A single layer of Mn in a GaAs quantum well: a ferromagnet with quantum fluctuations

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by  Roger G. Melko

A Single Layer of Mn in a GaAs Quantum Well:
a Ferromagnet with Quantum Fluctuations
Roger G. Melko, Randy S. Fishman, and Fernando A. Reboredo
Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6032
(Dated: February 4, 2008)
7
Some of the highest transition temperatures achieved for Mn-doped GaAs have been in δ-doped
0
heterostructures with well-separated planes of Mn. But in the absence of magnetic anisotropy,
0
theMermin-Wagner theorem implies that asingle plane of magnetic ions cannot beferromagnetic.
2
Using a Heisenberg model, we show that the same mechanism that produces magnetic frustration

n and suppresses the transition temperature in bulk Mn-doped GaAs, due to the difference between
a thelightandheavybandmasses,canstabilizeferromagneticorderforasinglelayerofMninaGaAs
J
quantumwell. This comes at the price of quantum fluctuations that suppress the ordered moment

1 from that ofafully saturated ferromagnet. Bycomparing thepredictions ofHeisenbergandKohn-
3 Luttingermodels, we conclude that theHeisenberg description of a Mn-doped GaAs quantumwell
breaks down when theMn concentration becomes large, but works quitewell in the weak-coupling
] limit of small Mn concentrations. This comparison allows us to estimate the size of the quantum
i
c fluctuationsin thequantumwell.
s
-
l
r I. INTRODUCTION wellwithanadditionalexchangeinteractionbetweenthe
t
m holesandtheMnspins,whicharetreatedclassicallyand
. The discovery of dilute-magnetic semiconductors confined to the central plane. Comparing the predic-
t
a (DMS)withtransitiontemperaturesabove170K1,2,3 has tions of the Heisenberg and KL plus exchange (KLE)
m models, we conclude that the Heisenberg description of
renewed hopes for a revolution in semiconductor tech-
- nologies based on electron spin4. However, the difficulty aMn-doped quantumwellworksquite wellforsmallMn
d concentrations or in the weak-coupling limit, but breaks
in producing DMS materials with ferromagnetic transi-
n
down when the Mn concentration becomes too large
tion temperatures above room temperature has stalled
o
c the development of practical spintronic devices. Some Thispaperisorganizedintofivesections. InSectionII,
[ of the highest transition temperatures to date have been weexamineseveralanisotropicS =5/2Heisenbergmod-
achieved in δ-doped GaAs heterostructures,5 where the els that might describe a single layer of Mn in a GaAs
4
Mn are confined to isolated planes. The Curie tempera- quantum well. Section III develops a more precise elec-
v
8 ture of digital heterostructures does not seem to vanish tronicdescriptionofthequantumwellbasedontheKLE
8 as the distance between layers increases6. Yet a single model. InSectionIV,theKLEmodelisusedtoestimate
2 layer of magnetic ions with spin S = 5/2 would not be the parameters of an anisotropic Heisenberg model. A
4 expected to become ferromagnetic due to the Mermin- brief conclusion is provided in Section V.
0
Wagner theorem, which states that gapless spin excita-
6
tions destroy magnetic order at any nonzero tempera-
0
/ ture in two dimensions. The magnetic anisotropy due to II. HEISENBERG DESCRIPTION OF A
at strain is commonly invoked7,8,9,10 to explain the ferro- QUANTUM WELL
m magnetism of heterostructures. However, strain changes
significantly as a function of film thickness and capping In bulk Mn-doped GaAs, the difference between the
-
d layers,andwouldbe negligibleforasinglelayerofMnin light (ml = 0.07m) and heavy (mh = 0.5m) band
n a GaAs quantum well. The questions addressed in this masses with ratio r = m /m 0.14 produces an
l h
o paper are: can a single layer of Mn in a GaAs quantum anisotropic interaction11 between a≈ny two S = 5/2 Mn
c
well be ferromagnetic and what kind of ferromagnet is ions. This anisotropy arises because the kinetic energy
:
iv it? K = k,αβǫ(k)αβc†k,αckβ is only diagonalized when the
X To answer these questions, we employ two comple- angulPar momentum j of the charge carriers is quantized
mentary approaches. First, we study the magnetic in- along the momentum direction kˆ. The heavy and light
r
a teractions between Mn moments embedded in a GaAs holes carry angular momentum j kˆ = 3/2 and 1/2,
quantum well using a two-dimensional (2D) S = 5/2 respectively. As shown by Zara´nd· and±Jank´o,11 th±e in-
Heisenbergmodel, withexchangeinteractionstakingthe teractionbetween Mn spins S and S (as in the inset to
i j
same form as in bulk Mn-doped GaAs.11 Remarkably, Fig. 1) can be written
the same anisotropic Heisenberg interactions that sup-
press the bulk transition temperature act to stabilize H = J(1)S S +J(2)S r S r . (1)
long-range magnetic order in 2D by producing a gap ij − ij i· j ij i· ij j · ij
in the spin-wave (SW) spectrum. Second, we estimate
For r = 1, K is diagonal in any basis and J(2) = 0.
the parameters of the Heisenberg model by construct- ij
ing a Kohn-Luttinger (KL) model for a GaAs quantum Since J(2) >0 for r<1, the Mn spins in GaAs prefer to
ij
2
align perpendicular to the vector r =(R R )/R spins on the square lattice. We then use SW theory to
ij i j i
R connecting spins i and j. For a tetrah−edron o|f M−n solve HZJ, assuming that S 1. At the mean-field
j
| ≫
spins, the interaction energy between every pair of spins (1/S0) level, the ferromagnetic alignment of the spins
cannot be simultaneously minimized and the system is along the z axis is unstable to an A-type antiferromag-
magnetically frustrated. As shown both in the weak- netic realignment of the spins in the xy plane (with
coupling, RKKY limit11 and more generally12,13 within lines of spins alternating in orientation) when J /J >
2 1
dynamicalmean-fieldtheory,this anisotropicinteraction 2+D/J . Toorder1/S,Eq.(1)maybewritteninterms
1
suppresses the Curie temperature compared to a non- of Holstein-Primakoff bosons:
chiralsystemwith r =1. Morenoet al.13 found that the
H J S2 S(a†a +a†a a†a a†a )
transitiontemperaturemaybeloweredbyabout50%for ij ≈ − 1n − i i j j − i j − j i o
r =0.14 compared to r=1. J S
2 (a a†)(a a†), (4)
Weconstructasimple2DHeisenbergmodelforasingle ± 2 i± i j ± j
layer of Mn-doped GaAs by including the effect of this
where is + for spin i and j separated by xˆ, and
anisotropy: ± −
for spin i and j separated by yˆ. The second term in
HZJ = 1 H D (Sz)2 (2) Eq.(2)willsimilarlycontribute2DS ia†iai. Sinceonly
2 ij − i the first terms in a 1/S expansion ofPthe spin operators
Xi6=j Xi have been retained, the interactions between SW’s have
been neglected. Because of the relatively large S = 5/2
wherethesumistakenoverasquarelatticeandS =5/2.
spinof the magnetic ions, however,the next orderin the
We also include a single-ion anisotropy term D that
expansion (1/S2) will be rather small for low tempera-
might be important in a quantum well due to elastic
tures,andweexpectthe linearapproximationto recover
strain and spin-orbit coupling. When J(2) > 0, the
ij alloftherelevantquantumphysics. WritingtheHolstein-
anisotropic interactions cause the Mn spins to point out
Primakoffbosons in a momentum representation,we get
of the plane, producing a gap ∆ in the SW spectrum
SW the useful form:
even when D = 0. By breaking the rotational invari-
ance of the spin, the anisotropic coupling would allow a HZJ = H0+ Ak a†kak+a†−ka−k
singlelayerofMnionstoorderferromagneticallyevenin Xk (cid:16) (cid:17)
tbheeloawb,stehnecesionfgslein-igolne-aionnisaontrisooptyroDpyi.saAlssodkisncouwssne1d5ftuorpthreor- + Xk Bk(cid:16)a†ka†−k+aka−k(cid:17), (5)
duce aSWgapandtostabilizeferromagneticlong-range
with coefficients given by
order in the 2D Heisenberg model.
Before proceeding to study HZJ using SW theory, we Ak = 2J1S[1 ξ(k)]+J2Sξ(k)+DS
notethatothertypesofHeisenbergmodelsmayalsosta- −
Bk = J2Sφ(k)/2, (6)
bilizeferromagneticorderin2D.Forexample,theHamil-
tonian where ξ(k) = (coskx + cosky)/2 and φ(k) = coskx
−
cosk (lattice constant set to unity). In this form, it is
y
H′ =−JhXi,jinSixSjx+SiySjy+(1+γ)SizSjzo (3) ethaesyHtaomseiletotnhiaatntthoecaonmismoturotepywJit2hdtehsetrtooytsatlhsepianbiSlitztoyt o=f
with an anisotropic coupling between neighboring spins PtoitSheiz g=roNunSd−stPatekam†kaagkn,ectaizuastiinogn.a quantum correction
was recently used10 to model a plane of Mn spins in a The SW Hamiltoniancanbe diagonalizedin the usual
GaAshost. Whenγ 6=0,the rotationalinvarianceofthe way by transforming to Bogoliubov bosons α†k and αk,
spins is broken and 2D ferromagnetism is stabilized at
whichobeythecanonicalcommutationrelations. Forcing
finite temperatures. However, H′ does not contain the theresultingHamiltoniantobediagonalinα†kαk,weob-
same anisotropic interactions between Mn spins that are
believed11 to be present in the bulk system. tain the SW energies ωk = 2 A2k−Bk2 with the energy
Due to the presence of the J(2) terms in H , HZJ gap ∆SW = 2(J2+D)S. Thepabsence of terms linear in
ij ij the boson operators and the positive definiteness of the
does not commute with the total spin S = S .
tot i i SW frequencies for D < J2 < 2J1+D guarantee that
Therefore, quantum fluctuations will suppress thePmag- the ferromagnetic st−ate with all spins aligned along the
netic momentofthe ZJmodel evenat zerotemperature.
z axis is stable against non-collinear rearrangements of
Suchquantumfluctuationsaretypicallyfoundinantifer- the spins14. The diagonalized SW Hamiltonian may be
romagnets but are rather unusual in ferromagnets away
used to estimate the quantum-mechanical correction to
from quantum critical points. Notice that the Hamilto-
the ground-state magnetization. Starting with the mag-
tnuiaantioHn′s.constructed above does not have quantum fluc- netizationwrittenashSizi=S−N−1 kha†kaki,we find
upon transforming to Bogoliubov bosoPns,
To gain a better idea of the size of the quantum fluc-
tbuoatthioJni(sj1)in≡thJe1ZaJnmdoJdi(ej2l), w≡eJsp2eccoiaulpizleetoonltyhenceaigshebwohrienrge hSizi=S− N1 Xk (cid:26)Aωkk − 12 + 2ωAkkhα†kαki(cid:27). (7)
3
2.5
3
2.45 D=0
i F)
zS 2.4 M
h i j (T/C2
r C
ij T D=0
2.35
D=0.1
D=0.5
2.3 J =0
0 0.5 1 1.5 2 2
1
J /J
2 1 0 0.5 1 1.5 2 2.5
J /J orD/J
2 1 1
FIG.1: (coloronline)Quantumcorrectiontothegroundstate
magnetization per spin (S = 5/2) in the limit D = 0. The
FIG. 2: (color online) The transition temperature of the
inset illustrates thedirection of theinteraction Ji(j2). nearest-neighbor HZJ model calculated in SW theory. For
D = 0, TC falls to zero at J2/J1 = 0 and 2 ; for J2 = 0,
TC = 0 for D = 0 only. The SW value for TC is normalized
Since nk = α†kαk = 1/(eβωk 1), Eq. (7) can be used bythe Weiss MF result.
h i −
to calculate the quantum correction to the ground-state
magnetization by simply setting α†kαk = 0 and evalu-
ating the integrals over k. The rhesult iis plotted versus We confirm this relationship by calculating TC in the
J2/J1 for D =0 in Fig. 1. HZJ model using the above SW theory, with J2 =0 and
Within linear SW theory, Eq. (7) also affords us the anonzeroD. AsshowninFig.2,theresultsforJ2 =0or
simplestestimateforthetransitiontemperatureTCofthe D =0agreequitewellasD orJ2 →0. ForlargerJ2 and
nearest-neighborHZJmodel. Clearly,thetransitiontem- D,thecurvesdeviatesignificantly,withtheJ2 =0curve
perature calculated within this formalism neglects the growing linearly in the limit of large D. Quite generally,
large quantitative corrections due to SW interactions. we find that as ∆SW 0,
→
But including such interactions to second-order in 1/S
4πJ S(S+1/2)
would require additional approximations to handle the 1
T . (9)
terms quartic in α†k and αk. Since the precise value of C → ln(8π2J1S/∆SW)
T does not play a role in the subsequent discussion, we
C
Thisgeneralexpressionreducestothecorrectlimitswhen
are satisfied that the correct qualitative trends (i.e. the
J orD =0. We concludethatthe behaviorofT inthe
stabilization of T at a finite value) are reproduced by 2 C
C
case of strong fluctuations is controlled by the isotropic
linear SW theory. Including now both terms of Eq. (7),
exchange J and the SW gap.
we can write the T-dependent order parameter as 1
With a finite D, the T versus J /J curve qualita-
C 2 1
Sz =S+ 1 1 Ak coth βωk . (8) tively resembles a scaled-up version of the D = 0 curve
h ii 2 − N Xk ωk (cid:18) 2 (cid:19) in Fig. 2. In that case however,the asymptotic behavior
at J =0 is removed (T becomes finite), the instability
2 C
WgiveenobsteatinofTpCarfarmometetrhseJc2r/oJs1sinogr Dpo/iJn1t16h.SizTih=e r0esufoltrias tcousisne-dplaabneovoer)d,earnindgtohcecumrsaxaitmJu2m/J1TC=i2n+creDa/seJs1.(Ians doiusr-
illustrated in Fig. 2, where TC is scaled by the Weiss SWcalculation,TC vanishesatthisinstabilityduetothe
mean-field (MF) result, T(MF) =4J S(S+1)/3. softening of the SW frequencies ωk with k = (π,0) and
C 1 k=(0,π).
Consider first the limiting case with D = 0. Be-
cause the SW stiffness is assumed to be independent
of temperature, the SW approach overestimates T for
C
J /J 1. Notice that T is symmetric on either side III. THE KOHN-LUTTINGER PLUS
2 1 C
≈ EXCHANGE MODEL
of J /J = 1 and drops to zero when J /J = 0 and 2.
2 1 2 1
The limit J /J 0 can be analyzed by considering the
2 1
→
behavior of the SW frequencies for small k. Expanding To estimate the size of the quantum fluctuations in
ωk to second order in k, we find that for small J2/J1, a Mn-doped GaAs quantum well, we model the quasi-
T 4πJ S(S+1/2)/ln(4π2J /J ). This inverse loga- 2D hole gas as a thin quantum well of width L ac-
C 1 1 2
→
rithm is similar to the form discussedin Ref. [15] for the counting for the Coulomb confinement potential arising
HZJ model with J = 0 and small single-ion anisotropy. from the ionized dopants17, central cell corrections, and
2
In that case, T T /ln(π2J /D), where T is the any additional epitaxial confinement. We use a spher-
C 3 1 3
bulktransitiontem→perature,approximatelyproportional ical approximation of the KL Hamiltonian18 with four
to J S(S+1). bands (j kˆ = 3/2 and 1/2) to evaluate the ener-
1
· ± ±
4
gies ǫ(k) for a GaAs quantum well with zero bound- terms in the KL Hamiltonian with matrix elements pro-
αβ
ary conditions at z = L/2. We express the Kohn- portionalto n(k ik )k m ,whichvanishesforn=m
x y z
± h | ± | i
Luttinger Hamiltonian in a reduced basis form by the but is givenby (8i/3L)(k ik ) for n=1 and m=2.
x y
− ±
lowest-energy wavefunctions ψ (z) = 2/Lcos(πz/L)
1 For any coupling constant J and orientation m =
c
andψ2(z)= 2/Lsin(2πz/L)ofthequapntumwell,with (sinθ,0,cosθ) of the Mn spins, the energy E(Jc,θ) of
hcnla|sksz2i|cmalily=an(npdπt/hLe)2Mδnnmi.mTpuhreitMiens aspreindsisatrreibnuotwedtrienatthede tmhaetKrixLEinmjo=de3l/is2oabntdainn,emdb=y1fi,r2stsdpiaacgeo:nalizingan8×8
z = 0 plane with concentration c. Since ψ (0) = 0, the
2
Mn spins only couple to the holes in the first wavefunc-
tion with projection ψ1(0)2 = 2/L. The exchange cou- HKLE = A1 B† . (10)
pling of the Mn spins| S =|Sm with the holes is given (cid:18) B A2 (cid:19)
i i
by V = 2J m j , where j = c† J c /2
− c i i · i i αβ i,α αβ i,β
are the hole spPins and Jαβ are the PauPli spin-3/2 matri- The diagonal block elements can be written in terms of
ces. Thestatesψ andψ arecoupledbytheoff-diagonal a generalized 4 4 matrix,
1 2
×
k2 nπ 2 1
⊥ + +bQ 0 d(k ik )2 0
ǫ x y
 2ma (cid:16) L (cid:17) 2mh − 
k2 nπ 2 1
 0 ⊥ + bQǫ 0 d(kx iky)2 
A0 = 2mb (cid:16) L (cid:17) 2ml − − ,
n  k2 nπ 2 1 
 d(k +ik )2 0 ⊥ + bQ 0 
 x y ǫ 
 2mb (cid:16) L (cid:17) 2ml − 
 k2 nπ 2 1 
 0 d(k +ik )2 0 ⊥ + +bQ 
 x y 2m L 2m ǫ 
 a (cid:16) (cid:17) h 
(11)
sothatA =A0 J m J. WeincludethestraintermQ ǫ (ǫ +ǫ )/2,whichismultipliedbythedeformation
potential1b= 11−.7ecVfo·rGaAs19. Theotherblockdiagoǫn≡alezlezm−entxxissimyyplyA =A0,sincetheholesinψ (z)donot
− 2 2 2
couple to the Mn spins at z =0. We have defined d=√3(1/m 1/m)/8, k =(k ,k ), m =4(3/m +1/m )−1
h l ⊥ x y a l h
and m =4(1/m +3/m )−1. The off-diagonal block elements co−upling ψ (z) and ψ (z) are
b l h 1 2
0 16id(k ik )/3L 0 0
x y
−
16id(k +ik )/3L 0 0 0
B = x y , (12)
0 0 0 16id(k ik )/3L
x y
 0 0 16id(k +ik )/3L − 0− 
 − x y 
which couples the j = 3/2 and 1/2 components. same role as the compressive strain in a thin film.20 For
z
± ±
AnexpressionforJ canbederivedbydirectlycompar- Λ < 20 and typical strains less than 0.5%, the contribu-
c
ingthepotentialV tothestandardspin-holeinteraction, tion to the band splitting from quantum confinement is
e.g. Eq. (1) of Ref. [19]. We estimate much larger than the contribution from strain. In the
limit Λ 0 of an infinitely narrow quantum well, the
S(βN0) →
J c, (13) effects of strain can be neglected entirely. For a nar-
c
≈ 2jΛ row quantum well with Λ < 20, ∆ 29(n/Λ)2 eV and
n
≈
where S = 5/2, j = 3/2, βN 1.2 eV,19, Λ = L/a is J /∆ Λc/29. Hence, a narrow quantum well with
0 c 1
≈ ◦ a small≈Mn concentration is in the weak-coupling limit
thenumberoflayersinthequantumwell,anda 4Ais
the lattice constant of Ga in the z = 0 plane. So≈we see with Jc ≪ ∆1. By contrast, GaAs films with Λ larger
than about 50 cannot be treated as quantum wells be-
that J 1ceV/Λis inverselyproportionalto the width
of thecq≈uantum well. For Λ=10, J 100c meV. cause too many wavefunctions ψn(z) would be required.
c ≈ The dominant contribution to the band splitting in such
Thesplittingbetweenthelightandheavybandmasses
films comes from strain8 rather than the confinement of
at the Γ point of ψ (z) is given by
n the holes.
1 nπ 2 1 1
∆ = 2bQ . (14) After obtaining the eigenvalues of the 8 8 matrix
n ǫ
2(cid:16)Λa(cid:17) (cid:18)ml − mh(cid:19)− HKLE, we evaluate the energy E(J ,θ) of ×a quantum
c
SincebothcontributionsarepositiveforQ <0,thecon- well by integrating over k . Since the hole filling p in
ǫ ⊥
finement of holes in the quantum well formally plays the theregionofperpendicularanisotropyisquitesmall,the
5
holes only occupy a very small portion of the Brillouin
L =10
zone centered around k = 0. This implies that each
⊥ )
hole interacts with many different Mn moments, so that 0 1.5
=
the precise geometry and location of the Mn impurities q c=0.175
(
within the z = 0 plane does not affect our results. Con- J1
/
) 1
sequently, the results of the KLE model for the energy
q0, c=0.350
do not depend on whether the Mn ions are randomly =
distributed within the central plane. The small area in q( 0.5
momentum space occupied by the holes also justifies our J2 c=0.525
use of a spherical approximation18 to the KL Hamilto-
nian. 0
Because each of the holes must interact with two Mn
moments in order to mediate their effective interaction,
1.5
ftuhnectrieosnultoifngJcq.uOanftcuomu-rwsee,llmeangenrgeyticEp(rJocp,eθr)tiiessalinkeevthene 0()J)/1
Kondoeffect do depend onthe signofthe exchangecou- 0 1
=
pling. But in the weak-coupling limit of small Jc, the q( c=0.175
Kondo temperature21 will be extremely small and the J1 c=0.350
0.5
Kondoeffectshallbe neglectedinthe subsequentdiscus- c=0.525
sion.
0
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07
p
IV. ESTIMATION OF THE HEISENBERG
PARAMETERS
FIG. 3: (color online) The top panel shows J (q = 0,θ =
2
0)/J (q = 0) (dashed) and J (q = 0,π/2)/J (q = 0) (solid)
BycomparingthepredictionsoftheKLEandZJmod- 1 2 1
versusp(holesperMn)forthreevaluesoftheMnconcentra-
els, we now estimate the ZJ exchange interactions be-
tion c. In the bottom panel, J (q=0)/J(0) is plotted versus
tween the Mn moments in a GaAs quantum well. In 1 1
p for the same concentrations.
general, obtaining the long-range interactions J(1) and
ij
J(2) that parameterize the ZJ Hamiltonian is not possi-
ij
ble. However, we can estimate the q = 0 component of two random configurations, we conclude that the single-
the isotropic exchange coupling, J (q = 0) = J(1), ion anisotropy D corresponding to the KLE model must
1 j ij
by consideringthe change in energy of the quantPum well vanish. Because the right-hand side of Eq. (16) is inde-
when the exchange with the Mn spins (all aligned along pendentofθwithintheHamiltonianEq.(2),theproxim-
the z direction with angle θ =0) is turned off. For HZJ, ity of J2(q = 0,θ 0) and J2(q = 0,π/2) is a measure
→
this gives the energy to align all of the spins. So we find of how well the KLE model can be approximatedby the
that Heisenberg model HZJ.
Analytic results for the q = 0 exchange parameters
1J (q=0)S2 DS2 = 1 E(J ,θ =0) E(J =0) . can be obtained in the weak-coupling limit of small Jc
1 c c
−2 − Nn − o if only ψ1(z) is occupied by holes and the hole chemical
(15) potential µ πpc/m a2 is also small compared to the
a
The q = 0 component of the anisotropic exchange cou- ≈
band splitting ∆ . For J µ ∆ , we find that
1 c 1
upalitnign,gJt2h(eqc=ha0n)ge=inPejnJei(rj2g)y, mwahyenbealelstoifmtahteedMbnysepvianls- 2JJ1(q(q==00))S,2w=her9eπWJc2/=Wπ2≡c/mJ≪1(0a)S22i≪satnhdeJb2a(nqdw=id0t,hθa)n=d
1 a
rigidly rotate awayfrom the z axis towards the xy plane m is the band mass in the xy plane. Both J (q = 0)
a 1
with angle θ: and J (q = 0,θ) are independent of the hole filling due
2
to the flatness of the 2D density-of-states7. Recall from
1 1 E(J ,θ) E(J ,θ =0)
J (q=0,θ)S2+DS2 = c − c . our discussion above that the system with only nearest-
4 2 N sin2θ neighborinteractionsis unstabletoanantiferromagnetic
(16)
realignmentofthespinsinthexy planewhenJ /J >2.
Within the KLE model, the average exchange vanishes 2 1
So in the limit J µ ∆ , a phase with aligned
when the moments are randomly oriented (whether per- c ≪ ≪ 1
moments is unstable when the exchange interactions are
pendicular or parallel to the plane) because holes with
short-ranged. Neglectingthe effectofanisotropy,J (q=
small momenta interact with many Mn moments. By 1
0) may be used to estimate T from the MF solution of
contrast, the single-ion anisotropy energy D (Sz)2 C
− i i a spin S Heisenberg model:
is large when the moments are randomly orientePd in the
z directionbutvanisheswhenthemomentsarerandomly 1 1 c
oriented in the plane. Comparing the energies of these TC(MF) = 3J1(0)S(S+1)= 3π(βN0)2maa2S(S+1)Λ2
6
c
0.18 eV, (17) So Fig. 3 suggests that the transition temperature of a
≈ Λ2
quantumwellwillbehighestforsomeholefillingslightly
which agrees precisely with the MF result of Lee et al.7 belowp1,whenthemomentsstillliealongthez axisand
J (q = 0) J (q = 0,θ). For c = 0.175, T will be go
Hence, the transition temperature of a quantum well in- 1 ≈ 2 C
through a maximum when p 0.048 holes per Mn.
creases as it becomes narrower. A departure of the con- ≈
V. CONCLUSIONS
centration c(z) of the Mn profile from a delta function
would increase the effective width of the confining po-
tential, thereby lowering the transition temperature. OurresultsimplythatasinglelayerofMn-dopedGaAs
More generally, when both wavefunctions ψ (z) and (or a very thin film with δ-doping) with perpendicular
1
ψ (z) are included, Eqs. (15) and (16) may be used to magneticmomentsmayachieveatransitiontemperature
2
evaluate J (q = 0,θ)/J (q = 0) as a function of p. As close to the MF resultof Eq. (17). Figure 2 suggestthat
2 1
illustrated in Fig. 3, J (q = 0,0)/J (q = 0) initially in- T willreachamaximumwithincreasingpwhenJ (q=
2 1 C 2
creaseswithfillingbeforedecreasingandbecomingnega- 0,θ) J (q = 0). At higher fillings, T is expected
1 C
≈
tiveabovep (indicatingthatthespinswithθ =0areno to decreaseuntil the Mn moments fall into the xy plane,
1
longer locally stable). We find that J (q = 0) increases whereuponcrystalfieldanisotropyisrequiredtostabilize
1
linearlyforsmallpandreachesavalueofabout1.4times long-range ferromagnetic order.
J(0) at a filling p 0.025, independent of c. The max- However, the description of a Mn-doped GaAs quan-
1 ≈
imum in J (q = 0,0)/J (q = 0) approaches 2 for small tum well by a Heisenberg model, even one with long-
2 1
Mn concentrations c and hole fillings p approaching 0, range interactions, is restricted to small Mn concentra-
which is just the limit J µ ∆ discussed above. tions and hole fillings that are above a threshold value.
c 1
≪ ≪
Bycontrast,J (q=0,π/2)/J (q=0)decreaseswithin- FromEq.(13), wesee thatthis is just the weak-coupling
2 1
creasingpandbecomesnegativewhenp>p2 >p1. This limit Jc ≪ ∆1 and Jc ≪ µ. As the Mn concentration
impliesthatthe momentsaretiltedawayfromthe z axis increases, the Heisenberg description is valid over a nar-
for p>p and only fall into the xy plane (E(J ,θ) has a rowerrangeofholefillingsandevenwithinthatrange,is
1 c
minimum at θ =π/2) above some higher filling p >p . not as accurate. Recent work22 on the double-exchange
3 2
Once the spins land on the xy plane above p , the ro- model also reached the conclusion that a mapping onto
3
tational invariance of the spins about the z axis would a Heisenberg model is only valid in the weak-coupling
destroy long-range magnetic order if not for crystal-field limit.
anisotropy within the plane20. ForsmallMnconcentrations,wheretheHeisenbergde-
As remarked earlier, the Heisenberg description of a scription works quite well, quantum fluctuations in the
quantum well requires that J (q = 0,θ) is independent total spin may be at least partially responsible for the
2
oftheangleθ. Hence,Fig.3impliesthatforanynonzero depression of the magnetic moment found in Mn-doped
Mnconcentration,theZJmodelfailswhentheholefilling GaAs epilayers.23 It is ironic that the same frustration
fallsbelowacriticalthreshold. TheZJmodelworksbest mechanism that suppresses the Curie temperature and
overthewidestrangeofholefillingsforsmallMnconcen- electronic polarization in bulk Mn-doped GaAs should
trations. For c = 0.175, the ZJ model is appropriate for permit ordering in a 2D quantum well. With these re-
holefillingsintherange0.025<p<0.07. Ascincreases, sults, we are in a position to address our originally pos-
the Heisenberg descriptiononly workswithin a narrower tulated questions: how can a single layer of Mn-doped
rangeofhole fillings,andeventhere notas well. ForMn GaAs be ferromagnetic,andwhat kindof ferromagnetis
concentrations that are insufficiently small, the interac- it? We conclude that a Mn-doped GaAs quantum well
tions betweenthe Mn momentswithin the quantumwell is ferromagnetic due to a SW gap produced by the dif-
are too complex to be accounted for by the ZJ model, ference between the light and heavy band masses, but
even when the exchange interactions are long-ranged. quantumfluctuationssuppresstheT =0momentofthis
Up to this point, we have not made any assumptions ferromagnet from its fully saturated value.
about the range of the interactions J(1) and J(2) in the It is a pleasure to acknowledge helpful conversations
ij ij
ZJmodel. WhentheMnconcentrationissufficientlylow, with Juana Moreno, Thomas Maier, and Adrian Del
we expectthe nearest-neighborinteractionsJ andJ to Maestro. This research was sponsored by the U.S. De-
1 2
be the dominant ones and our results for J (q = 0) and partmentofEnergyDivisionofMaterialsScienceandEn-
1
J (q=0,θ) can be used to estimate the transition tem- gineeringundercontractDE-AC05-00OR22725withOak
2
peratureof aquantum well. In SectionII, we foundthat Ridge National Laboratory, managed by UT-Battelle,
thetransitiontemperaturereachesapeakwhenJ =J . LLC.
2 1
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