PHYSICAL REVIEW B 1 MAY 2000-IIVOLUME 61, NUMBER 18

Theory for the bound magnetic polaron in diluted magnetic semiconductorsby a modified molecular-field approximation

Masakatsu UmeharaNational Institute for Research in Inorganic Materials, 1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan

~Received 30 August 1999; revised manuscript received 18 February 2000!

A theory for the bound magnetic polaron~BMP! in diluted magnetic semiconductors is presented. Thefluctuations of magnetization are taken into account by a modified molecular-field approximation with avariation method for magnetization. The spin-splitting and the spectrum for spin-flip Raman scattering calcu-lated for the electron-type bound magnetic polaron are compared to the experimental results with a goodagreement. The present method describes well the energetics of BMP from low temperatures to high tempera-tures. The method is also applicable to the hole-type bound magnetic polaron where the saturation effect of themagnetization becomes important.

aneo

p

onthtione

ope

thicatia

ait

ntlyho

Pte

thetheover

theeticthein-the

-

,ed

edopy

I. INTRODUCTION

In diluted magnetic semiconductors~DMS!, the thermo-dynamic fluctuations of magnetization become importsince there is no spontaneous magnetization even attremely low temperatures together with a small numbermagnetic ions in DMS. Actually, Dietl and Spałek1 ~DS! firstshowed that the molecular-field approximation~MFA! failsto explain the temperature dependence of the Zeeman sting, hereafter referred to as the spin-splitting~SS!, for theelectron~n!-type bound magnetic polaron~BMP! in DMS.DS proposed a theory which includes the fluctuationsmagnetization and succeeded in explaining the experimeresults. The theory, however, is valid for the case wheninduced magnetization is small compared to the saturavalue. After that, with a direct calculation of the partitiofunction for a classical spin system with no interaction btween magnetic ions, Warnock and Wolff2 ~WW! reached asimilar conclusion as obtained by DS and studied the prerties for the hole-type BMP in DMS. In spite of these thoretical approaches, a large number of studies for DMSstill on the usual MFA even at present. The advantage ofMFA is ease in dealing with and clearness in the physpicture. Thus in the present paper we propose an alternatheory which includes the fluctuations of magnetizationwell as the saturation effect of magnetization. The theorybased on a modified molecular-field approximation withvariation method for fluctuated magnetizations. The validof the method is examined by applying to then-type BMP;the calculated results are consistent with the experimeones, not only qualitatively but also almost quantitativeThe temperature range for application of the present metis also discussed.

II. METHODS AND CALCULATIONS

A. Molecular-field approximation „MFA …

We start from the MFA. In MFA, the free energy of BMmay be approximated as follows for a classical spin sysof magnetic ions:

PRB 610163-1829/2000/61~18!/12209~7!/$15.00

tx-f

lit-

ftalen

-

--isel

vesis

y

al.d

m

F@c~r !,M ~r !#

52\2

2m E c* ~r !¹2c~r !dr2E e2uc~r !u2

«ur udr

2kT lnH 2 coshS D@c~r !,M ~r !#

kT D J2

kT

VpxE G„M ~r !…dr

13S

2~S11!

kQ

VpxE @M ~r !#2dr . ~2.1!

The first term is the kinetic energy of the donor electron,second is the attractive Coulomb interaction betweenelectron and donor center, and the third, where the tracethe electron spin has been taken, is the free energy due toexchange interaction between the electron and magnions. The fourth is the magnetic entropy and the last ismagnetic strain energy due to a small antiferromagneticteraction between magnetic ions. In this paper, we treatcase without the external magnetic field.m is the effectivemass for the electron,« is the dielectric constant of the crystal, andD is the exchange interaction energy given by

D@c~r !,M ~r !#5ISxE M ~r !uc~r !u2dr ~2.2!

and thus is equal to half of the SS. HereI[ 12 (N0ac) is the

s-dexchange interaction constant,S is the magnitude of spinfor magnetic ion,c(r ) is the wave function for the electronand M (r ) is the magnetization for magnetic ion calculatby the Brillouin function: M (r )5B„h(r )… with the effec-tive magnetic fieldh(r ). Moreover,x is the effective con-centration of magnetic ion,Vp is the volume of the primitiveunit cell, andQ is the Curie-Weiss temperature obtainthrough susceptibility measurements. The magnetic entris approximated as

G„M ~r !…52h~r !M ~r !1 lnFsinh@h~r !~2S11!/2S#

sinh@h~r !/2S# G .~2.3!

12 209 ©2000 The American Physical Society

r

a

ohel

SmA1,Sls

a-d

m

-

ac-

h aow,take

tic

ndhehend

s

. Iniza-o beof

eldln,

o-

12 210 PRB 61MASAKATSU UMEHARA

Then, the variation of Eq.~2.1! with respect toM (r ) givesthe effective magnetic field acting on a magnetic ion as

kT3h~r !5Vp3ISuc~r !u2tanhS D

kTD23SkQ

~S11!M ~r !.

~2.4!

After elimination of M (r ) with use of Eq.~2.4!, the freeenergy becomes a functional ofc(r ) only. Then the varia-tion of the free energy with respect toc(r ) gives the mini-mum state of BMP. In this paperc(r ) is assumed to beca(r )5Aa3/p exp(2ar), wherea is a variation parameteto be determined. Then the SS~that is given by 2D! calcu-lated for n-type BMP in Cd0.95Mn0.05Se is shown in Fig. 1.The values of the parameters for the calculation arefollows:1,4 m50.13me , I[ 1

2 (N0ac)50.13 eV, «59.4, x50.03, andkQ50.98 K. This value forkQ is a littlesmaller than the value 1.2 K used by DS. In the case withthes-dexchange interaction, the effective Bohr radius of tdonor electron and the binding energy are, respectiv72.31 aB ~that is 38.25 Å! and 20.0 meV, whereaB is theBohr radius for the hydrogen atom. Experimentally, thehas been observed as the Raman shift in the spin-flip Rascattering.3–5 We see that the SS calculated by the MFwhich is shown by the open triangle with solid line in Fig.vanishes above about 2.5 K although the observed Spreserved up to much higher temperatures. This reveabreakdown of the MFA.

B. Modified molecular-field theory with thermal fluctuationsof magnetization

The MFA itself contains no fluctuations of the order prameter. Therefore, we have to modify the theory to inclufluctuations. Leth be the order parameter andF(h) be thefree energy in MFA like Eq.~2.1!. To include fluctuations,we need to consider all allowed values of the order paraeter. Then the partition function may be given by6

s

utey,

San,

isa

e

-

Z5E expF21

kTF~T,V,h!Gdh. ~2.5!

Thus the free energyF which includes fluctuations is obtained by

F52kT ln E expF21

kTF~T,V,h!Gdh. ~2.6!

Furthermore, since the probability density for valuesh isgiven as

v~h!5expF21

kTF~T,V,h!GYZ, ~2.7!

the mean value ofA(h) is calculated by

^A~h!&5E A~h!v~h!dh. ~2.8!

When the fluctuations of magnetization are taken intocount, the exchange interaction energyD given by Eq.~2.2!behaves as a vector

D@c~r !,M ~r !#5ISxE M ~r !uc~r !u2dr . ~2.9!

In DMS, since there is no magnetic ordering together witsmall number of magnetic ions, the electron spin can follthe motion of the SS vector, 2D, in BMP. Thus, hereafterwe regard the SS vector as the order parameter and thush5D in Eq. ~2.5!. In the case without the external magnefield, theD space is isotropic, thusD&50. According to Eq.~2.8!, the mean value of magnitude ofD5uDu is calculatedby

^D&51

Z E D expF21

kTF~D!G4pD2dD, ~2.10!

with

Z5E expF21

kTF~D!G4pD2dD. ~2.11!

Now, we are in the step to calculate the free energyF(D)considering the fluctuations of both the magnetization athe wave function of the electron. Since the motion of telectron is fairly faster than that of the magnetization, telectron state can follow the motion of the magnetization athus may be obtained by]F/]c(r )50 for givenM (r ). Thiseliminates the freedom of the motion of electron and thuFbecomes a functional of onlyM (r ): F5F@M (r )#. The nextstep is how to generate the fluctuations of magnetizationthis paper, instead of all the fluctuated states of magnettion, we take account of the fluctuated states that seem teffective for the problem. We assume that the fluctuationmagnetization is due to the fluctuation of the molecular fihb(r ) with a parameterb, which is the inverse of the spatiaextent of the fluctuation: the fluctuated magnetizatioMb(r ), is thus obtained byMb(r )5B„hb(r )…. We furtherassume hb(r )5g(j)uYb(r )u2, where g(j)5VpIS/@j(T)k(T1Q)# andYb(r ) is a trial function describing thefluctuation. ForYb(r ), we use the same function as the dnor electron wave function in the present paper:Yb(r )

-eithete

n

io

ira

ic

nitnchuhb

r

3,d

tua-

eecs-

th-

en-

theod

resi-

is3,d

eris

elyail-n-etrum. 4,

ion,.veFA.the

r

atc-

PRB 61 12 211THEORY FOR THE BOUND MAGNETIC POLARON IN . . .

5Ab3/p exp(2br). However,b is independent of the variation parametera for the electron wave function. This profilfor the fluctuated magnetization is analogy to the MFA wthe assumption that the BMP accompanies the ferromagncomplexes, except for introducing a variation paramej(T) in g(j). We determinej(T) by minimizing the freeenergyF ~j! that includes the fluctuations of magnetizatiothis process is performed by choosingj(T) so as to maxi-mize the partition functionZ5Z(j). Thus,j(T) is a pa-rameter to generate the proper magnitude of the fluctuatat each temperature within a given trial functionYb(r ).Since the fluctuations of the magnetization increases wtemperature,j(T) should decrease with increasing tempeture as shown later.@If j(T)51.0, the fluctuations of themagnetization generated at high temperatures are restrto only those close toMb(r );0.0.# In this way, giving afluctuation parameterb, we obtainMb(r ) andF@Mb(r )#; Dand F(D) are calculated as a function ofb. Owing to achange of variable fromD to b, Eqs. ~2.10! and ~2.11! arerewritten as

^D&51

Z E D~b!exp

F21

kTF„D~b!…G4pD~b!2 U dD~b!

db U db

~2.12!

and Z with the same transformation factor as in Eq.~2.12!.The integral in Eq.~2.12! is numerically performed for theinterval 0.0,baB,0.07. This upper valuebaB50.07 forthe integration corresponds toYb(r ) with an extension of7.56 Å, which is comparable to a side of the primitive ucell for CdSe. Such a fluctuation can scarcely occur, sithis extension is sufficiently smaller than the effective Boradius of the donor electron and thus the free energy for sa state is much higher than that in the equilibrium state. Tvalue of the integration obtained, thus, is considered tonumerically accurate.

First, we have to determinej(T) at each temperature. Foan example, we showZ(j)/Z(j51) calculated at 4 K in Fig.

FIG. 2. The partition functionZ(j) normalized byZ(j51),calculated at 4 K, is shown as a function of a variation parametej.The vertical arrow↓ points the value ofj that maximizesZ(j) at 4K. The j dependence of the SS at 4 K is also shown. The partitionfunction is shown by solid line, while the SS by broken line.

ticr

:

ns

th-

ted

erchee

2. From this figure, we seej50.282 at 4 K. In this way, weobtainedj(T) as 0.902, 0.800, 0.641, 0.448, 0.282, 0.200.159, and 0.129 atT50.25, 0.50, 1.0, 2.0, 4.0, 6.0, 8.0, an10.0 K, respectively. The decrease ofj(T) with increasingtemperature generates the proper magnitude of the fluctions at each temperature. Then, using Eq.~2.12! we calcu-late the SS for Cd0.95Mn0.05Se as a function of temperaturwith the obtainedj(T). The result is shown in Fig. 1 by thclosed triangle with solid line. The following characteristiare seen:~i! Below 1 K, the SS increases rapidly with decreasing temperature;~ii ! the SS decreases gradually wiincreasing temperature above;3 K; ~iii ! the temperature dependence of the SS becomes quite small above;6 K; the SSseems nearly constant for high temperatures. The experimtal data for the SS~or the Raman shift! in Cd0.95Mn0.05Se arealso shown in Fig. 1 for comparison. We can see thatcalculated result with the effect of fluctuations is in goagreement with the experimental result below 5 K.3–5 Above5 K, there are no available data for Cd0.95Mn0.05Se, to ourknowledge. The nearly constant SS for high temperatumentioned in~iii !, however, is consistent with the expermental data for Cd0.9Mn0.1Se above 15 K.5 Furthermore, thespectrum of the stokes shift for spin-flip Raman scatteringcalculated at 0.5 and 6 K and is shown in Fig.respectively:y axis in the figure indicates the calculatespectrum intensity proportional toD2 exp@2F(D)/kT#3exp(x)/@2 cosh(x)#, wherex5D/kT and exp(x)/@2 cosh(x)#is the probability that the donor electron occupies the lowelectron-spin level. The calculated spectrum peak at 0.5 Kat 9.2 cm21 with the half-value width of 2.45 cm21. This isconsistent with the experimental result~see Fig. 1 in Ref. 5!and clearly shows that the SS is thermally fluctuated largeven at 0.5 K. For 6 K, the experimental data are not avable in Refs. 4 and 5. However, judging from the experimetal data for Cd0.9Mn0.1Se at high temperatures in Ref. 4, thcalculated temperature dependence of the Raman spechas a strong resemblance to the experimental one. In Figthe calculated mean value of the electron-spin polarizat^tanh(D/kT)&, is shown together with that in the usual MFAThe electron-spin polarization in the MFA vanishes abo;2.5 K, which causes the disappearance of the SS in MWhen the fluctuation is included, the quantization axis ofelectron spin can follow the motion of the SS vector, 2D,and the electron spin is polarized along the direction ofD

FIG. 3. The spin-flip Raman scattering spectrum calculatedT50.5 and 6 K for Cd0.95Mn0.05Se is respectively shown as a funtion of the Stokes shift, which is equal to the SS, 2D.

vw

t

fhe

catio

iono

ha

ethethe

eand

omses

izafor

ag-orne,

12 212 PRB 61MASAKATSU UMEHARA

with the magnitude dependent onT and uDu at every instant.Thus, we see a nonvanishing electron-spin polarization efor much higher temperatures: in classical spin systemsstudy now, the electron-spin polarization is vanished aT5`. Furthermore, the calculated mean value of thes-d ex-change energy,D tanh(D/kT)&, is shown as a function otemperature together with that in MFA in Fig. 5. When tfluctuation is taken into account, the gain of thes-dexchangeenergy remains even at high temperatures. Finally, theculated mean value of the space profile for the magnetizain the BMP, ^M (r )&, is shown in Fig. 6 for relatively lowtemperatures as well as the donor-electron wave functWe can see the induced magnetization is fairly small evelow temperatures and the electron wave function is almtemperature independent for then-type BMP inCd0.95Mn0.05Se.

III. HIGH-TEMPERATURE EXPANSION

In this section, we shall confine our attention to the higtemperature region. At the high-temperature region,shown in Appendix A, the free energy in the MFA expressby Eq. ~2.1! may be approximated as follows:

F~a,D!>Ee~a!2kT ln 21kTW~T!D2, ~3.1!

FIG. 4. The calculated mean value of the electron-spin polartion with and without the fluctuation of magnetization is shownCd0.95Mn0.05Se as a function of temperature.

FIG. 5. The calculated mean value of thes-d exchange energywith and without the fluctuation of magnetization is shown asfunction of temperature.

ene

l-n

n.atst

-s

d

whereEe(a) is the first and second terms of Eq.~2.1!, thatis, the binding energy due to the nonmagnetic interaction,second term is the entropy due to the spin freedom ofbound electron whenD50, andW in the third term is

W~T!5H q21

2~kT!2J , ~3.2!

with

q512pa3

Vp~ IS!2x

S

S11

k~T1Q!

kT. ~3.3!

Here the fist term ofW comes from the entropy and thmagnetic strain energy for the substituted magnetic ions,the second term comes from the free energy due to thes-dexchange interaction; so the first term produces the randmotion of the localized spins, while the second term cauthe polarization of the localized spins. From Eq.~3.2! we cansee the usual MFA breaks down aboveTM given by

TM51

kA2q. ~3.4!

-

a

FIG. 6. The calculated mean value of the space profile of mnetization is shown atT50.5, 1.0, and 3.0 K, as well as the donelectron wave function. The magnetization is shown by solid liwhile the donor wave function is by dotted line.X axis shows thedistance from the donor center normalized byaB* that is the exten-sion of the donor electron without thes-d exchange interaction.

gy

-asobwo

retasorren

-oti

w

io

,

andis

sag-

uchII B

t ofaren,ical

sibleam-

a-that

ure

blee

ra-owtothec-era-thean-stss oflet. In

thatlex.tionpen-

in

sioic

do

PRB 61 12 213THEORY FOR THE BOUND MAGNETIC POLARON IN . . .

Usually, the extension of the bound electron,a, is a functionof D. In the high-temperature region with a smallD, how-ever,a may be approximated asa0 determined by minimiz-ing only Ee(a). In this way, we may regard the free enerEq. ~3.1! is a function of onlyD in the high-temperatureregion. Then, performing the integration ofD from 0 to` inEq. ~2.10!, we can obtainD& analytically as

^D&52

ApW~T!. ~3.5!

We calculate and show the SS~that is 2 D&) in Fig. 7 as afunction of T together withW(T): the thick solid and thethick broken lines for case ofW`50.13054 show, respectively, the SS andW(T) for the same values of parametersused in Sec. II. We can see from the figure that above ab5 K the SS calculated in Sec. II B agrees well with the SSthe high-temperature expansion method. Furthermore,see that the SS is nearly constant with temperature ababout 8 K together with the nearly constantW(T) in thesame temperature region. This suggests the strong cortion between the nearly constant SS and the nearly consW. The almost constantW(T) means that the localized spinof the substituted magnetic ions are not polarized any mbut are almost randomly fluctuated in that temperaturegion. Thus, we would like to conclude that the nearly costant SS~or the Raman shift! with temperature in the hightemperature region is caused from the almost randfluctuation of the localized spin of the substituted magneions. Since the electron spin follows the motion ofD adia-batically at every instant, the mean value ofD5uDu takes afinite value even in this case. In Appendixes B and C,will confirm this point further.

In the same manner, the root-mean-square fluctuatA^D2&, is obtained in the high-temperature region as

A^D2&5A 3

2W~T!. ~3.6!

FIG. 7. The calculated result by the high-temperature expanmethod in Sec. III is shown as a function of temperature. The thsolid and the thick broken lines, respectively, show SS andW(T)@Eq. ~3.2!# calculated for the same values of parameters as useFig. 1 (W`50.13054). The thin solid line shows SS for the caseW`50.530.13054 for comparison.W` is the value ofW(T) at T5`, that is the value ofq in Eq. ~3.3!.

utye

ve

la-nt

e,-

-

mc

e

n,

At T→`, ^D& andA^D2& approach to the following valuesrespectively:

^D&5~ IS!AxA Vp

3p2a3

S11

S, ~3.7!

A^D2&5~ IS!AxA Vp

8pa3

S11

S. ~3.8!

IV. DISCUSSION AND SUMMARY

~i! We first discuss the relevance between DS’s paperthe present one shortly. When the free energy in the MFAgiven instead of Eq.~3.1!,

F~a0 ,D!>Ee~a0!2kT lnH 2 coshS D

kTD J 1kTq~T!D2,

~4.1!

the SS calculated with Eq.~2.10! gives the same result aobtained by DS in essence. However, when the induced mnetization orD becomes large, Eq.~4.1! does not become agood approximation. Thus DS’s method cannot apply to sa case. On the other hand, our method proposed in Sec.is still applicable to the case where the saturation effecthe magnetization becomes important. Next, we compwith the method by WW. To calculate the partition functioWW take account of all accessible microstates for a classspin system with zero Curie-Weiss temperature,Q50. Onthe other hand, in the present paper, instead of all accesmicrostates, we sum over allowed values of the order pareter D @see Eq.~2.11!#. Actually, to calculate the allowedvalues ofD, we assume the molecular field described bytrial function Yb(r ) from which the fluctuated magnetizations effective to the present problem are generated. Attime, we introduce a variation parameterj(T) to generate theproper magnitude of the fluctuation at each temperatwithin a given trial functionYb(r ). We determinej(T) byminimizing the total free energyF with the effect of thefluctuation. After obtaining the value ofj(T), the fluctuatedmagnetization and thus the fluctuatedD are calculated. Asalready discussed in Secs. II B and III, there is consideravalidity to this modified molecular-field method, though thproper choice of the trial functionYb(r ) is demanded.

~ii ! Now, we would like to discuss the applicable tempeture range of the method proposed in Sec. II B. At quite ltemperatures nearT50 K, the present method is reducedthe usual MFA, since all the states fluctuated beyondminimum of F(D) can scarcely access to the partition funtion. On the other hand, the calculated result at high temptures in Sec. II B, for example the SS, agrees well withcalculated result obtained from the high temperature expsion method in Sec. III. This agreement strongly suggethat the present method in Sec. II B covers the energeticBMP from low temperatures to high temperatures. Next,us consider the magnetization probed by the electron spingenerating the fluctuation of magnetization, we assumethe BMP causes the formation of the ferromagnetic compThus the mean value of the space profile of magnetizacalculated at low temperatures gives the temperature dedence of the ferromagnetic complex properly, as showed

nk

inf

mthsc.caeniotat owned

ra

erov

heop

nb

laeit

ospeaheagceRfu

ri-

a

is

f

h a

-

to

on-zedmlyenmes

ren

lo-

anins

f thenor

12 214 PRB 61MASAKATSU UMEHARA

Fig. 6. At high temperatures, since the ferromagnetic coplex gets out of shape due to the random motion ofmagnetization, the mean value of the space profile losemeaning with increasing temperature, as discussed in Seand later in Appendixes B and C. The range of the applition is, thus, different between the energetics and the mvalue of the space profile of the magnetization. The reasodue to the variation method used in Sec. II B; the variatmethod is a better approximation to the energy than the sThe mean value of the electron-spin polarization and thathe s-d exchange energy shown in Figs. 4 and 5 are, hoever, valid up to the high temperature since both are a fution of D. In this way, except for the space profile of thmagnetization at high temperatures, the present methodscribes rather well the properties of BMP from low tempetures to high temperatures.

~iii ! In the final stage of this work, the author furthfound two articles: a model by Ryabchenk and Semen7

and a study by Hagstonet al.8 Both are not on the MFA. Theformer calculates the partition function on the model of tplane wave function for the bound electron with a methdifferent from WW’s. The latter is an extension and an aplication of WW’s method. The partition function by WWdepends only on the choice of the bound carrier wave fution. Thus the application of WW’s method to various prolems for the boundonecarrier in DMS might be powerful.

In this paper, we have presented the modified molecufield theory for BMP in DMS and have applied to thelectron-type BMP. The calculated result agrees well wthe experimental result not only qualitatively but also almquantitatively. This method is applicable to the hole-tyBMP in DMS where the saturation effect of the magnetiztion becomes important. The preliminary calculation for thole-type BMP shows certainly the saturation effect of mnetization in the several points, for examples, the indumagnetization, the SS, and the spectrum of the spin-flipman scattering. Detailed results will be published nearture.

ACKNOWLEDGMENTS

This work was partially supported by a Grant-in-Aid foScientific Research of Priority Areas ‘‘Spin Controlled Semconductor Nanostructures’’~Grant No. 09244105! from theMinistry of Education, Science, Sports and Culture in Jap

APPENDIX A

In Appendix A, we show the free energy Eq.~2.1! isreduced to Eq.~3.1! at the high-temperature region. For thaim, we first obtain the relation betweenD2 and@M (r )#2. Inthe discrete model,D is described as

D5~ IS!(i

M i uCi u2pi , ~A1!

whereM i5(Si /S) anduCi u2 is the occupation probability othe electron ati th site. Furthermorepi51.0 when the sitei isoccupied by the magnetic ion, otherwisepi50. In the MFA,D is given by Eq.~2.2! in the continuous model. ThenD2 inthe MFA becomes in the discrete model as

-eitsIII-

anis

nte.f-

c-

e--

d-

c--

r-

ht

-

-da--

n.

D25~ IS!2(i

(j

M iM j uCi u2uCj u2pipj , ~A2!

whereMi is the magnetization at sitei induced by thes-dexchange interaction. In the high-temperature region witsmall magnetization, M j is expressed as M j5x(T,Q)ISuCj u25Mi(uCj u2/uCi u2), where x(T,Q) is anappropriate susceptibility. Then Eq.~A2! is reduced to

D25~ IS!2(i

M i2pi(

juCj u4pj

5~ IS!2x2E @M ~r !#2drE uc~r 8!u4dr 8, ~A3!

with the concentration of the magnetic ions,x. On the otherhand, since G(M )52@3S/2(S11)#M22O(M4) for asmall M, the last two terms in Eq.~2.1! becomes

3S

2~S11!k~T1Q!

x

VpE @M ~r !#2dr . ~A4!

Then, by using Eqs.~A3! and the hydrogen-type wave function for c(r ), Eq. ~A4! is rewritten askTqD2, whereq isgiven by Eq.~3.3!. After expanding the free energy duethe s-d exchange interaction, the third term in Eq.~2.1!, toorder of D2, the free energy in the MFA, Eq.~2.1!, can bereduced to Eq.~3.1! in the high-temperature region.

APPENDIX B

In Sec. III, we showed that the SS becomes nearly cstant at the high-temperature region where the localispins of the substituted magnetic ions are almost randofluctuated. In this appendix, we directly confirm that whthe localized spins are randomly fluctuated, the SS becothe constant given by Eqs.~3.7! and ~3.8!. Here, instead ofthe mean value ofD, we discuss the root of the mean-squafluctuation ofD. Then^D2& may be given for a classical spias

^D2&5~ IS!2(i

(j

^M iM j&uCi u2uCj u2 pipj . ~B1!

We now discuss the high-temperature region, where thecalized spins are randomly fluctuated, that is^M iM j&5d i j .Then

^D2&5~ IS!2(i

uCi u4pi5~ IS!2xVpE uc~r !u4dr . ~B2!

In the case ofc(r )5A(a3/p) exp(2ar) employed in Sec.II, we obtain just the same expression forA^D2& as Eq.~3.8!without any further approximation. In this manner, we cconfirm that the random fluctuation of the localized spproduces the SS given by Eqs.~3.7! and ~3.8! at high tem-peratures. The above analysis also shows the adequacy odiscussion in Sec. III. On the other hand, when the dowave function is expressed asuCi u251/N with box normal-ization,A^D2& approaches

A^D2&5~ IS!Ax

N~B3!

dstntpl

ofitehta

o

wesedti-

i-f

tates

PRB 61 12 215THEORY FOR THE BOUND MAGNETIC POLARON IN . . .

at high-temperature limit. HereN is the number of the cationsite in the box. In Appendix C, Eq.~B3! is obtained from theensemble average of the simple case.

APPENDIX C

In Appendix B, we confirmed that when the localizespins are randomly fluctuated, the SS becomes the concalculated in Sec. III. In Appendix C, we study this poifurther on the ensemble average of the following simcase. We take the Ising model withS51/2, and assumeuCi u251/N for the donor wave function as in the last partAppendix B. Now, we discuss the case that only three sof the cation are substituted by the magnetic ion. For tcase, the accessible states are restricted to the eight sshown in Table I. The probability of finding each state atT5` is equal for these eight states. Then we can easily cfirm ^Si

z&50 and^SizSj

z&50 for iÞ j ; so the localized spinsare randomly fluctuated. The average ofD is obtained by

^D&5IS32S 3

81

1

833D Y N5ISS 1.5

N D , ~C1!

and in a similar way

A^D2&5~ IS!)

N. ~C2!

With x53/N, Eqs.~C1! and ~C2! are rewritten as

^D&5ISAxA 3

4N, ~C3!

ys

as

ys

ant

e

sistes

n-

A^D2&5ISAx

N. ~C4!

Note Eq.~C4! is just the same form as Eq.~B3!. In this way,from the discussion in Sec. III and Appendixes B and C,confirm that the SS in the high-temperature region is caufrom the random fluctuation of the localized spin of substuted magnetic ions.

TABLE I. Schematic representation for the accessible mcrostates is shown in the second column for the Ising model oS5

12 with 3 substituted magnetic ions. The1~or 2! sign in the first

column means that the SS vector calculated for the accessible sshown in the right is directed to1z ~or 2z) direction. The uparrow in the second column meansSz5

12 , while the down arrow

Sz5212 . Furthermore,A5(nMn

zuCnu2pn , andB is the probabilityof finding each microstate atT5`. The average ofD at T5` is,thus, given by the sum ofuA3Bu over the eight states.

P.

1T. Dietl and J. Spałek, Phys. Rev. Lett.48, 355 ~1982!; Phys.Rev. B28, 1548~1983!.

2J. Warnock and P. A. Wolff, Phys. Rev. B31, 6579~1985!.3M. Nawrocki, R. Planel, G. Fishman, and R. R. Galazka, Ph

Rev. Lett.46, 735 ~1981!.4D. L. Peterson, D. U. Bartholomew, U. Debska, A. K. Ramd

and S. Rodriguez, Phys. Rev. B32, 323 ~1985!.5E. D. Isaacs, D. Heiman, M. J. Graf, and B. B. Goldberg, Ph

.

,

.

Rev. B37, 7108~1988!.6For example, see L. D. Landau and E. M. Lifshitz,Statistical

Physics, 3rd ed.~Nauka, Moscow, 1975!, Chap. 147.7S. M. Ryabchenko and Yu. G. Semenov, Zh. Eksp. Teor. Fiz.84,

1419 ~1983! @Sov. Phys. JETP57, 825 ~1983!#.8W. E. Hagston, T. Stirner, P. Harrison, O. F. Holbrook, and J.

Goodwin, Phys. Rev. B50, 5264~1994!.