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Low-temperature ferromagnetism in (Ga, Mn)N: Ab initio calculations
K. Sato*
The Institute of Scientific and Industrial Research, Osaka University, Osaka 567-0047, Japan and
Institut für Festkörperforschung, Forschungszentrum Jülich, D-52425 Jülich, Germany
W. Schweika and P. H. Dederichs
Institut für Festkörperforschung, Forschungszentrum Jülich, D-52425 Jülich, Germany
H. Katayama-Yoshida
The Institute of Scientific and Industrial Research, Osaka University, Osaka 567-0047, Japan
(Received 23 April 2004; revised manuscript received 24 June 2004; published 3 November 2004)
The magnetic properties of dilute magnetic semiconductors (DMSs)are calculated from first-principles by
mapping the ab initio results on a classical Heisenberg model. By using the Korringa–Kohn–Rostoker
coherent-potential approximation (KKR-CPA)method within the local-density approximation, the electronic
structure of (Ga, Mn)N and (Ga, Mn)As is calculated. Effective exchange coupling constants Jij’s are deter-
mined by embedding two Mn impurities at sites iand jin the CPA medium and using the Jij formula of
Liechtenstein et al. [J. Magn. Magn. Mater. 67,65(1987)]. It is found that the range of the exchange
interaction in (Ga, Mn)N, being dominated by the double exchange mechanism, is very short ranged due to the
exponential decay of the impurity wave function in the gap. On the other hand, in (Ga, Mn)As, where p-d
exchange mechanism dominates, the interaction range is weaker but long ranged, because the extended valence
hole states mediate the ferromagnetic interaction. Curie temperatures (TC’s)of DMSs are calculated by using
the mean-field approximation (MFA), the random-phase approximation, and the, in principle exact, Monte
Carlo method. It is found that the TCvalues of (Ga, Mn)N are very low since, due to the short-ranged
interaction, percolation of the ferromagnetic coupling is difficult to achieve for small concentrations. The MFA
strongly overestimates TC. Even in (Ga, Mn)As, where the exchange interaction is longer ranged, the perco-
lation effect is still important and the MFA overestimates TCby about 50%–100%.
DOI: 10.1103/PhysRevB.70.201202 PACS number(s): 75.50.Pp
Dilute magnetic semiconductors (DMSs), such as (In,
Mn)As and (Ga, Mn)As discovered by Munekata et al. and
Ohno et al., have been well investigated as hopeful materials
for spintronics.1Curie temperatures (TC’s)of these DMSs are
well established1–3 and some prototypes of spintronics de-
vices have been produced based on these DMSs. The mag-
netism in these DMSs are theoretically investigated and it is
known that the ferromagnetism in these systems, as well as
(Ga, Mn)Sb, can be well described by Zener’s p-dexchange
interaction, due to the fact that the majority of dstates lies
energetically in the lower part of the valence band.4Dietl
et al.5and MacDonald et al.6explained many physical prop-
erties of (Ga, Mn)As based on the p-dexchange model, and
first-principles calculations by Sato et al. showed that the
concentration dependence of TCin (Ga, Mn)As was well
understood by the p-dexchange interaction if a correction to
the local-density approximation (LDA)is simulated by the
LDA+Umethod with U=4 eV.4
While these p-dexchange systems, in which the dstates
of Mn impurities are practically localized, are well under-
stood, there exist an even larger class of systems where the d
levels lie in the gap exhibiting impurity bands for sufficiently
large concentrations. To these impurity band systems belong
(Ga, Mn)N, (Ga, Cr)N, (Ga, Cr)As, (Zn, Cr)Te, (Zn, Cr)Se,
and many others, as shown by first-principles calculations.7
Most of these systems are controversially discussed in the
literature, and an unambiguous determination of the ferro-
magnetism has only been reported for (Zn, Cr)Te with a
relatively high Cr concentration of 20% and a Curie tempera-
ture of 300 K.8In particular, in this class of materials (Ga,
Mn)N has been frequently mentioned as the most promising
high-TCDMS referring to the prediction of model calcula-
tions by Dietl et al.5and ab initio results by Sato et al.9
Many groups have tried to fabricate ferromagnetic (Ga,
Mn)N, but the experimental results are very controversial
and confusing. After the observation of the ferromagnetism
of (Ga, Mn)N,10 many experiments followed; however, re-
ported TC’s are scattered between 20 and 940 K.10–14 More-
over, recently Ploog et al. observed spin-glass behavior in
7% Mn-doped GaN and suggested that the ferromagnetism
observed in 14% Mn-doped GaN originated from Mn-rich
clusters.15 Thus, the ferromagnetism in (Ga, Mn)N is still an
open question which we reconsider in this paper. Ab initio
calculations by Akai16 and others4,9,17–20 show that the mag-
netic properties of the above impurity band systems are
dominated by the double exchange mechanism and that the
ferromagnetism is stabilized by the broadening of the impu-
rity band. In the mean-field approximation (MFA)high TC
values have been predicted [e.g., 350 K for (Ga, Mn)N with
5% of Mn, 500 K for (Ga, Cr)N with 5% of Cr, 400 K for
(Zn, Cr)Te with 5% of Cr, and so on]and the 冑cdependence
of TCon concentration chas been explained by band
broadening.4,17 Similar high, although slightly smaller, TC
values have also been obtained in the random-phase approxi-
mation (RPA).
In this paper, we will show that a general obstacle for
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ferromagnetism exists in these dilute systems, in particular in
(Ga, Mn)N. Due to the large band gap, the wave function of
the impurity state in the gap is well localized, leading to a
strong, but short-ranged exchange interaction, being domi-
nated by the nearest neighbors. Therefore, for low concen-
trations, the percolation of a ferromagnetic cluster through
the whole crystal cannot be achieved, so that a ferromagnetic
alignment of the impurity moments cannot occur. Thus, a
paramagnetic or disordered, spin-glass-like, state is ob-
served, in particular for low concentrations.
The electronic structure of DMS is calculated based on
the local-density approximation (LDA)by using the
Korringa–Kohn–Rostoker (KKR)method. In this paper we
focus on (Ga, Mn)N and (Ga, Mn)As as typical examples for
the double exchange and the p-dexchange systems, respec-
tively. In these systems, Mn impurities distribute randomly at
Ga sites in the host semiconductor being described as
共Ga1−c,Mnc兲X, where cis the Mn concentration and Xrefers
to N or As. To describe the substitutional disorder, we use the
coherent-potential approximation (CPA). In this framework,
all Mn impurities are equivalent and consequently, we sup-
pose a ferromagnetic alloy. It has already been shown that
the magnetic properties of metallic ferromagnetic alloys are-
well described within the CPA.21 While the CPA describes
the electronic structure in the mean-field approximation, we
go beyond this approximation and explicitly calculate the
exchange interaction Jij between two impurities at sites iand
j, which are embedded in the ferromagnetic CPA medium.
For the evaluation of Jij we use the frozen potential
approximation22 and apply a formula by Liechtenstein
et al.23 According to this formula, the total energy change
due to infinitesimal rotations of the two magnetic moments at
site iand jis calculated using the magnetic force theorem,
and the total-energy change is mapped on the (classical)
Heisenberg model H=−⌺i⫽jJije
ជ
ie
ជ
j, where e
ជ
iis a unit vector
parallel to the magnetic moment at site i, thus resulting in the
effective exchange coupling constant Jij. This approach is
already employed to estimate magnetic interactions in DMSs
by Turek et al.24 and Bouzerar et al.25 For the present KKR-
CPA calculations, we use the package MACHIKANEYAMA2000
coded by Akai.26 We assume muffin-tin potentials and
use the experimental lattice constants of the host
semiconductors.27 It has already been shown that the lattice
relaxations in (Ga, Mn)N and (Ga, Mn)As are very
small.20,28,29 Zinc-blende structures are assumed both for
GaN and GaAs. In reality, GaN has a Wurtzite structure.
However, results for both structures are practically identical,
because splitting of impurity bands due to symmetry lower-
ing is small20 and disorder-induced bandwidth always over-
comes the splitting. The angular momenta are cut off at
l=2 in each muffin-tin sphere. All calculations are performed
for the neutral charge state of Mn, so that doping effects are
not included.
Figure 1 shows the calculated exchange interactions Jij in
(Ga, Mn)N and (Ga, Mn)As. As shown in Fig. 1(a),in
(Ga, Mn)N the interaction strength is strong, but the interac-
tion range is short, so that the exchange coupling between
nearest neighbors dominates. For example, nearest-neighbor
interaction J01 in 1% Mn-doped GaN is about 13.5 mRy,
while the other interactions are almost two orders of magni-
tude smaller than J01 except for J04. Therefore, in this case
the very large mean-field value of TCis mostly determined
by J01. For higher concentrations, J01 is suppressed and the
interaction between next-nearest neighbors becomes nega-
tive, resulting in a complicated structure in the distance de-
pendence of the exchange interaction. Concerning the
mechanism of the ferromagnetism, it has already been
pointed out that the double exchange mechanism dominates
in (Ga, Mn)N, where pronounced impurity bands appear in
the gap.4,7,9,17 It is intuitively understood that the exchange
interaction in (Ga, Mn)N becomes short ranged due to the
exponential decay of the impurity wave function in the gap.
In contrast to (Ga, Mn)N, the exchange interaction has long
tails in (Ga, Mn)As, in particular for low concentrations, as
shown in Fig. 1(b). The qualitative difference in the interac-
tion range between (Ga, Mn)N and (Ga, Mn)As is apparent
from the figure. In (Ga, Mn)As, the p-dexchange interaction
becomes important, as shown in Ref. 4. Since the extended
hole state mediates the ferromagnetic interaction,5the inter-
action range is long ranged in p-dexchange systems, essen-
tially. Actually, the interaction extends farther than three lat-
tice constants (20th shell). For higher concentrations, due to
the screening of the pair interaction by the other impurities,
the interaction range becomes slightly shorter.
As is well known, the LDA predicts the position of local-
ized dlevels at too high energy. However, according to re-
cent calculations by Shick et al.,30 the LDA+Ucalculations
only slightly affect the impurity bands at the Fermi level in
(Ga, Mn)N due to the extended nature of the antibonding t2
FIG. 1. Calculated exchange interaction Jij in (a)(Ga, Mn)N
and (b)(Ga, Mn)As as a function of distance.
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201202-2
states of the impurity bands. Therefore, the LDA provides a
fairly good description of the magnetic properties of (Ga,
Mn)N. Even if the nearest-neighbor interactions are changed
in the LDA+Ucalculations, this will not affect much the
Curie temperatures for low concentrations, because only the
longer ranged interactions are relevant due to the percolation
effects. On the other hand, as we have already shown in Ref.
4, the LDA+Ucalculations with U=4 eV yield a different
description of the magnetism in (Ga, Mn)As. This effect
could change the calculated TCvalues slightly, however, the
exchange interaction in (Ga, Mn)As still remains long ranged
and the basic argument of the following discussion is not
affected.
It is well known that the Curie temperature in the
mean-field approximation TC
MFA is calculated as kBTC
MFA
=
共
2
3
兲
c⌺i⫽0J0i, where kBis Boltzmann constant. As shown in
this equation, evaluation of TC
MFA does not require any infor-
mation on the interaction range, because only the sum of the
coupling constants appears in the equation. This simplifica-
tion leads to significant errors in the calculated TCof a dilute
system with low concentrations. This fact is easily under-
stood by simple consideration and is known as the percola-
tion problem.31 Let us suppose a Heisenberg model with a
ferromagnetic exchange interaction only between nearest
neighbors (nearest-neighbor Heisenberg model), and con-
sider what happens when the system is diluted with nonmag-
netic sites. When the concentration of magnetic sites is
100%, we have a perfect ferromagnetic network. Due to the
dilution, the network is weakened, and for a concentration
below a percolation threshold the ferromagnetism cannot
spread all over the system, leading to a paramagnetic state,
since due to missing longer ranged interactions the moments
can no longer align. Obviously, this effect is not counted in
the mean-field equation for TC, because the dilution effect is
included only as a concentration factor cin the equation. In
the case of the nearest-neighbor Heisenberg model, the per-
colation threshold cpfor the fcc structure is 20%.31 In real
cases the exchange interaction could reach beyond the near-
est neighbors and the percolation threshold might be lower.
However, in this paper we are interested in the concentration
range well below the nearest-neighbor threshold cp. There-
fore, the exact TCvalues could be much lower than the
mean-field values, in particular for the double exchange sys-
tems, like (Ga, Mn)N, where the exchange interaction is very
short ranged [Fig. 1(a)].
In order to take the percolation effect into account, we
perform Monte Carlo simulations (MCS)for the effective
classical Heisenberg model. The thermal average of magne-
tization Mand its powers are calculated by means of the
Metropolis algorithm.32 Due to the finite size of super cells
used in the simulation, it is difficult to determine TCfrom the
temperature dependence of 具M共T兲典. In particular, when con-
sidering dilute systems, finite-size effects and appropriate
finite-size scaling are of particular importance for a correct
and efficient evaluation of TCby Monte Carlo simulations.
Therefore, we use the cumulant crossing method as proposed
by Binder.32 The fourth order cumulant U4(a linear combi-
nation of 具M4典/具M2典2兲has been shown to have a size-
independent, universal fix-point at TC. We calculated U4
for diffferent cell sizes (6⫻6⫻6, 10⫻10⫻10, and 14
⫻14⫻14 conventional fcc cells)as a function of tempera-
ture. For each temperature, we perform 240 000 Monte Carlo
steps per site, while configuration averages were taken every
20th step per site.
First, as a pedagogical example we show the calculated
TCfor the dilute fcc nearest-neighbor Heisenberg model as
calculated by MFA, RPA, and MCS in Fig. 2. For MCSs for
dilute systems, we take 20 different random configurations of
FIG. 2. Curie temperatures of nearest-neighbor Heisenberg
model in fcc structure. TC’s are calculated by the mean-field ap-
proximation (solid line), the random-phase approximation (dotted
line), and the Monte Carlo simulation (crosses). The percolation
threshold is 0.20 for the fcc structure.
FIG. 3. Curie temperatures of (a)(Ga, Mn)N and (b)(Ga,
Mn)As calculated by the MFA (solid lines), the RPA (dotted lines),
and the MCS (filled squares). For the MCS, the exchange interac-
tions up to 15th shell are taken into account.
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magnetic sites for the ensemble average. As shown in Fig. 2,
it is found that both MFA and RPA give reasonable estima-
tions of TCfor c=1, with the RPAbeing closer to exact MCS
results. It has been analytically shown that for this model
MFA gives an upper limit of TCand RPA gives a lower
limit.33 However, for c艋0.7, MCS results are below RPA
values and in particular, below the percolation threshold
共cp=0.20兲the Curie temperature vanishes: TC=0. Thus the
serious deficiency of both MFA and RPA in the dilute con-
centration range is evident.
Next, we show the calculated TCvalues of (Ga, Mn)N
[Fig. 3(a)] and (Ga, Mn)As [Fig. 3(b)] as obtained by the
MCS from the Jij values in Fig. 1. Thirty configurations of
Mn atoms are considered for averaging and Jij interactions
up to 15 shells are included; on the other hand, for the MFA
and the RPA estimations, interactions are included up to 72
shells. As shown in Fig. 3(a), very small TCvalues are pre-
dicted for low concentrations in (Ga, Mn)N. MFA and RPA
values are almost two orders of magnitude too large. Thus
we find that the magnetism is strongly suppressed due to the
missing percolation of the strong nearest-neighbor interac-
tions. Only the weak, longer ranged interactions satisfy the
percolation requirement, leading to small but finite Curie
temperatures for 5%, 10%, and 15% of Mn. As shown in Fig.
3(b), due to the longer ranged interaction in (Ga, Mn)As, the
reductions from the MFA are not very large, but still signifi-
cant. Naturally, these changes are larger for smaller concen-
trations. The TCvalues of 103 K obtained for 5% Mn is in
good agreement with the experimental values of 118 K re-
ported by Edmonds et al.2This value refers to measurements
in thin films, which are free of Mn interstitials representing
double donors. Including interactions beyond the 15th shell,
MCS could give slightly higher TCvalues for low concentra-
tions, where the interactions do not converge within the 15th
neighbors. At very high concentrations we expect that the
MFA and RPA values will be in better agreement with the
MCS results.
In this communication, we have shown by ab initio cal-
culations that (Ga, Mn)N shows no high-temperature ferro-
magnetism for low Mn concentrations. The strong ferromag-
netic interaction of Mn nearest-neighbor pairs does not
become effective below the nearest-neighbor percolation
limit. The weak longer ranged interaction leads to a ferro-
magnetic phase with very low TCof several tens Kelvin.
Therefore, the experimentally observed very high TCvalues
do not refer to a homogeneous ferromagnetic phase, but have
to be attributed to small ferromagnetic MnN clusters and
segregated MnN phases. Our results are of relevance for all
DMS systems with impurity bands in the gap. To obtain
higher Curie temperatures one needs longer ranged interac-
tions and/or higher concentrations. The latter requirement
naturally points to II-VI semiconductors, having a large solu-
bility for transition-metal atoms. The observation of a TC
value of 300 K for (Zn, Cr)Te with 20% Cr8is in line with
these arguments. Similar results as presented above have
been recently reported by a Swedish–Czech collaboration.34
This research was partially supported by JST-ACT,
NEDO-Nanotech, a Grant-in-Aid for Scientific Research on
Priority Areas A and B, SANKEN-COE, and 21st Century
COE from the Ministry of Education, Culture, Sports, Sci-
ence and Technology. This work was also partially supported
by the RT Network Computational Magnetoelectronics (Con-
tract No. RTN1-1999-00145)of the European Commission.
*Electronic address: ksato@cmp.sanken.osaka-u.ac.jp
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