Magnetism in (III, Mn)-V Diluted Magnetic Semiconductors: Effective Heisenberg Model

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by  S. Hilbert

Magnetism in (III, Mn)-V Diluted Magnetic Semiconductors: Effective Heisenberg
Model
S. Hilbert1,∗ and W. Nolting2
1Max-Planck-Institut fu¨r Astrophysik, Karl-Schwarzschild-Straße 1, D-85741, Garching, Germany
2Institut fu¨r Physik, Humboldt-Universit¨at zu Berlin, Newtonstraße 15, D-12489 Berlin, Germany
(Dated: February 2, 2008)
5
0 The magnetic properties of the diluted magnetic semiconductors (DMS) (Ga, Mn)As and (Ga,
0
Mn)N are investigated by means of an effective Heisenberg model, whose exchange parameters
2
are obtained from first-principle calculations. The finite-temperature properties of the model are

n studiednumericallyusingamethodbasedupontheTyablikovapproximation. Themethodproperly
a incorporatestheeffectsofpositionaldisorderpresentinDMS.TheresultingCurietemperaturesfor
J (Ga, Mn)As are in excellent agreement with experimental data. Due to percolation effects and

7 noncollinear magnetic structures at higher Mn concentrations, our calculations predict for (Ga,
Mn)N verylow Curie temperatures compared to mean-field estimates.

]
i
c PACSnumbers: 75.50.Pp,75.10.Nr,85.75.-d
s
-
l
r I. INTRODUCTION expensive and usually assume classical spins. However,
t
m a proper treatment of the positional disorder of the lo-
calized moments and their quantum nature is needed to
. Ferromagnetic (III, Mn)-V diluted magnetic semicon-
t make reliable predictions about the magnetic properties
ma damucotnogrssc(iDenMtiSst) dhuarvinegatthtreapctaesdt yceoanrssi1d,2e.raTblheeiartitnevnetsiotin- of DMS17,18.
In a previously published article9, the exchange pa-
- gationhasbeendrivenbythe ideaofusingtheir coupled
d rameters of an effective (classical) Heisenberg Hamil-
electronic and magnetic degrees of freedom to construct
n tonian have been calculated from first-principles for
electronic devices ranging from fast nonvolatile memo-
o Ga Mn As and Ga Mn N. There, however, these
c ries to quantum computers3. To date, however, techni- 1−x x 1−x x
had only been used to calculate Curie temperatures
[ cal applicability has been limited by the fact that most
within MFA. More recently, results of classical MC
known DMS have Curie temperatures T below room
1 C simulations on the basis of these exchange parameters
v temperature2,4,5,6,7. have been presented15. Here, we employ a different
3 ForthedevelopmentofferromagneticDMSwithhigher approach19,20 toinvestigatethepropertiesoftheeffective
4
Curie temperatures, it is important to understand theo- Heisenberg Hamiltonian. This approach generalizes the
1
reticallythemagnetisminthesematerialsandtodevelop Tyablikov approximation21 to systems with positional
1
0 theories which provide reliable qualitative and quantita- disorder, which is treated numerically exactly. Further-
5 tive predictions. The magnetism in these materials is more, the method assumes quantum spins. The quan-
0 due to magnetic moments localized at magnetic impuri- tum fluctuations ofthe spins aretreatedwithinrandom-
/ ties, which interact with each other indirectly via holes phase approximation, which goes beyond MFA and the
t
a in the valence and impurity band of the host semicon- classical-spin approximation. It should be mentioned
m
ductor. Therefore,forthedescription,oneoftenemploys that a similar approachhas been proposed in22.
- an effective Heisenberg model, whose exchange parame-
d
tersaredeterminedbythe interactionbetweenthelocal-
n
o ized moments and the holes8,9,10,11,12,13,14,15. However, II. MODEL
c themagneticimpuritiesaremainlyrandomlydistributed
: over the sites of the crystal lattice. This positional dis-
v Details of the electronic-structure calculation for
order breaks the translational symmetry of the crystal
i Ga Mn AsandGa Mn Nandtheextractionofthe
X and thus greatly complicates the theoretical description 1−x x 1−x x
exchangeparametersJ(R)asfunctionoftheMn-Mndis-
ar opfrotxhime amtiaotner(iMal.FAS)8tu,9doiersthbeasreadndoonmt-hpehamseeaanp-pfireoldximapa-- tance R can be found in Ref.9. Here, these exchange
parameters are used as input for a ‘diluted’ Heisenberg
tion (RPA) combined with the virtual-crystal approxi-
model
mation(VCA)10 neglecteffectsofthepositionaldisorder
inDMS.Approachesbasedonpercolationtheory11,12 ac- N
count for the randomness of the impurity positions, but H =− J e ·e , (1)
ij i j
require a simple functional dependence of the exchange
i,j=1
X
parametersontheinter-spindistanceandtreatthemag-
netismitselfonlyonamean-fieldlevel. Monte-Carlosim- inwhichonlyafractionofthelatticesitesisoccupiedby
ulations (MC)13,14,15,16 seem to provide a better way to aspin. Hence,iandjlabeltheoccupiedlatticesitesonly,
includethepositionaldisorder,butthesearenumerically whose total number is N, and e =(Sx,Sy,Sz)/(~S)
i i i i
2
is the normalized spin operator of the localized mag-
netic moment at lattice site i with lattice vector R and 3 (a) x = 0.02
i
Jij =J(|Ri−Rj|). The magnitude S of the spins is ab- 2.5 0.05
sorbed by the exchange parameters due to the particu- )
y
lar way in which these are calculatedfrom the electronic R 2 0.10
m
structure. (1.5

The finite-temperature properties of Hamiltonian )
R 1
(1) are studied using a generalization of the Tyab- (
J
likov approximation to systems without translational 0.5
symmetry19,20. The generalization treats the positional
0
disorder in the spin system numerically exactly except
that a uniform magnetization is assumed. Further-
1 1.5 2 2.5 3 3.5 4
more, the effects of low-energy quantum excitations, i.e. R / a
magnons, are included. Within this approximation, the
10
local magnon spectral density is given by19,20:
(b) x = 0.02
8
1 N 2~hSzi 0.05
S (E)=2~2hSzi δ E− E , (2) )
ii N r=1 (cid:18) ~2S2 r(cid:19) Ry 6 0.10
X m
(
where the E are the eigenvalues of the Hamilton 4
r )
matrix H, which is defined by its matrix elements R
(
H =δ N J −J . These eigenvalues also deter- J 2
ij ij n=1 in ij
mine the Curie temperature:
P 0
−1
2S(S+1) 1 1 1 1.5 2 2.5 3 3.5 4
k T = . (3)
B C 3 S2 N E R / a
r r!
X
To evaluate this expression for a given set of E ’s, the FIG.1: ExchangeinteractionsJ(R)betweenMnionsofdis-
r
value of S has to be fixed. For Mn ions in Ga Mn As tanceRin(a)Ga1−xMnxAsand(b)Ga1−xMnxNforvarious
1−x x concentrations x (from9,23)
andGa Mn N,S =5/2shouldbeappropriate2. How-
1−x x
ever, this choice is not consistent with the calculation of
the exchange parameters from the electronic structure,
whereclassicalspinsareassumed. Therefore,wewilluse Figure 2 shows the resulting magnon spectral densi-
Eq. (3) in the limit S → ∞, which yields TC values a ties. For Ga1−xMnxAs, the spectrum is smooth and
factor 5/7 less than for S =5/2. continuous. For Ga1−xMnxN, one can recognize rem-
Due to the positional disorder of the spins present in nantsofpeakstypicalfornearest-neighborinteractionat
DMS, the eigenvalues cannot be computed by Fourier low concentrations, which are broadened by small long-
transformation of H. However, the eigenvalues may be rangedinteractions. ComparedtoGa1−xMnxAs,thereis
obtained by the numerical diagonalization of the Hamil- a large spectral density at low energies for Ga1−xMnxN.
ton matrix for a finite system. In our calculations, we For concentrations x ≥ 0.08, antiferromagnetic interac-
used systems of ∼ 10000 spins, which were randomly tions come into play and negative magnon energies ap-
distributed over the lattice sites of a cubic section of an pear indicating a ground state which is not a saturated
fcc lattice with periodic boundary conditions. For each ferromagnet20.
concentrationxofMnions,weaveragedthespectralden- The Curie temperatures calculated using Eq. (3) are
sities over eight random configurations. shown in Fig. 3. For Ga Mn As, the calculated val-
1−x x
ues agree remarkably well with the experimental values
of optimally annealed samples5,6,24. Furthermore, the
III. RESULTS calculatedcurvesuggeststhatslightly higherT ’s might
C
be achieved by further increasing the Mn content x, but
In Fig. 1, the Mn-Mn exchange interactions J(R) in values above 300K seem rather unlikely.
Ga Mn AsandinGa Mn Nareshownasfunctions Since experimental values for T in Ga Mn N are
1−x x 1−x x C 1−x x
of the Mn-Mn distance R for severalconcentrationx. In quite controversial (reported values range from 0K to
Ga Mn As,thefalloffoftheinteractionwithRiscom- 940K4,26,27,28,29,30), we refrain from a comparison here.
1−x x
parably slow. In Ga Mn N, the interaction between However,the Curietemperatureswecalculatedarequite
1−x x
nearest neighbors is much larger than in Ga Mn As, low compared to earlier mean-field estimates (e.g. in
1−x x
butMnmomentsfurtherapartareonlyveryweaklycou- Ref.2). These low T values despite the high values of
C
pled. the nearest-neighbor exchange may be explained as fol-
3
) 200
1
- 0.5 (a) x = 0.01 (a)
y
R 0.02
(m 0.4 0.05 150 calc.
0.10 exp.
2 )
S 0.3 K
( 100
/ C
0.2 T
)
E
( 0.1 50
i
i
S 0
0 2 4 6 8 10 12
0
2 z 0.02 0.04 0.06 0.08 0.1
(S E)/(2<S >) (mRy) x
1 ) 0.7 (b) x = 0.01 50 (b)
- 0.05
y 0.6 0.10
R
m 0.5 0.15 40
(
)
2 0.4 K
S ( 30
/ 0.3 C
) T
(E 0.2 20
i
i0.1
S 10
0
0 2 4 6 8 10 12
0
2 z 0.020.040.060.08 0.1 0.120.14
(S E)/(2<S >) (mRy) x
FIG. 2: Local magnon spectral density Sii(E) for (a) FIG. 3: Calculated Curie temperature TC of (a)
Ga1−xMnxAsand(b)Ga1−xMnxNforvariousconcentrations Ga1−xMnxAs (compared with experimental values of an-
x of Mn nealed samples5,6,24,25) and (b) Ga1−xMnxN for various con-
centrations x of Mn ions
lows: Forconcentrationswellbelowthenearest-neighbor
percolation threshold c ≈ 0.231, even a large nearest- should not play an important role for the ferromagnetic
P
neighbor exchange does not contribute substantially to stability,whichcanbeeasilyseenbyconsideringthecase
the stability of the magnetic phase. Since the exchange of nearest-neighbor interaction only20.
parameters for larger inter-spin distances are very small
in Ga Mn N, ferromagnetic order can only be estab-
1−x x
lished at very low temperatures. Note that the drop of IV. SUMMARY
T for x ≥ 0.08 may be due to the used approximation:
C
As indicated by the magnon spectra seen in Fig. 2, the In this paper, we presented a method for calculat-
system’s groundstate is different froma saturatedferro- ing the magnetic properties of ferromagnetic DMS. The
magnet,butasuchuniformmagneticstateisassumedin method applies a Tyablikov-like approximation for sys-
the approximation. tems with positional disorder to an effective Heisenberg
Figure 4 presents a comparison of the the Curie tem- Hamiltonian,whoseexchangeparameterswhereobtained
peratures calculated using different approximations for by first-principle calculations. Unlike in MFA or VCA-
the effective Heisenberg model. The T values ob- RPA, no approximations with respect to the positional
C
tained by MC simulations are slightly higher than the disorderare made apartfromthe simplificationof a uni-
ones calculated by the presented approach, whereas form magnetization. As the main advantage over clas-
both MFA and VCA-RPA yield much higher T ’s. For sical MC simulations, the presented treatment of the ef-
C
Ga Mn As, the difference is about a factor two to fectiveHeisenbergmodeladmitsquantumspinsandthus
1−x x
eight. For Ga Mn N, the difference is even much may open up a way towardsa fully quantum-mechanical
1−x x
larger. This is due to the fact that the MFA and VCA- treatment of magnetism in DMS. Furthermore, the nu-
RPA do not take into account percolation effects: Large merical effort is fairly low compared to MC simulations.
nearest-neighbor interactions yield large Curie temper- OurcalculationsofT forGa Mn Asshowexcellent
C 1−x x
atures even for concentrations well below the nearest- agreement with experimental data. For Ga Mn N,
1−x x
neighbor percolation threshold. However, for such con- we obtained very low Curie Temperatures despite high
centrations, the nearest-neighbor interaction strength effective nearest-neighbor exchange parameters, which
4
we found are much lower than MFA and VCA-RPA val-
ues. These results support recent findings obtained by
350 usingMCsimulationsincombinationwithfirst-principle
methods15,16.
300
The presented model should be improved by using a
self-consistent method describing the electronic degrees
250 offreedomatfinitetemperature(suchas,e.g.,in32,33). In
)
K ordertoobtainafullyquantummechanicaltheory,quan-
( 200
C tum spins should be used instead of classical spins in in
T
thecalculationoftheeffectiveexchangeparametersfrom
150
the electronic structure. This will also remove the ambi-
100 guity in the choice of S. Furthermore, the treatment of
the effectiveHeisenbergmodelmaybe extendedtoallow
50 for a site-dependent hSzi. In addition, the model might
i
be improved in order to handle systems with a ground
0 state deviating from a saturated ferromagnet. Further-
0.020.040.060.08 0.1 0.120.14
x more,clusteringandotherformsofshort-rangechemical
orderingmay alsobe included into the model inorderto
investigatetheireffectsonthemagneticstability. Finally,
FIG. 4: Comparison of the Curie temperatures TC of the method should be applied to other DMS.
Ga1−xMnxAs (diamonds) and Ga1−xMnxN (squares) ob-
tained by the presented approach (solid line, filled symbols),
VCA-RPA (dashed line, filled symbols), MFA (dotted line,
Acknowledgments
filled symbols) and MC (dash-dotted line, open symbols,
taken from15).
This work benefited from the support of the SFB290
of the Deutsche Forschungsgemeinschaft. Thanks to J.
shows the importance of percolation effects. Moreover, Kudrnovsky´ for supplying the exchange parameters and
for both Ga Mn As and Ga Mn N, the T values for helpful discussion.
1−x x 1−x x C
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