationofLDA+SIC2pstatesthanLDA.And trations. In addition, our calculations for higher C con-
the local magnetic moment of C increases from 1.242µB centrationsshow that the ferromagnetismin MgO1−xCx
(LDA) to 1.482µB (LDA+SIC). The exchange splitting is most stabilized at x ≈ 0.10 (also refer to Fig. 5).
inthe caseofLDA+SICincreasesapproximately2times Figures 2(b)−2(d) demonstrate the DOS of CaO0.9C0.1,
as compared to that in the case of LDA, resulting in the SrO0.9C0.1, and BaO0.9C0.1, respectively. As seen from
possibility of the suppression of super-exchange interac- thefigures,thebandgapbecomesnarrowerandthelocal-
tion and the enhancement of the ferromagnetic double izationofthe 2pstates becomes strongerwith increasing
exchange mechanism. the distance between atoms (or lattice constant). The
Figure 2 depicts the DOS of AO1−xCx (A = Mg, Ca, majority spin states are located in the valence band and
Sr,Ba)calculatedwithinLDA+SICatx=0.10. Asseen causeasmallbroadeningoftheband. TheFermilevellies
3
FIG. 3: Local magnetic moment of substitutional C atoms
in MgO1−xCx, CaO1−xCx, SrO1−xCx, and BaO1−xCx at
x = 0.10.
intheimpuritybandinducedbyminorityspinswith1/3
of this impurity band being occupied by electrons. The FIG. 4: Exchange coupling constant vs. distance between C
exchangesplittinginthesematerialsisapproximately2.3 atoms in units of the lattice constant a in MgO1−xCx (a),
eV, but the C’s 2p states in BaO0.9C0.1 is the most lo- CaO1−xCx (b), SrO1−xCx (c), and BaO1−xCx (d) at several
concentrations of C (x=0.05,0.10,0.15, and 0.20).
calized. This change in the localization with respect to
the lattice constant is also indicated by the value of the
localmagneticmomentofCwhichisshowninFig.3. As
a result, one can expect the ferromagnetism to be stabi-
lized by the predominantdouble exchangemechanismin the1st-nearestneighbors,whiletheinteractionsbetween
these materials even at higher C concentrations. the2nd−nearestatomseasilyaccurthroughthemediate
atoms such as Sr or Ba which have large ionic radii.
The exchange interactions Jij in MgO1−xCx,
In short, the role of the nearest neighbors in the ex-
CaO1−xCx, SrO1−xCx, and BaO1−xCx are shown
changeinteractionchangeswiththelatticeconstant. The
in Fig. 4. As seen from Fig. 4(a), the exchange inter-
higher the lattice constant, the greater is the contribu-
action of the 1st-nearest neighbors in MgO1−xCx is
tion from the 2nd-nearest neighbors in the induction of
the strongest. The interaction strength is considerably
themagnetisminthehostmaterials. Theexchangeinter-
strong (about 140 meV for 5% of the C concentration)
actionisveryshortrangedandthe ferromagneticdouble
and most of the contributions come from the nearest
exchange mechanism is expected to be predominant for
neighbor interaction J01. Except J01 and J04 (at 10%
of the C concentration), contributions from other J all materials in question. Except MgO1−xCx, the mag-
ij
are very small and ignorable. Thus, the exchange netism in CaO1−xCx, SrO1−xCx, and BaO1−xCx can be
more stabilized at higher C concentrations. This is also
interaction in MgO1−xCx can be considered as a typical
consistentwiththeresultsoftheMonteCarlosimulation
case of short ranged interactions. In addition, the
for the estimation of T .
exchange interaction is considerably suppressed as the c
C concentration increases. At x = 0.20, the exchange In order to evaluate Tc, we perform the Monte Carlo
interaction becomes very small and the magnetism is simulations. The Metropolis algorithm [22] is applied
fully suppressed. Interestingly, by increasing the lattice to calculate the thermal average of the magnetization
constant, the role of the 2nd-nearest neighbors (J02) M and its powers. Then, the cumulant crossing method
becomes more important. For CaO1−xCx (Fig. 4(b)), proposedbyBinder[22]isemployedandthefourthorder
J01 is approximately two times smaller than that in cumulant U4 (a linear combination of <M4>/<M2>2)
MgO1−xCx, but the contributions J02 from the 2nd- is calculated as a function of temperature for different
nearest neighbors becomes important. The dominance cell sizes (14×14×14,16×16×16, and 18×18×18
of the contributions J02 of the 2nd-nearest neighbor conventional fcc cells) to find the universal fix-point at
pairs in SrO1−xCx and BaO1−xCx is shown in Fig. 4(c) Tc. We estimate Tc for four values of the substitutional
and Fig. 4(d), respectively. In these former materials, C concentrations: x = 0.05,0.10,0.15, and 0.20. The
the exchange interactions of the 2nd-nearest neighbors obtainedresultsaredemonstratedinFig.5. Somepoints
aremoreimportantthanthe1st-nearestneighbors. This should be clarified here.
canbe causedby the sufficiently largedistances between First,asdiscussedaboveforFig.2(a)andFig4(a),the
4
change interaction J01 between the 1st-nearest neighbor
pairs considerably decreases with the increasing lattice
constant, owing to the contributions J02 from the 2nd-
nearestneighbors,T monotonouslyincreaseswithxand
c
can gain a value higher than room temperature if the C
concentration is high enough. However, a question that
arises here is the solubility of C in the materials.
In conclusion, we have presented the results of the
study on the origin of ferromagnetism and predicted T
c
ofalkaline-earth-metals-oxide-basedDMSswithouttran-
sition metal elements. The electronic structures and ex-
changecouplingconstantsarecalculatedbyapplyingthe
pseudo-SIC approach. The dominant exchange mecha-
nismandtheroleofthenearestneighborsintheseDMSs
are discussed, and T is also evaluated by employing the
c
Monte Carlo simulation. In short, some comments can
be made as follows:
(i) All DMSs in question have a half-metallic ferro-
magnetism. The ferromagnetic double exchange mecha-
nism is predominantfor all materials in question, except
forMgO1−xCx whenthe C concentrationsishigherthan
10%.
FIG. 5: Curie temperature Tc vs. substitutional impurity (ii) The exchange interaction in these materials is
concentration in MgO1−xCx (solid line), CaO1−xCx (dashed
strong but short ranged. While the contributions
line), SrO1−xCx (dashed-dot-dot line), and BaO1−xCx
come mostly from the 1st-nearest neighbor pairs in
(dashed-dot line). The circles, crosses, and triangles denote
thecalculated points. MgO1−xCx, the important role played by the 1st- and
2nd-nearestneighborpairsinCaO1−xCx,SrO1−xCx,and
BaO1−xCx is comparable. However, while J01 becomes
weaker, J02 gets considerably stronger with the lattice
ferromagnetism of MgO1−xCx is the most stabilized at constant being larger and becomes the dominant con-
x ≈ 0.10 and will be suppressed at higher C concentra- tribution to the stabilization of the ferromagnetism in
tionsduetotheanti-ferromagneticsuper-exchangeinter- SrO1−xCx and BaO1−xCx.
action and the suppression of the majority and minority (iii) Correspondingly, Tc of CaO1−xCx, SrO1−xCx,
spins. Tc of MgO1−xCx (solid line) increases with x in andBaO1−xCx increasesmonotonouslywiththeincreas-
the rangeof xfrom0.05to 0.10andhas a peak of108K ing C concentration, while Tc of MgO1−xCx gains the
at x ≈ 0.10, and then it sharply drops as a function highestvalueat10%oftheCconcentration,thenitdrops
of −x at higher x. As shown in Fig. 2(a), the impu- sharply and tends to reachzero at higher concentrations
rity band formed by C’s 2p electrons is strongly broad- due to the anti-ferromagnetic superexchange interaction
ened. ThebandwidthW oftheimpuritybandinthegap and the compensation of majority and minority spins.
can be much larger than the effective correlation energy
U =E(N+1)+E(N−1)−2E(N),andtheStoner’scondi-
tionfortheexistenceofmagnetismmightbecontravened
Acknowledgments
[11, 12]. Correspondingly, the magnetism might be fully
suppressed at the C concentrations higher than 16% be-
cause of the strong broadening of the impurity bands of This research was partially supported by a Grant-in-
both majority and minority spins, resulting in the com- Aid for Scientific Research in Priority Areas “Quantum
pensationofspins;therefore,thematerialinquestionhas Simulators and Quantum Design” (No. 17064014) and
a nonmagnetic behavior at higher C concentrations. “Semiconductor Nanospintronics,” a Grand-in-Aid for
Second,contrarytoMgO1−xCx,theimpuritystatesin Scientific Research for young researchers, JST-CREST,
CaO1−xCx,SrO1−xCx,andBaO1−xCx arelocalizedand NEDO-nanotech, the 21st Century COE, and the JSPS
thebandwidthsoftheimpuritybandsformedbyminority core-to-core program “Computational Nano-materials
spins in the gap are sufficiently small to satisfy Stoner’s Design.” We aregratefulto Prof. H. Akai (Osaka Univ.)
condition; hence, the ferromagnetism is stabilized by a for providing us with the MACHIKANEYAMA2000
predominant ferromagnetic double exchange mechanism packageand to Prof. A. Yanase (OsakaUniv.) for many
and is more stabilized as x increases. Although the ex- valuable discussions.
5
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