Table Of ContentPolaron percolation in diluted magnetic semiconductors
A. Kaminski and S. Das Sarma
Department of Physics, University of Maryland, College Park, Maryland 20742-4111
We theoretically study the development of spontaneous magnetization in diluted magnetic semi-
conductors as arising from a percolation of bound magnetic polarons. Within the framework of
2
a generalized percolation theory we derive analytic expressions for the Curie temperature and the
0
magnetization,obtainingexcellentquantitativeagreementwithMonteCarlosimulation resultsand
0
good qualitative agreement with experimental results.
2
n PACSnumbers: 75.50.Pp,75.10.-b,75.30.Hx
a
J
4 Diluted magnetic semiconductors (e. g. In1−xMnxAs, conductor have employed the simple Weiss mean field
1 Ga1−xMnxAs), which are materials with some fraction theory incorporating the band structure details (e.g.
of the nonmagnetic lattice atoms replaced by magnetic spin-orbit coupling, etc.), without taking into account
]
i atoms (i. e. a fraction x of Ga or In atoms being ran- effects of disorder. While this approach is claimed to
c
s domly replaced by Mn atoms), have attracted a great providequantitativelyaccuratevaluesofTc,itfailsqual-
- deal of attention [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] fol- itativelyinaccountingforexperimentallyobservedtrans-
l
r lowing the recent discovery [11] of ferromagnetism in port properties [13] of GaMnAs where the resistivity is
t
m Ga1−xMnxAs, with a Curie temperature Tc 100K in very high and seems to obey a Mott variable range hop-
≈
. the x=0.03 0.07range. The subjectis of considerable ping behavior at low temperatures reminiscent of an in-
t −
a fundamental and technological interest. Technologically, sulating system. In addition, the resistivity always de-
m a semiconductor, which is also a ferromagnet, raises the creases with increasing temperature above T , which is
c
- exciting potential of spintronic applications [12] where again typical of a localized insulating system. The free-
d
logicandmemory operationscouldinprinciple be seam- carrier mean-field approach also fails to explain the re-
n
lessly integrated on a single device. From a fundamen- centlyreported[14]dependenceofT onannealingwhich
o c
c talperspective,understandingferromagnetisminanovel indicates a crucial role for disorder. A recent dynamical
[ material(which is alsoasemiconductor)is animportant meanfieldtheorycalculation[10]concludedthatthe fer-
challenge, particularly so in this case in light of the ex- romagnetic GaMnAs may actually belong to a situation
1
v tensive existing earlier work involving doped II-VI semi- where the carriers are just on the edge of being strongly
9 conductor materials (as against the systems of current localized. The approach [5, 6, 7] treating the carriers
2 interest, which are magnetically doped III-V systems) as localized carriers in a semiconductor impurity band
2
whichwereneverconvincinglyfoundtobeferromagnetic (thus being opposite to the the free-carrier mean-field
1
except perhaps with extremely low values of T . It is approach), which has so far been explored only numer-
0 c
2 therefore not surprising that there has been a great deal ically, also leads to reasonable agreement with experi-
0 of theoretical activity [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] trying ments and indicates a very strong dependence of T on
c
/ tounderstandtheferromagneticmechanisminGaMnAs. disorder. In this paper we provide an analytical the-
t
a Although no theoretical consensus on the precise ferro- ory which takes into account both disorder and strong
m
magnetic mechanism has yet been reached in the liter- magnetic interaction starting with localized carriers and
- ature it is now established that the interaction between using the physically appealing magnetic polaron perco-
d
n the magnetic atoms, which leads to the ferromagnetic lation picture. Our theory has a starting point similar
o phase at low enough temperatures, is induced by charge to that in Refs. [5, 6, 7] except that ours is a completely
c carriers (holes in the case of GaMnAs) in the semicon- analytical physical theory in contrast to the numerical
:
v ductor host. Theoretical investigation of these systems approach used in the works [5, 6, 7]. Where applica-
i is hampered by the fact that both disorder and inter- ble our analytical results agree well with the numerical
X
actions are strong and must be taken into account non- results of Refs. [5, 6, 7].
r
a perturbatively. This problem has been approached in a
numberofways,includingtheoreticalapproximationsas- In this paper we consider a system in which transi-
suming charge carriers being almost free [1, 2, 3, 4] and tion to the insulating state due to localization of the
numerical studies in the opposite limit of strongly local- charge carriers occurs at temperatures higher than the
ized charge carriers [5, 6, 7]. However, a comprehensive CurietemperatureTc. Thecarrierswillbecalled“holes”
understanding of the physics of diluted magnetic semi- throughout the text of this paper since in GaMnAs the
conductors has not been achieved yet. carries are holes although the theory to be developed
in this paper applies equally to the situation where the
Theoretical approaches [1, 2, 3, 4] treating the charge carriers are electrons. Exchange interaction of localized
carriers as free carriers in the valence band of the semi- holes with magnetic impurities leads to the formation of
2
bound magnetic polarons [15, 16]. Since the concentra-
tion of magnetic impurities is much larger than the hole
concentration [17], most likely due to compensation by
Asantisitedefects,aboundmagneticpolaronconsistsof
onelocalizedhole,andalargenumberofmagneticimpu-
rities around the hole localization center. Even though
the direct exchange interaction of the localized holes is
antiferromagnetic, the interaction between bound mag-
neticpolaronsmaybeferromagnetic[16]atlargeenough
concentrations of magnetic impurities. To understand
the physics of this phenomenon, one may consider two
neighboring polarons (Fig. 1). The localized holes of
these polarons both act on the impurities surrounding
themthusproducinganeffective magneticfieldforthese FIG. 1: Interaction of two bound magnetic polarons (after
impurities. The energy minimum is reached by this sys- Ref.[16])Thepolaronsareshownwithgraycircles, smalland
large arrows show impurity and hole spins respectively.
tem when the impurity spins are parallel to this effec-
tive field, and the magnitude of the field is maximum.
The maximum ofthis effective magnetic fieldis achieved
when the spins of the localizedholes areparallel. There- respectively,Sˆk/ˆsj are the impurity/hole spinoperators.
fore at low temperatures the system should eventually Matrix elements Jkj of the impurity-hole exchange in-
reach the state where the spins of all holes point in the teraction decay exponentially with the distance between
same direction, and all impurity spins point in the same the interacting impurity and hole, Jkj =J0exp( 2rk
orintheoppositedirection,dependingonthesignofthe rj /aB). The direct (antiferromagnetic) excha−ng|e in−-
|
impurity-hole exchange interaction. teraction between the magnetic impurities is neglected,
If the hole localization radius is much less than the since their relative concentration x in the lattice of the
characteristic distance between the localized holes, the hostsemiconductorismuchlessthanunityalthoughthis
disorder in the hole positions must have dramatic ef- may be important for larger values of x, leading to the
fectonthewholepictureofthe ferromagnetictransition. suppression of Tc. We will not discuss possible mech-
This notionhas been confirmedby BerciuandBhatt [6], anisms of hole localization, since the properties of our
whohaveshown,bymeansofnumericalsimulations,that model hold for any of them as long as the decay of the
both the Curie temperature and the shape of the mag- localized hole’s wave function is exponential.
netization curve M(T) are indeed strongly affected by At some temperature T, magnetic impurities that are
disorder. It has been known that the percolation theory at distances r <R (T) (a /2)ln(sS J /T) from hole
p B 0
≡ | |
[18]providesmanyadequatetoolstodealwithferromag- localization centers have their spins strongly correlated
netism in disordered systems with strong localization of with the spins of the corresponding holes (here s and S
carriers [19]. In this paper we present a quantitative de- are the absolute values of the hole or impurity spin re-
scription of the spontaneous magnetization in magnetic spectively). Thespinswhichdonothavealocalizedhole
semiconductors within the framework of the percolation within a circle of radius R (T) around them are essen-
p
theory. tiallyfree. ThequantityR (T)istheeffectiveradiusofa
p
In our model, the charge carriers are localized. The magneticpolaron;itgrowsasthetemperatureislowered.
hole wave function is assumed to fall off exponentially Clearly at low enough temperatures neighboring mag-
awayfromlocalizationcenters,withdecaylengtha . We netic polarons overlap and interact with each other via
B
considerthelowcarrierdensityregimeinwhichthemean interaction with impurities between them [16]. This in-
distance between the localized holes is much larger than teractionproducesalignmentofthepolaronspins. When
the hole localizationradius,a3n 1. We note thatfor the cluster of correlated polarons having the size of the
B h ≪
GaMnAs,a 10˚A,andthereforeourtheoryappliesin sample (the infinite cluster) appears, the ferromagnetic
B
theregimen ≈ 1021cm−3,withexperimentaln values transition occurs.
h h
currentlybeing≪around1019 cm−3. The localizationcen- Even though the picture presented above is qualita-
ters are distributed randomly in the sample. Magnetic tivelycorrect,oneneeds tofindthemaximumcharacter-
impurities are distributed within the sample with con- istic temperature T (r) at which the spins of two po-
2p
centration ni nh randomly as well. The Hamiltonian larons at a given distance r from each other are still
≫
of the system has the form strongly correlated. The Hamiltonian of a two-polaron
subsystem is given by Eq. (1), where hole index j takes
Hˆ = J Sˆ ˆs , (1)
kj k j only two values j and j corresponding to the two po-
1 2
kj
X larons under consideration. Our present goal is to find
whereindiceskandj labelmagneticimpuritiesandholes the characteristic spin-correlationtemperature T (r) at
2p
3
given r, a , J , and n . Since we are interested in the Hamiltonian to the following form:
B 0 i
system behavior only at and below the percolationtran-
N
sition, we may limit our consideration to polaron pairs θ
with r & n−1/3 a . A rough preliminary estimate, Hˆ =sJ(r/2)cos2 Sk(z) . (3)
h ≫ B k=1
which canbe obtainedfromstraightforwarddimensional X
analysis, reads Here the angle between the spins of the two polarons
is denoted θ, and the direction of the z axis is chosen
along the direction of the vector s +s . We have ne-
j1 j2
T2p(r) A(r,aB,ni)sS J0 exp( r/aB) , (2) glected spatial variation of the polaron exchange field in
∼ | | −
the interaction region, since the relative magnitude of
this variation does not exceed unity.
where dependence of the dimensionless prefactor func-
The partition function of the system described by
tion A(r,a ,n ) on its arguments is weaker than expo-
B i Hamiltonian (3) can easily be found, and after straight-
nential. Since at T sS J exp( r/a ) the two po-
∼ | 0| − B forward algebra we arrive at the expression for the av-
larons must be correlated, A 1. At temperatures of
≥ erage cosine of the angle θ between the spins of the two
the order of the right-hand side of Eq. (2), each polaron
interacting polarons:
already has a large number of impurity spins near its
center strongly polarized along the direction of the spin
N[sSJ(r/2)/T]2, √NsS J(r/2) T
of its hole. Due to this coupling, the spin of the po- cosθ | |≪
laron’s hole becomes “massive,” and may be considered h i∼( 1− N[sSJ(r1/2)/T]2 , √NsS|J(r/2)|≫T .
a classical field as far as its interaction with more re- (4)
mote impurities is concerned. This effective field pro-
Therefore,thespinsofthetwopolaronsatdistancer are
duces polarization of the remote impurities; the charac-
correlated at temperatures below √NsS J(r/2). Using
teristic value of the projection S(j1) of the spin of kth | |
k the estimate N a2rn , we finally arrive at
impurity onto the spin direction of the j -th polaron is ≈ B i
1
irnotuegrhalcytioSnmoifn{th1,issJimkjp1/uTri}ty. Twhitehcthhaeraocttheerrisptioclaernoenrgiys ooff T2p(r)∼aB√rnisS|J0|exp(−r/aB) . (5)
the order of sSJ , therefore the contribution of the The maximum possible distance r (T) between two
kj2 corr
kth impurity to the interaction of two polarons is of the polarons with correlated spins at a given temperature T
order of S2sJ min 1,sJ /T [here we assume that is therefore
kj2 { kj1 }
sSJ ,T > sSJ since T (r) sS J exp( r/a )].
Thiskje1stimate allkojw2 s us to2dpeterm≥ine|w0h|ich im−puriBties r (T) a lnsS|J0| + 1ln a3n lnsS|J0| .
corr ∼ B T 2 B i T
are important for the interaction of two given polarons. (cid:20) (cid:18) (cid:19)(cid:21)
(6)
Namely, these impurities are the ones in the lens-shaped
region between the two localization centers, see Fig. 1.
Now we can use result (5) to get a better estimate for
Thediameterofthisregionisoftheorderof√raB;more the thickness of the lens-shaped interaction regionin or-
remote impurities are interacting too weakly with both
der to perform the next iteration. It is determined by
polarons to be of any importance. Finding the thickness
the condition J <T, so the thickness is of the order
of the lens-shaped region is less trivial. As we consider kj1,2
impurities away from the middle point and closer, say, of aBln(aB√rni). The resulting correction to T2p(r) is
just a factor of ln(a √rn ). Taking this correction into
to j , decrease of J is fully compensated by the in- B i
1 kj2 account is clearly beyond the accuracy limits of the ap-
crease of J , until we enter the region where the spins
kj1 proximationusedabove,soweneglectit,anduseEq.(5)
of the impurities are saturated due to their proximity to
as our final result.
j . Thus the size of the region of interest in the direc-
1 Nowweconsiderasystemofrandomlyplacedmagnetic
tion along r r depends on the temperature. Since
j1 − j2 polarons. As the temperature is being lowered,the spins
the characteristic correlation temperature T (r) is the
2p of neighboring polarons become aligned, and clusters of
quantity to find, we will proceed by iterations, starting
polarons with the same spin appear. At any given tem-
with the value given by the right-hand side of Eq. (2)
peratureT,the polaronsseparatedbya distancesmaller
withA(r,a ,n )=1(sothethicknessofthelens-shaped
B i than r (T) are joined into magnetic clusters. The
interaction region equals a ). As we will see, just one corr
B lowerthetemperature,themoresuch“links”betweenpo-
iteration yields the answer with good precision.
larons are established, and the larger the averagecluster
We will take into account only the impurities in the size. Finally, at low enough temperatures, a cluster hav-
interaction region; the total number of these impurities ing the dimensions of the sample, the so called “infinite
equals N a2rn in the first iteration. The coupling of cluster”appears,andthemagnetizationofthesampleac-
∼ B i
each of these impurities to either hole is approximately quiressomefinitevalue. Theproblemoffindingthetran-
J(r/2) J exp( r/a ). It allows us to reduce the sition temperature is identical to the problem of finding
0 B
≡ −
4
thecriticalpercolationradiusintheproblemofrandomly 1.0
placed overlapping spheres [18]. The latter problem has
0.8 2
been solved numerically, and it has been demonstrated )0
thattheinfiniteclusterformsinthesystemwhenthelink M( 0.6 1
lteuntgintghtrheiaschexesprtehsesiovanluinetorpEerqc.≈(5)0,.8w6e/g√3enthth[e20e]x.pSreusbssiotin- /) 0.4 2
T
for the ferromagnetic transition temperature (
M 0.2
1
T a √n n−1/6sS J exp 0.86 . (7) 0.0
c ∼ B i h | 0| −a √3n 0.0 0.2 0.4 0.6 0.8 1.0 1.2
(cid:18) B h(cid:19) T/T
c
The limit ofapplicability ofEq.(7) is determinedby the
condition a3n 1. A similar exponent was obtained
B h ≪ FIG. 2: The solid lines show the relative magnetization of
in Ref. [21] for ferromagnetic transition in PdFe alloys.
themagneticimpurities[Eq.(9)]for a3Bnh =5×10−3 (curve
Itis instructive to pointoutthat ourEq.(7) is consis- 1)and2×10−2 (curve2). Thedashedlinesshowtherelative
tent with the mean-field result derived in the literature
magnetization of localized holes, whose contribution to the
[1, 2, 3, 4] for the opposite limit of almost free holes, total sample magnetization is small.
T n J2 ξ(T ),
c ∼ i mf c
where J is relatedto J of Eq.(1) byJ a3J , and To conclude we have developed an analytic polaron
mf 0 mf ∼ B 0
ξ(T) n /max ε ,T is the magnetic susceptibility of percolation theory for DMS ferromagnetism in the limit
h F
∼ { }
holes,withε beingthe Fermienergy. Inthecaseoflow oflowcarrierdensityorequivalentlystrongcarrierlocal-
F
electron density, T ≫εF ization a3Bnh ≪1. Interestingly our polaron percolation
theory reproduces the free carrier Weiss mean field the-
Tc ∼√ninh|Jmf|, ory in the limit of a3Bnh ∼ 1. Our analytic results are
in good agreementwith existing numericalresults in the
which matches the result (7) of the bound-polaron pic-
strongly localized limit [5, 6, 7]. The experimental DMS
turSeinactetnhehl≪imintio,foanpeplpicoalabriolintyinocfltuhdeeslamttaern,yam3Bnahgn∼et1ic. couryrremnatlrygihnaavlleyaa3Bpnphli∼cab1l0e−t1o−t1h0e−3exwpheircihmmenatkaelssoyusrtetmhes-,
impurities, and the total magnetization of the sample is
and we get reasonable agreement with experimental re-
that of impurities:
sults for T and for M(T).
c
M(T)=Sn [r (T,n ,n ,a )√3n ], (8) This work was supported by the US-ONR, the LPS,
i corr i h B h
V
andDARPA.WethankB.I.Shklovskiiforusefuldiscus-
where r is definedby Eq.(6), the anduniversalfunc-
corr sion on percolation theory.
tion (y) is the infinite cluster’s volume in the model
V
of overlapping spheres; it depends only on the product
y of the spheres’ diameter and the cubic root of their
concentration.
Using Eqs. (6) and (7), we cast Eq. (8) in the form [1] J.K¨onigetal.,cond-mat/0111314andreferencestherein.
[2] T. Dietl et al.,Phys. Rev.B 63, 195205 (2001).
MM((T0)) =V 0.86+ a3Bnh 31 lnTTc , (9) [[34]] TJ..KD¨oientilgeettaal.l.,,SPciheynsc.eR2e8v7.,L1e0t1t.98(42,00506)2.8 (2000); M. F.
(cid:18) (cid:19) Yang et al.,Phys. Rev.Lett. 86, 5636 (2001).
(cid:0) (cid:1)
withtheCurietemperatureT givenbyEq.(7). Onecan [5] R. N.Bhatt et al., cond-mat/0111184.
c
see that the shape of the magnetization curve is deter- [6] M.Berciu andR.N.Bhatt,Phys.Rev.Lett.87, 107203
mined by only one dimensionless parameter a3n , while (2001).
B h [7] Xin Wan and R. N.Bhatt, cond-mat/0009161.
the expression (7) for T is more complicated and de-
c [8] H. Akai,Phys. Rev.Lett. 81, 3002 (1998).
pends on all parameters of the model. Fig. 2 shows the [9] V. I. Litvinov and V. K. Dugaev, Phys. Rev. Lett. 86,
temperaturedependenceofthemagnetizationattwoval- 5593 (2001).
ues of a3Bnh; the curve is more concave at smaller values [10] A. Chattopadhyay et al., Phys. Rev. Lett. 87, 227202
of this parameter, which is precisely the experimental (2001).
observation. Our magnetization results agree with the [11] H. Ohno, Science 281, 951 (1998); H. Ohno, J. Magn.
Magn. Mat. 200, 110 (1999).
numerical results of Ref. [6] and are consistent with the
[12] S. Das Sarma, Am. Scientist 89, 516 (2001).
experimental magnetization data in GaMnAs, particu-
[13] A. van Esch et al.,Phys. Rev.B 56, 13103 (1997).
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carrierlocalizationassociatedwithlowervaluesofa3Bnh. Teor.Fiz.56,1013 (1969)]; T.Kasuyaet al.,SolidState
5
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