Ferromagnetic properties of p-(Cd,Mn)Te quantum wells: Interpretation of
magneto-optical measurements by Monte Carlo simulations
A. Lipinska,1 C. Simserides,2 K. N. Trohidou,2 M. Goryca,3 P. Kossacki,3 A. Majhofer,3 and T. Dietl1, 4
1Institute of Physics, Polish Academy of Science,
al. Lotnikow 32/46, PL 02-668 Warszawa, Poland
2Institute of Materials Science, NCSR Demokritos, GR-15310 Athens, Greece
3Institute of Experimental Physics, University of Warsaw, ul. Ho_za 69, PL 00-681 Warszawa, Poland
4Institute of Theoretical Physics, University of Warsaw, ul. Ho_za 69, PL 00-681 Warszawa, Poland
(Dated: March 2, 2009)
In order to single out dominant phenomena that account for carrier-controlled magnetism in p-
Cd1xMnxTe quantum wells we have carried out magneto-optical measurements and Monte Carlo
simulations of time dependent magnetization. The experimental results show that magnetization
relaxation is faster than 20 ns in the paramagnetic state. Decreasing temperature below the Curie
temperature TC results in an increase of the relaxation time but to less than 10 s. This fast
relaxation may explain why the spontaneous spin splitting of electronic states is not accompanied
by the presence of non-zero macroscopic magnetization below TC. Our Monte Carlo results repro-
duce the relative change of the relaxation time on decreasing temperature. At the same time, the
numerical calculations demonstrate that antiferromagnetic spin-spin interactions, which compete
with the hole-mediated long-range ferromagnetic coupling, play an important role in magnetization
relaxation of the system. We nd, in particular, that magnetization dynamics is largely accelerated
by the presence of antiferromagnetic couplings to the Mn spins located outside the region, where
the holes reside. This suggests that macroscopic spontaneous magnetization should be observable
if the thickness of the layer containing localized spins will be smaller than the extension of the hole
wave function. Furthermore, we study how a spin-independent part of the Mn potential aects TC.
Our ndings show that the alloy disorder potential tends to reduce TC, the eect being particularly
strong for the attractive potential that leads to hole localization.
PACS numbers: 75.50.Pp,05.10.Ln,75.30.Et,78.55.Et
I. INTRODUCTION
Modulation-doped p-type (Cd,Mn)Te/(Cd,Mg,Zn)Te
quantum wells (QWs) remain a unique medium allow-
ing to probe carrier-induced Ising-like ferromagnetism in
the two-dimensional (2D) case,1,2,3,4,5 as in this system
the mean free path is longer than the QW width. The
presence of ferromagnetism was revealed by the obser-
vation of spontaneous splitting of the photoluminescence
(PL) spectrum, which indicated that local spin order-
ing is stable, below the Curie temperature TC, for times
longer than the exciton lifetime.2,6 Theoretical analyzes
have shown that the magnitude of TC as a function of
the Mn concentration x and the hole areal density p can
be quantitatively described provided that in addition to
carrier-mediated ferromagnetic spin-spin couplings, the
presence of competing short-range antiferromagnetic su-
perexchange interactions as well as hole correlation ef-
fects are taken into account.1,6,7,8,9 These experiments
and their quantitative interpretations were carried out
prior to later theoretical studies.10,11,12
A surprising result of both optical and magnetic mea-
surements is the absence of hysteresis loops and, hence,
of macroscopic spontaneous magnetization below TC.2,3
This nding questions our understanding of the actual
ground state of the system. It has been suggested, in par-
ticular, that the formation of spin density waves, driven
by the q-dependent carrier susceptibility, may account for
the absence of spontaneous magnetization.2 This possi-
bility has recently been considered in the context of the-
oretical search for carrier-mediated nuclear magnetism in
2D systems.13
In order to determine the importance of various
phenomena that control magnetism in such reduced-
dimensionality magnetically disordered systems we have
carried out time-resolved magneto-optical measurements.
Furthermore, with the goal to obtain information on
mechanisms controlling spin dynamics, we have extended
our previous9 Monte Carlo (MC) simulations of magneti-
zation in p Cd1xMnxTe QW. The experimental nd-
ings provide a dependence of magnetization relaxation
on temperature and carrier density. Above TC the relax-
ation time is found to be shorter than 20 ns, while below
TC magnetization persists up to a few microseconds, a
time scale consistent with the absence of spontaneous
magnetization in the static measurements. Interestingly,
we have obtained a similar relative prolongation of the
relaxation time below TC from MC simulations based on
the Metropolis algorithm and on the determination of
the one-particle hole eigenfunctions at each MC sweep.
The simulations explain also the rate of increase of the
relaxation time with the carrier density. We conclude
that the absence of magnetic hysteresis in this 2D Ising
system can be explained without taking carrier-carrier
correlation into account.
At the same time, by analyzing results of our Monte
Carlo simulations, we are able to show that short range
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2spin-spin antiferromagnetic (AFM) interactions play a
crucial role in accelerating magnetization dynamics. This
corroborates the outcome of the previous results showing
a narrowing of the hysteresis loop by AFM interactions.9
However, we realize now, based on much more exten-
sive simulations, that the eect of AFM interactions be-
comes much reduced if the thickness of the layer contain-
ing Mn spins is taken to be smaller than the extent of
the hole wave function. This nding, from the one hand,
substantiates theoretical considerations of Boudinet and
Bastard,14 suggesting that magnetization relaxation of
bound magnetic polarons in p-type CdTe/(Cd,Mn)Te
QWs occurs owing to the AFM coupling to the Mn spins
located outside the relevant Bohr radius. On the other,
this result implies that macroscopic spontaneous magne-
tization and magnetic hysteresis should be recovered in
quantum wells, in which the thickness of the Mn layer
would be smaller than the region visited by the holes.
Furthermore, we study how a spin-independent part
of the potential introduced by Mn impurities aects TC.
Our simulations show that alloy disorder tends to reduce
TC. The eect is particularly dramatic for the attrac-
tive alloy potential which, if suciently large, leads to
a strong hole localization. This result substantiates the
notion that delocalized or weakly localized carriers are
necessary to generate a sizable ferromagnetic coupling
between diluted localized spins.
This article is organized as follows. In Sec. II we
present experimental ndings of magneto-optical studies
on magnetization dynamics, which have motivated the
theoretical eort. The theoretical model that describes
the carrier-induced ferromagnetism in diluted magnetic
semiconductor (DMS) quantum wells is introduced in
Sec. III. In Sec. IV we present MC investigations of the
dependence of the critical temperature on the carrier den-
sity and spin-independent alloy potential. Section V con-
tains results of our MC studies of spin dynamics and
of spin ordering, which serve to interpret the ndings
of magneto-optical measurements. Specically, subsec-
tion V A is devoted to magnetization dynamics, while
subsection V B contains a discussion of possible domain
sizes. Our conclusions are summarized in Sec. VI.
II. EXPERIMENTAL RESULTS
The experimental results have been obtained for sam-
ples grown in Grenoble by molecular beam epitaxy on
Cd0:88Zn0:12Te (001) substrates.1,15 The modulation-
doped p-type structures contain a single 8 nm wide
Cd1xMnxTe QW embedded by Cd1yzMgyZnzTe bar-
riers, in which the Mg content (y = 0.25-0.28) results in a
sizable valence band oset, while the presence of Zn (z =
0.07{0.08) ensures a good lattice match to the substrate.
The front barrier is doped by nitrogen acceptors. The
distance between the QW and the doping layer is 20 nm,
which results in the hole density up to 3  1011cm2.15
These holes occupy the ground state subband whose scat-
tering broadening is much smaller than the distance to
the next subband, making the system to be truly 2D.
According to the well established procedure,
magnetic properties of the QW are probed by
magnetospectroscopy.1,2,6 The photoluminescence
(PL) is excited by a cw laser with the photon energies
below the absorption edge of the barrier material. A typ-
ical PL spectrum of a paramagnetic p-type QW consists
of a single line related to the radiative recombination of
charged excitons (X+). Depending on its polarization
chirality, the emission line exhibits a downward or
upward shift in the magnetic eld. Owing to a strong
sp-d exchange interaction specic to DMSs, this Zeeman
shift is giant (typically over 20 meV in 1 T), and its
magnitude is proportional to the local magnetization
of the Mn spins. Importantly, a combination of strain,
connement, and spin-orbit interaction makes that the
giant shift vanishes virtually entirely for the in-plane
magnetic eld. Below a critical temperature, which we
identify as the Curie temperature TC, a spontaneous
splitting of the PL line into two components is observed
in the absence of an external magnetic eld. The energy
distance between these two lines, or the downward shift
of the lower line, provides information on an average
value of spontaneous magnetization in the regions visited
by the holes at given temperature.
Surprisingly, the emitted light at the wavelength corre-
sponding to either of these two lines remains unpolarized,
even after ramping the eld down to zero below TC. A
magnetic eld of the order of 20 mT, much higher than
the demagnetization eld for this diluted alloy, is nec-
essary to achieve a full circular polarization of the lines
at T  0:5TC.2 This indicates the absence of a macro-
scopic spontaneous magnetization at these low tempera-
tures, the conclusion conrmed by a direct magnetization
measurements.16 Moreover, even focusing detection on
the spot with diameter below 1 m does not result in the
circularly polarized emission. This could be explained ei-
ther by a small size of the relevant magnetic domains or
by fast
uctuations of the magnetization direction. Al-
ternatively, a spin-density wave could be considered as
the relevant ground state of this reduced dimensionality
correlated system. However, a selective excitation of PL
by circularly polarized light leads to circularly polarized
PL.3 This indicates that local spin ordering is stable for
times longer than the exciton lifetime, estimated to be in
the 100 ps range.
Some of us have recently developed a technique for
studies of magnetization dynamics by means of time re-
solved PL measurements performed after a short pulse
of the magnetic eld.4,5 Pulses of the magnetic eld are
produced by a magnetic coil mounted at the surface of
the sample. The illumination with a laser beam and the
collection of the PL signal are performed along the axis
of the coil (Faraday conguration). The coil diameter of
0.5 mm results in a small value of inductance and allows
to obtain short rise and fall times of about 10 ns. A
2 A current produces a magnetic eld of about 40 mT.

3FIG. 1: Experimental: Temporal decay of magnetization in
the Cd0:96Mn0:04Te quantum well at various temperatures
determined as a dierence in photoluminescence intensities
collected for two circular polarizations after a pulse of the
magnetic eld. The Curie temperature is 2.5 K.
The time evolution of the PL during and after the pulse
is probed with resolution down to 10 ns. Magnetization
dynamics is studied by analyzing the dierence in PL in-
tensities for two circular polarizations after the pulse of
the magnetic eld. Results of measurements shown here
have been obtained at dierent temperatures below and
above TC, which is 2.5 K for the studied Cd0:96Mn0:04Te
QW.
Above TC, in the paramagnetic state, a single PL line
is observed at zero eld, and the signal induced by the
magnetic pulse is related to Zeeman splitting of this line.
Its relaxation time is found to be shorter than 20 ns. Such
a value is just to be expected for (Cd,Mn)Te containing
4% of Mn.17,18
Typical temporal proles for three dierent tempera-
tures below TC = 2:5 K are presented in Fig. 1. The
relaxation time becomes much slower when decreasing
temperature (see Fig. 1). Magnetization persists up to a
few microseconds when T  0:5TC. However even for the
lowest temperatures, the relaxation time is shorter than
10 s, and under all experimental conditions the rema-
nent magnetization vanishes totally after 20 s. This
time scale is much shorter than an acquisition time of
standard optical measurements, which accounts for un-
measurably small values of the coercive eld observed
experimentally.
It is interesting to note that the decays are not mono-
exponential. The relaxation begins with a rapid decay,
faster than 100 ns, but it also contains a slower compo-
nent, in the microsecond range. One could expect that
regions of higher hole density have higher local magne-
tization resulting in higher barriers for the magnetiza-
tion reversal. In order to examine this possibility we
carried out relaxation measurements for a constant tem-
FIG. 2: Experimental: Temporal decay of magnetization in
the Cd0:96Mn0:04Te quantum well determined as a dierence
in photoluminescence intensities for two circular polarizations
collected at various wavelengths after a pulse of the magnetic
eld. The Curie temperature is 2.5 K.
perature (below TC), and dierent detection wavelength
which was tuned over the low energy component of the
PL line. As mentioned above, this line shows a red shift
when decreasing temperature below TC. This shift is
proportional to Mn magnetization, whose magnitude is
primarily determined by the hole density.6,15 Due to spa-
tial
uctuations in the hole density,5 the line is signi-
cantly broadened. The low energy side of the line comes
from the regions containing higher hole densities, while
the high energy side corresponds to locations visited by
fewer holes. This provides an opportunity to trace the
relaxation time as a function of the hole concentration.
The obtained temporal magnetization proles are pre-
sented in Fig. 2. It is seen that the contribution of the
slow component increases for the regions containing a
high hole density. This nding as well as the observed
temperature variation of the magnetization decay time
will be compared to results of numerical simulations in
the next section.
III. THEORY AND SIMULATION PROCEDURE
To understand the experimental results described in
Sec. II we carried out Monte Carlo simulations for the
subsystem of Mn spins which are at the same time subject
to short-range antiferromagnetic and long-range hole-
mediated ferromagnetic spin-spin interactions. Accord-
ing to our results, the presence of these two kinds of com-
peting interactions accounts for a non-standard character
of ferromagnetism in the studied system. Since we are
mainly interested in modulation-doped p-Cd1xMnxTe
quantum wells, all our results are obtained using the
parameter values characteristic for this system, but the

4methodology we use may as well be applied to other 2D
p-type II-VI DMS structures.
The antiferromagnetic interactions originate from su-
perexchange, the mechanism specic to undoped charge
transfer insulators, such as (Cd,Mn)Te. The character
and magnitude of these interactions are known from pre-
vious extensive experimental studies of magnetic proper-
ties of (Cd,Mn)Te. In order to nd out how the presence
of holes aects the system we compute, in the spirit of the
adiabatic approximation, the total energy of the holes at
a given conguration of the Mn spins. In this way our
approach encompasses automatically the description of
carrier-mediated exchange interactions within either p-d
Zener model or Ruderman-Kittel-Kasuya-Yosida theory.
The carrier total energy is evaluated neglecting hole cor-
relations, the approximation that leads to an underesti-
mation of the ferromagnetic coupling.7 Accordingly, our
computed Curie temperatures are expected to be system-
atically lower than those observed experimentally. Fur-
thermore, in accord with the eective mass theory, and
owing to a relatively small disorder in our structures,
we assume that the in-plane and the perpendicular hole
motions can be factorized, i.e., that the total envelope
function assumes the form (R) = (r)'(z), where
R = (r; z) with in-plane and perpendicular positions de-
scribed by r = (x; y) and z respectively; '(z) is the wave
function of a hole residing in the ground state subband of
the QW in question. For the biaxial strain under consid-
eration, the relevant subband is of the heavy hole origin,
so that the spin-orbit interaction xes the hole spin in
the direction of the growth axis. Furthermore, since the
heavy holes penetrate only weakly the barriers, '(z) in
the form corresponding to an innite QW is adequate for
our computations.
Within the above model and in the absence of an ex-
ternal magnetic eld, the Hamiltonian H of our system
takes the form,
H =
p2
2m
+Hpd +Hdd: (1)
This Hamiltonian describes noninteracting carriers in the
heavy hole parabolic valence subband for which the in-
plane eective mass is m = 0:25m0 for the QW in
question.19 Since the coupling between the holes and the
Mn ions is weak in (Cd,Mn)Te,20 we can consider the p-d
interaction in a simple contact form,
Hpd =
X
i
1
3
jzSzi(r ri) j'(zi)j
2 ; (2)
where the hole spin j = 3=2, jz =  3/2, is coupled to the
Mn ions randomly distributed over the fcc lattice sites ri.
We treat Mn spins Si as classical vectors with S = 5=2.
While this approximation is certainly qualitatively valid
for the large spin in question, quantitatively { according
to the mean-eld theory { it leads to an underestima-
tion of the TC value by (S + 1)=S. For the p-d exchange
integral we take  = 5:96 102 eV nm3, which corre-
sponds to the exchange energy N0 = 0:88 eV, where
N0 is the cation concentration.21
We have also tested how our results are aected by
the spin-independent part of the potentials introduced
by Mn ions. To take this alloy disorder into account we
add to Eq. 2 a term
HAD = V
X
i
(r ri) j'(zi)j
2 (3)
with V = 3:93  102 eV nm3 which corresponds to
the valence band oset WN0 = V N0 = 0:58 eV in
(Cd,Mn)Te.22,23 It is known that the eect of the Mn
isoelectronic impurities on the carrier wave function is
governed by the ratio of the total Mn potential U to its
critical value Uc < 0 at which a bound state starts to
form,20
U=Uc = 6m
[W (S + 1)=2]=(3~2b); (4)
where b is the potential radius. In the case under consid-
eration, the hole spin is xed along z-axis, and Mn po-
larization hSzi is typically below 10%. Accordingly, for
S+ 1! hSzi the total Mn potential is expected to be ef-
fectively repulsive for the holes in (Cd,Mn)Te, U=Uc < 0.
Interestingly, in many important systems, such as oxides
and nitrides, the spin-independent part of the magnetic
impurity potential is actually attractive.20 Accordingly,
we have also performed some simulations of magnetiza-
tion for V = 3:93  102 eV nm3 which corresponds
to the valence band oset WN0 = V N0 = +0:58 eV.
However, it should be noted that by using the -like po-
tentials, which corresponds to b! 0, we actually overes-
timate U=Uc and determine an upper limit of the eect
of the alloy potential upon TC. With this in mind, we
have carried out all MC dynamics studies presented in
Sec. V for V = 0.
The short-range intrinsic antiferromagnetic (AFM) in-
teraction between the Mn spins is given by,
Hdd = 2kB
X
ij
JijSi
Sj ; (5)
with Jij = 6:3;1:9;0:4 K for the nearest, next near-
est, and next next nearest neighbors, respectively.24 Our
MC simulations conrm that these values of Jij repro-
duce correctly the temperature dependence of the spin
susceptibility in the absence of holes but we note that
other sets of Jij values have also been proposed in the
literature.25
The above approach disregards the presence of long-
range dipole-dipole interactions between Mn spins. Both
theoretical evaluations and MC simulations we have car-
ried out show that the dipole-diploe interactions can be
safely ignored in the case under consideration.
In order to determine hole eigenfunctions and eigen-
values for a given conguration of Mn spins we assume

5periodic boundary conditions in the QW atomic lay-
ers and diagonalize H in a plane-wave basis with 2D
wave vectors truncated at a radius kc = 5:52(2=L) or
kc = 8:10(2=L).26 This part of the 2D k-space cor-
responding to 97 or 213 k-states, respectively, is su-
cient to ensure the convergence for all hole density values
considered here. The energy of the holes is determined
by summing up the lowest eigenvalues corresponding to
a given number of holes Nh. This procedure assumes
that the hole gas is degenerate, i.e., neglects thermal
broadening of the hole distribution, the eect considered
previously,6 and found to be small. In order to keep k-
space shells lled up, we have considered Nh = 1, 5, 9,
13, 21, 25, 29, 37, 45, 49, and 57, which correspond to
values of hole concentrations p up to 1:11  1011 cm2
for the chosen size of the simulation box.
In our MC simulations we employ the Metropolis al-
gorithm in which the hole eigenfunctions and eigenvalues
are updated at each MC sweep (in one MC sweep all Mn
spins are rotated). This procedure if applied after each
single spin rotation would be computationally excessively
time consuming and, therefore, we followed the idea of
the so-called perturbative Monte Carlo method.27,28,29
We typically keep 2000 initial MC sweeps (to thermalize
the system), followed by 104-105 MC sweeps used for a
further analysis. Most of the numerical results presented
in this paper have been obtained for the magnetic eld
put to zero. However, our software allows as well simula-
tions in non-zero magnetic elds. We have occasionally
used this opportunity to obtain some of the initial con-
gurations.
For the computations presented here we adopt the sim-
ulation box of dimensions LLLW , where L  350ao
and LW = 8ao, with the fcc lattice constant ao =
0:647 nm in our case. Hence, the quantum well width
is LW = 5:2 nm and the area L2 =(226 nm)2. We
note that a typical QW width of experimental samples
is somewhat grater, LexpW = 8 nm. However, the adop-
tion of a smaller value of LW allows us to treat systems
with a larger area and, at the same time, since within
the mean-eld theory7 TC / 1=LW , results in a partial
compensation of a systematic error stemming from treat-
ing the spins classically, LexpW S=[LW (S + 1)] = 1:1. We
randomly occupy 4% of cations sites, which corresponds
to 156800 Mn spins. In view of the required accuracy,
this large system size makes averaging over the Mn dis-
tribution and extrapolation to an innite system size un-
necessary. The number of Monte Carlo sweeps needed
for the convergence is of the order of 104 to 105. Hence,
we have to use a (pseudo)random number generator with
a suciently long period. After testing various genera-
tors, the Mersenne Twister developed by Matsumoto and
Nishimura has been selected, because of its huge period
and its very high order of dimensional equidistribution.30
IV. CURIE TEMPERATURE
The critical temperature TC has been determined in
the standard way: by MC simulations we have calculated
the temperature dependence of the spin projections and
the spin susceptibility for both hole and magnetic ion
spins. To denote spins we use the notation  = s (for
holes) and  = S (for Mn ions). Similarly, N = Nh (for
the number of holes) and N = NMn (for the number of
Mn ions). At each MC sweep, the spin projections per
hole or per Mn ion are given by:
j =
PN
i=1 ij
N
; j = x; y; z: (6)
We symbolize statistical averages of the above quantities
by h:::i.31 In other words, e.g.
hji =
Pnsum
n=1 j
nsum
(7)
and
hjj ji =
Pnsum
n=1 jj j
nsum
; (8)
where n denotes successive MC sweeps used for the statis-
tical average and nsum denotes the total number of MC
sweeps used for the statistical average. The spin suscep-
tibilities per spin are dened in the following manner:31
j =
1
T
[h2j i hji
2]: (9)
We associate the temperature at which the magnitude
of spin susceptibility reaches a maximum with the Curie
temperature TC of the system. Usually, the MC com-
putations have been performed with the step of 0.2 K.
Therefore, an uncertainty in TC values is usually at least
0:1 K. Some examples of the temperature dependence
of the spin projections and of the spin susceptibilities are
presented in Fig. 3. Specically, we have plotted hjSzji
and S for the Mn ions (Figs. 3(a) and 3(c)) as well
as hjshji and h for the holes (Fig. 3(b) and 3(d)), for
the hole numbers Nh = 13 and 29, which correspond to
p = 2:54  1010 and 5:66  1010 cm2, respectively. In
Fig. 3 the case when the alloy disorder is absent (noAD)
is compared to the situation of the repulsive (+AD) al-
loy disorder potential. In Fig. 4 we show the eect of
the attractive (-AD) alloy disorder potential by referring
to the data obtained without alloy disorder (noAD), for
Nh = 13 and 45, i.e., p = 2:54  1010 and 8:78  1010
cm2, respectively.
In Fig. 5 we summarize our MC results for TC as a
function of the hole concentration obtained neglecting
alloy disorder (V = 0) as well as for V = 3:93 
102 eV nm3, corresponding to the repulsive and at-
tractive alloy potential introduced by Mn, respectively.

6FIG. 3: Monte Carlo simulations: the z-component of the absolute spin projections hjSzji and hjszji (open symbols - dotted
lines; left scale) and the spin susceptibilities S and s (full symbols - solid lines; right scale) for the Mn ions (a,c) and for the
holes (b,d), in the absence of alloy disorder (noAD, V = 0), and for the repulsive (+AD, V > 0) alloy disorder potential. A
maximum in the temperature dependence of spin susceptibility is identied as the Curie temperature below which the holes
induce a long range ferromagnetic order. Here the number of holes Nh = 13 (a,b) and 29 (c,d), i.e., p = 2:54  1010 and
5:66 1010 cm2, respectively.
We see that the attractive potential washes the ferro-
magnetism virtually entirely out. A contour plot of hole
eigenfunctions demonstrates that holes get localized for
the chosen magnitude of V . Our results substantiate
therefore the view that delocalized or weakly localized
carriers are indispensable to set a long-range order be-
tween diluted spins. Obviously, smaller amplitude of the
attractive potential or greater values of hole concentra-
tions will lead to carrier delocalization and the reentrance
of a ferromagnetic order.
Interestingly, as seen in Fig. 5, the repulsive potential
V > 0, the case of (Cd,Mn)Te, also leads to reduced
magnitudes of TC comparing to the values determined
for V = 0. We interpret this nding by noting that in
the presence of a repulsive potential, the amplitude of
the wave function at Mn ions is diminished comparing to
the case V = 0. This eectively reduces the p-d cou-
pling and shifts the appearance of the carrier-mediated
ferromagnetic order to lower temperatures.
It is instructive to compare these MC ndings
to the expectations of the mean-eld approximation
(MFA),1,6,7,9 derived neglecting entirely chemical and
thermal
uctuations. We see in Fig. 5. that, somewhat
fortunately, the MC data for V = 0 and MFA results
for classical spins (solid line) are in a good quantitative
agreement at high hole densities. In particular, both ap-
proaches predict that the magnitude of TC does not vary
with the hole density, the result re
ecting the energy in-
dependence of the density of states in the 2D case. How-
ever, at low hole densities, TC values obtained from the
MC simulations tend to decrease, the eect associated
with a broadening of the density of states induced by
hole scattering on Mn spins6,7 and encompassed by our
MC simulations. The scattering-induced lowering of TC
is even more apparent if V > 0.
Owing to mutual compensations between ferromag-

7FIG. 4: Monte Carlo simulations: the z-component of the absolute spin projections hjSzji and hjszji (open symbols - dotted
lines; left scale) and the spin susceptibilities S and s (full symbols - solid lines; right scale) for the Mn ions (a,c) and for the
holes (b,d), in the absence of alloy disorder (noAD, V = 0), and for the attractive (-AD, V < 0) alloy disorder potential. A
maximum in the temperature dependence of spin susceptibility is identied as the Curie temperature below which the holes
induce a long range ferromagnetic order. Here the number of holes Nh = 13 (a,b) and 45 (c,d), i.e., p = 2:54  1010 and
8:78 1010 cm2, respectively.
netic and antiferromagnetic interactions, the magnitude
of TC is rather sensitive to the presence of hole corre-
lations. This is shown by dashed line in Fig. 5, which
has been obtained by incorporating into theory within
the MFA the Fermi liquid Landau parameter AF = 2.
This procedure increases the ferromagnetic contribution
to TC by a factor of two but since carrier correlations do
not aect the compensating AFM term, the resulting in-
crease of TC is actually much greater. Therefore, in order
to compare MC results to experimental ndings we treat
AF as a tting parameter. The theoretical values of TC
displayed in Fig. 6 has been determined from,1,6,7,9
(TC) = (T
(MC)
C )=AF : (10)
Here (T )  T is the spin susceptibility of the an-
tiferromagnetically coupled Mn spins in the absence of
the holes, where  = 0:79 according to our MC simula-
tion, and T (MC)C is the Curie temperature at a given hole
concentration, determined from the MC simulations that
neglect carrier correlations (AF = 1). We see that for
p = 1  1011 cm2 by taking AF = 1:5 and AF = 3 for
the case V = 0 and V = 3:93  102 eV nm3, respec-
tively we obtain a good description of the experimental
ndings. In particular, a decrease of TC values with low-
ering of the hole density observed experimentally6 and
also visible in our MC simulations, results from scatter-
ing broadening of the density of states,7 particularly rel-
evant at low Fermi energies. However, the experiment
implies that this lowering extents to higher concentra-
tions than the range predicted by the simulations, which
indicates that additional scattering mechanisms operate
in real samples.
Having discussed the ferromagnetic ordering tempera-
ture TC, we turn to the temperature dependence of Mn
and hole spin projections, as depicted in Figs. 3 and 4.

8FIG. 5: Curie temperature TC as a function of the hole area
density p. Solid symbols show results of Monte Carlo simula-
tions carried out for classical spins, QW width LW  5:2 nm,
and Mn content 4% neglecting carrier correlations in the ab-
sence of alloy disorder (V = 0, squares) as well as for re-
pulsive and attractive alloy potentials (V > 0, triangles
and V < 0 circles, respectively). Results obtained within
the mean-eld approximation (i.e., disregarding thermal and
chemical
uctuations) are shown by the solid line. The dashed
line presents the MFA results taking into account carrier cor-
relations with the Landau parameter AF = 2.
We see that the holes become entirely spin polarized,
hjszji ! 1=2, immediately below TC. On the other hand,
the increase of the Mn spin projection hjSzji on lower-
ing temperature is much slower. This is because, the
molecular eld produced by the spin polarized carriers is
typically below 1 kOe, much too small, even at 0.1 K,
to saturate Mn spins which are coupled by AFM inter-
actions.
In Fig. 7 we compare the magnitudes of the Mn spin
projection hjSzji from our MC simulations at 0.1 K with
the values expected from the MFA for the saturated car-
rier spins,6,7
hjSzji(T ) =
21=2S2p(T )
6kBTo(T )LW
; (11)
where (T ) and o(T ) are the values of Mn spin sus-
ceptibilities per Mn ion computed without holes, in the
presence and in the absence of the AF interactions, re-
spectively. We see that the MFA quite correctly repro-
duces the computed magnitudes of hjSzji as a function of
the areal hole concentration p. Furthermore, while the
values of TC are systematically smaller in the presence
of the repulsive alloy potential comparing to the case
V N0 = 0, the magnitude of hjSzji is seen to be larger
in this case. We assign this surprising result to the fact
that the repulsive alloy potential delocalizes majority-
spin holes, which can therefore embrace more eectively
FIG. 6: Curie temperature TC as a function of the hole areal
density p. Solid symbols show results of Monte Carlo simula-
tions carried out for classical spins, QW width LW  5:2 nm,
and the Mn content 4% in the absence of alloy disorder
(V = 0, squares) and for repulsive alloy potentials (V > 0,
triangles) obtained with the Fermi liquid Landau parameters
AF = 1:5 and AF = 3. Experimental results6 for QW width
LW  8 nm and Mn content 3-5% for two samples (open
circles and triangles).
FIG. 7: Monte Carlo simulations: z-component of the abso-
lute spin projection of Mn spins hjSzji at 0.1 K in the absence
of alloy disorder (V = 0, squares) and for repulsive alloy
potential (V > 0, triangles). The dashed straight line repre-
sents a dependence expected within the MFA.
the magnetic impurities. However, as the hole density p
increases, the data with and without alloy potential tend
to approach. This is because the dense majority-spin
holes can embrace more eectively the magnetic impuri-
ties even without help from the repulsive potential. In
the other extreme, when the values of p decreases to-
wards zero, the data with and without alloy potential

9FIG. 8: Monte Carlo simulations: Time evolution of
SzMn(t)=S at dierent temperatures in a Cd0:96Mn0:04Te
quantum well. The hole concentration is p = 0:41011 cm2
(Nh = 21). Time is measured in the Monte Carlo steps per
Mn site.
approach, too. Here, the TC magnitude becomes so low
that the carriers' spins ceases to be entirely saturated in
the presence of the repulsive potential, which lowers the
corresponding hjSzji value.
V. INTERPRETATION OF
MAGNETO-OPTICAL MEASUREMENTS
In this section we present our MC results on Mn mag-
netization dynamics as a function of temperature and
carrier concentration. We also discuss whether it is pos-
sible within the proposed model to estimate the typical
size of magnetic domains.
A. Magnetization dynamics: autocorrelation
function
We have traced time dependence of Mn magnetization
in the absence of an external magnetic eld for very long
runs up to 105 MC sweeps. We have repeated our sim-
ulations for dierent initial congurations including the
one with all Mn spins aligned along the z-axis. From
our ndings we conclude that except for the rst few
MC sweeps the whole time evolution of magnetization
does not depend on the initial conguration. The typi-
cal time-evolution of magnetization at various tempera-
tures is shown in Fig. 8. For temperatures well below TC
the system remains almost unchanged during the whole
simulation time. For higher temperatures one can ob-
serve occasional global spin inversions: the system
ips
over between two states of equal energy and spontaneous
magnetization that dier only in sign.
FIG. 9: Monte Carlo simulations: The magnetization au-
tocorrelation function G() of Mn ions in a Cd0:96Mn0:04Te
quantum well calculated at dierent temperatures. The con-
centration of holes p = 0:4  1011 cm2 (Nh = 21). Time is
measured in the Monte Carlo steps per one Mn site. The thin
solid lines describe exponential decays tted to the Monte
Carlo results
The typical time scale of the observed evolution de-
pends on many factors including temperature and car-
rier concentration. To determine this characteristic time
scale we calculate the time-autocorrelation G() of mag-
netization given by31
G() =
hm(t)m(t+ )i hmi2
hm2i hmi2
; (12)
where m(t) is the value of magnetization at time t and
hmi is its average value. The autocorrelation function
dened by Eq. 12 depends only on the time dierence
 . To calculate this autocorrelation function for up to
 = 104 MCS computation runs as long as at least 105
MCS are needed. The total computer time depends on
numerical parameters of which the most important is the
k-space cut-o.
Figure 9 shows the magnetization autocorrelation func-
tions calculated for the studied system at various tem-
peratures below and above TC. Above TC, i.e., in the
paramagnetic state, one can observe a very fast decay
of the autocorrelation function whereas for temperatures
below TC correlations persist for times orders of magni-
tude longer. Within the present MC approach, it is not
possible to translate numbers of MC steps into physical
time units in a reliable quantitative way. The qualita-
tive relations between the time scales of the investigated
magnetic relaxation processes should nevertheless be re-
produced in a correct way. Therefore, we undertake only
a qualitative comparison with the experimental data of
Sec. II.

10
FIG. 10: Monte Carlo simulations: The magnetization auto-
correlation function G() for selected values of hole concen-
trations in a Cd0:96Mn0:04Te quantum well at T = 0:7 K.
Time is measured in the Monte Carlo steps per Mn site.
In order to better understand mechanisms that account
for magnetization dynamics in a system of competing
spin-spin AFM and carrier-mediated ferromagnetic inter-
actions we have also calculated autocorrelation functions
for a variety of carrier concentration values keeping the
number of Mn ions unchanged. In Fig. 10 we display the
corresponding results calculated for three dierent carrier
concentrations at the same temperature T = 0:7 K. As
one could expect we do observe a signicant slowing down
of magnetization dynamics with increasing the number of
holes in the QW. Such behavior is understandable within
the proposed model. Taking into account the magnetic
anisotropy { all hole-spins align along z-axis { and the
fact that only the z-component of each Mn-spin couples
via the p-d exchange to the system of holes, one can see
that if more holes are present in the system, the value
of local magnetization and, hence, the barrier height
for magnetization reversal become higher. Note that
for the carrier concentrations p = 4:10
1010 cm2 and
p = 8:78
1010 cm2 the system is in the ferromagnetic
state at T = 0:7 K. In contrast, for p = 1:75
1010 cm2
the Curie temperature lies below T = 0:7 K (cf. next
sections), so that the corresponding points describe the
dynamics of magnetization in a paramagnetic QW.
A next important aspect one must understand in order
to interpret properly the experimental ndings is the way
in which the AFM interactions in
uence the magnetiza-
tion dynamics. Therefore, we have calculated the magne-
tization autocorrelation function for the system where all
short-range AFM interactions are disregarded. In Fig. 11
we compare the obtained magnetic autocorrelation func-
tions for systems with and without AFM interactions
calculated at T = 0:7TC (for each system its own TC
as previously determined by MC simulations9 had been
used). The results presented in Fig. 11 clearly indicate
FIG. 11: Monte Carlo simulations: The magnetization au-
tocorrelation function G() of Mn ions in a Cd0:96Mn0:04Te
quantum well at T = 0:7TC. The hole density is p =
0:4  1011 cm2 (Nh = 21). Open circles and full squares:
antiferromagnetic interactions are neglected (Jij = 0) and
taken into account (Jij 6= 0), respectively. Time is measured
in the Monte Carlo steps per Mn site.
that short-range AFM interactions may strongly accel-
erate the decay of the ferromagnetic order. In accord
with the above observation one can expect that the ef-
fect of AFM interactions is considerably reduced when
the layer of Mn ions is thinner than the extend of the
hole wave function, in other words, when Mn ions are
concentrated close to the maximum (probability) den-
sity of holes. To check this we repeated our calculations
with all Mn ions occupying only the four central layers
of the investigated QW (note that it is eight layers' thick
and that for holes the ground state of the innite well
is taken as the "perpendicular" part of their wave func-
tion, '(z)). The results of our calculations are shown in
Fig. 12. One can clearly see that indeed if Mn ions reside
only in the central layer, the magnetization dynamics re-
sembles the one for the case when all AFM interactions
are switched o. We may as well formulate our conclu-
sion the other way round: more Mn spins close to the
QW edges, faster relaxation of magnetization. This nd-
ing provides a support for theoretical considerations,14
suggesting that magnetization relaxation of bound mag-
netic polarons in p-type CdTe/(Cd,Mn)Te QWs occurs
owing to the AFM coupling to the Mn spins located out-
side the relevant Bohr radius.
B. Measuring the domain size: the two-point
spin-spin connected correlation function
Let us now look at the spatial distribution of magne-
tization and, in particular, whether the presence of com-

11
FIG. 12: Monte Carlo simulations: The magnetization au-
tocorrelation function G() of Mn ions in a Cd0:96Mn0:04Te
quantum well at T = 0:7TC. The hole density is p =
0:4  1011 cm2 (Nh = 21).Open circles and full squares:
Mn located in the central layers of the QW only and evenly
distributed within the whole QW, respectively. Time is mea-
sured in the Monte Carlo steps per Mn site.
peting short-range antiferromagnetic and long-range fer-
romagnetic interactions results in the formation of mag-
netic domains. One of the standard ways to estimate the
typical domain size is to calculate the spin-spin two-point
connected correlation function G2c(R) given by
31
G2c(R) =
hm(r)m(r + R)i hmi2
hm2i hmi2
; (13)
We have calculated spin-spin correlation functions for
both: the localized Mn ions and the holes for a number
of temperature values and various carrier concentrations.
Figure 13 shows the correlation functions G2c(R) obtained
for the hole subsystem. We observe long-range correla-
tions (longer than the size of the simulation box) in the
ferromagnetic state. For temperatures below TC the hole
liquid becomes completely spin-polarized. This has al-
ready been predicted in the frame of the mean-eld ap-
proximation for the 2D hole gas in a QW.7 In Fig. 14 and
15 the spin-spin correlation functions for the Mn subsys-
tem are shown. In Fig. 14 the scale has been stretched
to make visible how the short-range AFM interactions
in
uence the spin-spin correlation function at small dis-
tances. In this way we can clearly see that for the nearest
neighbor Mn spins the AFM coupling prevails (the cor-
relation function becomes negative). At long distances,
however, below TC, the long-range ferromagnetic inter-
actions dominate as may be seen in Fig. 15 that shows
the spin-spin correlation functions on a larger scale. As
the distance R between the Mn spins becomes large we
observe (small) positive long-range correlations for tem-
FIG. 13: Monte Carlo simulations: The two-point connected
correlation function G(2)c (R) of valence band holes at various
temperatures in a Cd0:96Mn0:04Te quantum well. The hole
concentration is p = 0:4 1011 cm2 (Nh = 21).
FIG. 14: Monte Carlo simulations: The two-point connected
correlation function G(2)c (R) of Mn ions in a Cd0:96Mn0:04Te
quantum well at T = 0:2TC. The hole concentration is p =
0:4 1011 cm2 (Nh = 21).
peratures below TC. At the same time the correlations
decrease to zero when temperature increases towards TC.
The positive correlations observed in the ferromagnetic
phase and the lack of correlations in the paramagnetic
phase may indicate that magnetic domains do exist, but
their typical size is greater than the simulation box. To
verify this conjecture further massive simulations would
be needed.

12
FIG. 15: Monte Carlo Simulations: The long-distance parts of
two-point connected correlation functions G(2)c (R) of Mn ions
at various temperatures in a Cd0:96Mn0:04Te quantum well.
The hole concentration is p = 0:4 1011 cm2 (Nh = 21).
VI. CONCLUSIONS
We have examined magnetization dynamics in a p-
Cd0:96Mn0:04Te quantum well by magneto-optical stud-
ies and by Monte Carlo simulations taking into ac-
count the presence of competing short-range antiferro-
magnetic superexchange and long-range carrier-mediated
ferromagnetic interactions. Single-particle hole energies
and eigenfunctions for particular Mn spin congurations
have been updated numerically after each Monte Carlo
sweep. In addition to the p-d exchange interaction, spin-
independent alloy disorder has been taken into account
and found to be important. In particular, a suciently
strong attractive potential introduced by the magnetic
ion leads to hole localization and to the corresponding
disappearance of a long-range order mediated by the car-
riers. Also a strong repulsive potential, by reducing the
magnitude of the carrier wave function at the magnetic
ion, may diminish the magnitude of TC. This eect is
particularly strong in the case of a quantum well, where
the direction of the hole spin is xed rather by the spin-
orbit interaction than by the p-d coupling, so that the
repulsive potential may not be compensated by the p-d
interaction.
Both experimental and theoretical results demonstrate
that the transition from the paramagnetic to ferromag-
netic phase increases the magnetization relaxation time.
In particular, the relaxation time determined from trac-
ing the time evolution of magnetization after a pulse of
the magnetic eld is shorter than 20 ns in the paramag-
netic state. Lowering temperature below TC results in
an increase of the relaxation time up to only 2 s. Our
numerical results indicate that antiferromagnetic inter-
actions between Mn spins account for this relatively fast
magnetization
uctuations and, thus, might be responsi-
ble for the absence of spontaneous magnetization below
TC, as found in static measurements. While our Monte
Carlo simulations of the two-point spin-spin connected
correlation function show a huge in
uence of the short-
range antiferromagnetic interactions on the magnetiza-
tion dynamics, they do not reveal the formation of do-
mains that would be smaller than our simulation cell,
L = 226 nm.
At the same time, our Monte Carlo results reveal that
magnetization relaxation in the ferromagnetic phase pro-
ceeds primarily due to antiferromagnetic couplings to Mn
spins residing at the quantum well boundary, as they are
weakly polarized by the carriers. Accordingly, magneti-
zation dynamics would be much slow down if the width
of the region containing Mn spins were been narrower
than the extent of the carrier wave function. We predict,
therefore, that spontaneous magnetization and the asso-
ciated magnetic hysteresis could be observed in quantum
wells containing the Mn layer only in its center.
Acknowledgments
The work in Warsaw was supported in part by Tech-
nology Agency, by SPINTRA Project of European Sci-
ence Foundation (ERAS-CT-2003-980409), and by the
FunDMS Advanced Grant within the European Research
Council "Ideas" Programme of EC 7FP. Part of the nec-
essary computer time in Athens was provided by the Na-
tional Grid Infrastructure HellasGrid. We thank N. Pa-
panikolaou for useful discussions.
1 A. Haury, A. Wasiela, A. Arnoult, J. Cibert, S. Tatarenko,
T. Dietl, and Y. Merle d'Aubigne, Phys. Rev. Lett. 79,
511 (1997).
2 P. Kossacki, D. Ferrand, A. Arnoult, J. Cibert,
S. Tatarenko, A. Wasiela, Y. Merle d'Aubigne, J.-
L. Staehli, J.-D. Ganiere, W. Bardyszewski, K. Swi atek
, M. Sawicki, J. Wrobel, and T. Dietl, Physica E 6, 709
(2000).
3 P. Kossacki, A. Kudelski, J.A. Gaj, J. Cibert, S. Tatarenko,
D. Ferrand, A. Wasiela, B. Deveaud, T. Dietl, Physica E
12, 344 (2002).
4 P. Kossacki, D. Ferrand, M. Goryca, M. Nawrocki, W. Pa-
cuski, W. Maslana, S. Tatarenko, and J. Cibert, Physica
E 32, 454 (2006).
5 W. Maslana, P. Kossacki, P. P lochocka, A. Golnik, J. A.
Gaj, D. Ferrand, M. Bertolini, S. Tatarenko, and J. Cibert
Appl. Phys. Lett. 89, 052104 (2006).
6 H. Boukari, P. Kossacki, M. Bertolini, D. Ferrand, J. Cib-
ert, S. Tatarenko, A. Wasiela, J. A. Gaj, and T. Dietl,
Phys. Rev. Lett. 88, 207204 (2002).

13
7 T. Dietl, A. Haury, and Y. Merle d'Aubigne, Phys. Rev. B
55, R3347 (1997).
8 T. Dietl, J. Cibert, D. Ferrand, and Y. Merle d'Aubigne,
Materials Sci. Engin. B 63, 103 (1999).
9 D. Kechrakos, N. Papanikolaou, K. N. Trohidou, and
T. Dietl, Phys. Rev. Lett. 94, 127201 (2005).
10 B. Lee, T. Jungwirth, and A. H. MacDonald, Phys. Rev.
B 61, 15606 (2000).
11 L. Brey and F. Guinea, Phys. Rev. Lett. 85, 2384 (2000).
12 D.J. Priour, E.H. Hwang, and S. Das Sarma, Phys. Rev.
Lett. 95, 037201 (2005).
13 P. Simon, B. Braunecker, and D. Loss, Phys. Rev. B. 77,
045108 (2008).
14 P. Boudinet and G. Bastard, Semicond. Sci. Technol. 8,
1535 (1993).
15 P. Kossacki, H. Boukari, M. Bertolini, D. Ferrand, J. Cib-
ert, S. Tatarenko, J. A. Gaj, B. Deveaud, V. Ciulin, and
M. Potemski, Phys. Rev. B 70, 195337 (2004).
16 M. Sawicki, P. Kossacki, and T. Dietl, unpublished.
17 D. Scalbert, J. Cernogora, and C. Benoit A La Guillaume,
Solid State Commun. 66, 571 (1988).
18 T. Dietl, P. Peyla, W. Grieshaber, and Y. M. d'Aubigne,
Phys. Rev. Lett. 74, 474 (1995).
19 G. Fishman, Phys. Rev. B 52, 11132 (1995).
20 T. Dietl, Phys. Rev. B 77, 085208 (2008).
21 J. A. Gaj, R. Planel, and G. Fishman, Solid State Com-
mun. 29, 435 (1979).
22 J. A. Gaj, W. Grieshaber, C. Bodin-Deshayes, J. Cibert,
G. Feuillet, Y. Merle d'Aubigne, and A. Wasiela, Phys.
Rev. B 50, 5512 (1994).
23 T. Wojtowicz, M. Kutrowski, M. Surma, K. Kopalko,
G. Karczewski, J. Kossut, M. Godlewski, P. Kossacki, and
Nguyen The Khoi, Appl. Phys. Lett. 68, 3326 (1996).
24 Y. Shapira and V. Bindilatti, J. Appl. Phys. 92, 4155
(2002).
25 B. Hennion, W. Szuszkiewicz, E. Dynowska, E. Janik, and
T. Wojtowicz, Phys. Rev. B 66, 224426 (2002).
26 J. Schliemann, J. Konig, and A. H. MacDonald, Phys. Rev.
B 64, 165201 (2001).
27 T. N. Troung and E. V. Stefanovich, Chem. Phys. Lett.
256, 348 (1996).
28 T. N. Troung and E. V. Stefanovich, Chem. Phys. 218, 31
(1997).
29 M. P. Kennett, M. Berciu, and R. N. Bhatt, Phys. Rev. B
66, 045207 (2002).
30 M. Matsumoto and T. Nishimura, ACM Trans. Model.
Comput. Simul. 8, 3 (1998).
31 M. E. J. Newman and G. Barkema, Monte Carlo Methods
in Statistical Physics (Colorado Press, Oxford, 1999).