Ferromagnetic transition in diluted magnetic semiconductors
A summary is given of recent theoretical works on effects of the Ruderman- Kittel-Kasuya-Yosida (RKKY) interaction between the localized spins in various dimensionality systems of doped diluted magnetic semiconductors (DMS). Since this interaction is long-range, its influence on the temperat...
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Інститут фізики конденсованих систем НАН України
1999
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irk-123456789-1205382017-06-13T03:05:37Z Ferromagnetic transition in diluted magnetic semiconductors Dietl, T. A summary is given of recent theoretical works on effects of the Ruderman- Kittel-Kasuya-Yosida (RKKY) interaction between the localized spins in various dimensionality systems of doped diluted magnetic semiconductors (DMS). Since this interaction is long-range, its influence on the temperature and magnetic field dependencies of magnetization and spin splitting of the bands is evaluated in the mean field approximation, but by taking into consideration disorder-modified carrier-carrier interactions. Theoretical evaluations show that the hole densities, which can presently be achieved, are sufficiently high to drive a paramagnetic- ferromagnetic phase transition in bulk and modulation-doped structures of II-VI DMS. The results of recent magnetooptical studies on MBE-grown samples, containing a single, modulation-doped, 8 nm quantum well of Cd₁₋xMnxTe/Cd₁₋y₋zMgyZnzTe:N are shown to corroborate the theoretical expectations. These studies reveal the presence of a ferromagnetic transition induced by the two-dimensional hole gas. The transition occurs between 1.8 and 2.5 K, depending on the Mn concentration x, in agreement with the theoretical model. Зроблено огляд недавніх теоретичних робіт про вплив взаємодії Рудермана-Кіттеля-Касуї-Іоміди між локалізованими спінами у системах різної вимірності в розведених магнітних напівпровідниках (РМН). Оскільки ця взаємодія далекосяжна, її вплив на температурну і польову залежності намагніченості і спінове розщеплення зон оцінюється в наближенні середнього поля, беручи до уваги змінену безладом взаємодію носій-носій. Теоретичні оцінки показують, що експериментально досяжні густини дірок достатньо високі для того, щоб спричинити фазовий перехід парамагнетик-феромагнетик у тривимірних модульованих структурах II–VI РМН. Показано, що результати теоретичних обчислень підтверджуються недавніми магнетооптичними дослідженнями вирощених методом НВЕ зразків Cd₁₋xHnxTe/Cd₁₋₄₋₇HgyZnz:N, що містять одну модульовану квантову яму глибиною 8 nm. Ці дослідження свідчать про наявність феромагнітного переходу, спричиненого двовимірним дірковим газом. Перехід відбувається при температурі між 1.8 і 2.5 K, залежно від концентрації марганцю x, що узгоджується з теоретичною моделлю. 1999 Article Ferromagnetic transition in diluted magnetic semiconductors / T. Dietl // Condensed Matter Physics. — 1999. — Т. 2, № 3(19). — С. 495-508. — Бібліогр.: 50 назв. — англ. 1607-324X DOI:10.5488/CMP.2.3.495 PACS: 75.50.Rr, 75.30.Hx, 75.50.Dd, 78.55.Et http://dspace.nbuv.gov.ua/handle/123456789/120538 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| description |
A summary is given of recent theoretical works on effects of the Ruderman-
Kittel-Kasuya-Yosida (RKKY) interaction between the localized spins in
various dimensionality systems of doped diluted magnetic semiconductors (DMS). Since this interaction is long-range, its influence on the temperature and magnetic field dependencies of magnetization and spin
splitting of the bands is evaluated in the mean field approximation,
but by taking into consideration disorder-modified carrier-carrier interactions. Theoretical evaluations show that the hole densities, which can
presently be achieved, are sufficiently high to drive a paramagnetic-
ferromagnetic phase transition in bulk and modulation-doped structures of
II-VI DMS. The results of recent magnetooptical studies on MBE-grown
samples, containing a single, modulation-doped, 8 nm quantum well of
Cd₁₋xMnxTe/Cd₁₋y₋zMgyZnzTe:N are shown to corroborate the theoretical expectations. These studies reveal the presence of a ferromagnetic
transition induced by the two-dimensional hole gas. The transition occurs
between 1.8 and 2.5 K, depending on the Mn concentration x, in agreement
with the theoretical model. |
| format |
Article |
| author |
Dietl, T. |
| spellingShingle |
Dietl, T. Ferromagnetic transition in diluted magnetic semiconductors Condensed Matter Physics |
| author_facet |
Dietl, T. |
| author_sort |
Dietl, T. |
| title |
Ferromagnetic transition in diluted magnetic semiconductors |
| title_short |
Ferromagnetic transition in diluted magnetic semiconductors |
| title_full |
Ferromagnetic transition in diluted magnetic semiconductors |
| title_fullStr |
Ferromagnetic transition in diluted magnetic semiconductors |
| title_full_unstemmed |
Ferromagnetic transition in diluted magnetic semiconductors |
| title_sort |
ferromagnetic transition in diluted magnetic semiconductors |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
1999 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120538 |
| citation_txt |
Ferromagnetic transition in diluted magnetic semiconductors / T. Dietl // Condensed Matter Physics. — 1999. — Т. 2, № 3(19). — С. 495-508. — Бібліогр.: 50 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT dietlt ferromagnetictransitionindilutedmagneticsemiconductors |
| first_indexed |
2025-07-08T18:05:54Z |
| last_indexed |
2025-07-08T18:05:54Z |
| _version_ |
1837103015986724864 |
| fulltext |
Condensed Matter Physics, 1999, Vol. 2, No. 3(19), pp. 495–508
Ferromagnetic transition in diluted
magnetic semiconductors
T.Dietl
Institute of Physics and College of Science, Polish Academy of Sciences
al. Lotników 32/46, PL-02668 Warszawa, Poland
Received July 6, 1998
A summary is given of recent theoretical works on effects of the Ruderman-
Kittel-Kasuya-Yosida (RKKY) interaction between the localized spins in
various dimensionality systems of doped diluted magnetic semiconduc-
tors (DMS). Since this interaction is long-range, its influence on the tem-
perature and magnetic field dependencies of magnetization and spin
splitting of the bands is evaluated in the mean field approximation,
but by taking into consideration disorder-modified carrier-carrier interac-
tions. Theoretical evaluations show that the hole densities, which can
presently be achieved, are sufficiently high to drive a paramagnetic-
ferromagnetic phase transition in bulk and modulation-doped structures of
II-VI DMS. The results of recent magnetooptical studies on MBE-grown
samples, containing a single, modulation-doped, 8 nm quantum well of
Cd1−xMnxTe/Cd1−y−zMgyZnzTe:N are shown to corroborate the theoret-
ical expectations. These studies reveal the presence of a ferromagnetic
transition induced by the two-dimensional hole gas. The transition occurs
between 1.8 and 2.5 K, depending on the Mn concentration x, in agreement
with the theoretical model.
Key words: semimagnetic semiconductors, RKKY interaction, quantum
wells
PACS: 75.50.Rr, 75.30.Hx, 75.50.Dd, 78.55.Et
1. Introduction
Magnetic properties of diluted magnetic semiconductors (DMS) are known to
be dominated by antiferromagnetic superexchange interactions between the local-
ized spins [1]. It has been known for a long time that the compensation of these
interactions by a ferromagnetic coupling would result in a dramatic enhancement of
the sensitivity of DMS to the temperature and to the magnetic field, particularly
in the vicinity of the ferromagnetic phase transition. Search for the ferromagnetic
transition has been successful in the case of Mn-based p-type IV-VI [2] and III-V
compounds [3], in the latter case a critical temperature as high as 110 K has recently
c© T.Dietl 495
T.Dietl
been reported for Ga0.95Mn0.05As [4,5]. In the case of II-VI DMS, a tight binding
model [6] suggests that the superexchange in Cr-based DMS might be dominated
by a ferromagnetic contribution. Accordingly, an attempt has been undertaken [7]
to overcome the well-known small solubility of Cr in II–VI compounds by means of
MBE growth of Cd1−xCrxTe.
A rapid progress in doping of II–VI wide gap semiconductors by substitutional
impurities has recently been achieved. For instance, electron and hole concentra-
tions in the excess of 1019 cm−3 have been reported for ZnSe:I [8] and ZnTe:N [9].
At the same time, modulation doping of II–VI quantum wells by either electrons
[10] or holes [11] as well as patterning of conducting quantum wires [12] has been
successfully performed. Motivated by this progress a theoretical analysis of the na-
ture and strength of the carrier-mediated spin-spin interactions in bulk, layered,
and nanostructured II–VI compounds has been undertaken by the present authors
and co-workers [13]. The results of that work [13] are described here. Two alter-
native approaches, the RKKY and self-consistent models, are discussed and their
equivalence is demonstrated in the mean field approximation (MFA). The role of
disorder and carrier-carrier interactions is examined, and shown to be important.
Physical arguments supporting the validity of the MFA, even in the reduced di-
mensionality systems, are also discussed. Quantitative estimations based on the
theoretical results are then presented. They indicate that even for the highest avail-
able electron density no transition to the ferromagnetic phase is expected above 1
K in II–VI DMS. By contrast, such a transition is predicted for p-type materials,
either in the bulk or modulation-doped form. It has recently been demonstrated
[14,15] that indeed the hole liquid in single modulation-doped quantum wells of
Cd1−xMnxTe/Cd1−y−zMgyZnzTe:N induces the foreseen transition. The correspond-
ing experimental results, together with conclusions and outlook, are presented in the
final sections of the present paper. Another review of works devoted to studies of
ferromagnetism in II-VI DMS [16] gives a fuller account of the experimental aspects.
The influence of delocalized or weakly localized carriers on the interaction be-
tween magnetic ions is of interest here. The dimensionality d of the subsystem of
the carriers is determined by the shape of the potential V (ζ) that leads to their
confinement. Accordingly, the case d = 1 or d = 2 corresponds to a two- or one-
dimensional potential well, respectively. Because of a short magnetic correlation
length [1] the localized magnetic moments gµBSi are assumed, to form a macro-
scopic 3D system. Thus, according to experimental studies [1,17] their magnetiza-
tion in the absence of the carriers can be described by a modified Brillouin function,
M(T,H) = gµBx̃NoSBS(T + To, H), where effective spin concentration x̃No < xNo
and temperature T +To > T account for the influence of antiferromagnetic superex-
change interactions [1,6,17].
2. Ruderman-Kittel-Kasuya-Yosida model
The carrier-mediated spin-spin coupling is usually described in terms of the
Ruderman-Kittel-Kasuya-Yosida (RKKY) model, which provides the energy J ij
496
Ferromagnetic transition
of the exchange coupling, Hij = −JijSi · Sj, between two spins located at Ri
and Rj as a function of the density-of-states of the carriers at the Fermi level,
ρd(εF) = π1−d(2/π)(d−2)(d−3)/2m∗kd−2
F /~2, and the exchange integral I of their inter-
action with the spins, Hi = −Is ·Siδ(R−Ri). Following the well known procedure
[18] and adopting the one-band effective-mass approximation we obtain to the sec-
ond order in I
Jij =
ρd(εF)k
d
FI
2
2π
Fd(2kF|ri − rj|)|ϕo(ζi)|
2|ϕo(ζj)|
2. (1)
Here r is the vector in the d dimensional space, ϕo(ζ) is the ground-state envelope
function of the carriers in the confining potential V (ζ), and
F1(y) = −πsi(y)/2, (2)
F2(y) =
∫ ∞
1
dt
J1(yt)
yt(t2 − 1)1/2
, (3)
F3(y) = [sin(y)− y cos(y)]/y4, (4)
where si(y) is the sine-integral and Jn(y) is the Bessel function. The asymptotic
behaviour of Fd(y) for large y is π cos(y)/2y, sin(y)/y2, and − cos(y)/y3, while for
y → 0, Fd tends to π2/4, [1/2 − γ + ln(4/y)]/2, and 3/y for d = 1, 2, and 3,
respectively, where γ = 0.57721... is the Euler constant. For the same sequence
of d, the first zero of Fd(y) occurs for y ≈ 1.7, 3.5, and 4.5, respectively. The
formula for d = 3 reproduces the result first obtained by Ruderman and Kittel [19],
and for d = 1 that of Yafet [20], whereas the expression for the case of d = 2 is
equivalent to the formula given recently by Aristov [18]. Knowing the dependence
Jij on the distance between the spins r we can calculate the mean-field value of
the Curie-Weiss temperature of spins located at ζ i, Θ(ζi) = S(S + 1)
∑
j Jij/3kB.
Since in semiconductors, unlike in metals, the value of y that corresponds to the
distance r between the nearest neighbouring spins is much smaller than the period
of the oscillatory functions in equations 2–4, ynn = 2kFrnn ≪ 1, we replace the
summation over the ion positions by an integration extending from y = 0 to ∞.
Under the assumption that the distribution of the magnetic ions is random we
obtain Θ, averaged over ζ, in the form,
Θ = S(S + 1)NoI
2ρd(εF)
∫
dζx̃(ζ)|ϕo(ζ)|
4/12kB, (5)
which shows that the net RKKY interaction is ferromagnetic, Θ > 0. This is in
contrast to the case of metals where a spin glass phase is observed. In semiconductors,
however, where the mean distance between magnetic ions r̄ = (4πx̃No/3)
−1/3, is
much smaller that the electron wavelength λF = 2π/kF, the spin-spin interaction
mediated by the carriers is merely ferromagnetic. Since Θ is proportional to the
effective mass, to the degree of confinement as well as to the square of the exchange
integral I we expect much greater magnitudes of Θ in the presence of the holes than
for the electrons in II–VI DMS.
497
T.Dietl
The above approach neglects intervalley or intersubband virtual transitions.
Since those terms in Jij, which result from such transitions, contain products of
orthogonal envelope functions, ϕ∗
ν(ζj)ϕν′(ζj), their contribution to Θ vanishes, pro-
vided that the concentration of magnetic ions is uniform. Moreover, in such a case
each of the occupied valleys or subbands gives an independent contribution to Θ,
described by the relevant ϕν(ζ) and k
(ν)
F .
3. Self-consistent model
In this section we present results which demonstrate the equivalence – on the
level of the MFA and for a random distribution of the magnetic ions – of the RKKY
approach and a simple self-consistent Vonsovskii model of the ferromagnetism [22,
23].
We start recalling the molecular-field relation between the magnetization of the
localized spins, M(T,H), and the spin-splitting of the relevant band,
∆(T,H) = I
∫
dζM(T,H, ζ)|ϕo(ζ)|
2/gµB + g∗µBH, (6)
where g and g∗ are the Landé factors of the localized spins and the carrier spins,
respectively. In the MFA, M is induced by the external field H and the molecular
field produced by the carriers, H∗ so that M(T,H) = gµBx̃NoSBS(T +To, H+H∗),
where H∗ = I(n↓ − n↑)/2gµB. Here n↓,↑ is the density of spin down and spin up
carriers, respectively, which, at εF ≫ kBT , is given by n↑,↓ =
1
2
|ϕo(ζ)|
2
∫ εF
±∆/2
dε ρd(ε),
where the dependence of εF on ∆ is to be determined from the condition n↑+n↓ = n,
with n being the total carrier density. A similar model of ferromagnetism, developed
for the case of a nondegenerate gas of carriers thermally activated from the impurity
levels, was put forward by Pashitskii and Ryabchenko [23].
The above set of coupled equations makes it possible to determine in a self-
consistent way the mean-field values of ∆ as functions of temperature and magnetic
field. In particular, in order to evaluate ∆(T,H) above the phase transition we take
into account the terms which are linear in H +H ∗. Such a procedure leads to,
∆(T,H) =
∆o(T,H)
[
1− I2ρd(εF)
4g2µ2
B
∫
dζχo(T, ζ)|ϕo(ζ)|4
] , (7)
where ∆o(T,H) and χo(T ) are the spin-splitting and the magnetic susceptibility in
the absence of the free carriers.
We see that for the magnetic susceptibility of the form χo(T ) = C/(T + To),
∆(T ) diverges at Tc = Θ− To, where Θ turns out to coincide with the Curie-Weiss
temperature determined by the RKKY interactions, displayed in equation 5.
4. Applicability of the mean-field approximation
The long range character of the ferromagnetic interaction, λF ≫ r̄, has also
important consequences for the nature of the phase transition. It has been shown
498
Ferromagnetic transition
[24] that as long as σ < d/2 in the dependence J(r) ∼ 1/rd+σ, the mean-field ap-
proach to the long wavelength susceptibility χ(T ) is valid, a conclusion non-affected
presumably by disorder in the spin distribution. By contrast, the critical exponents
[24] η = 2 − σ and ν = 1/σ point to much faster decay of χ(q) with q than that
expected from the classical Ornstein-Zernike theory [25]. This means that our mean-
field model of the ferromagnetic transition driven by the RKKY interactions should
remain valid down to at least |T − Tc|/Tc ≈ (r̄kF)
2. At the same time, unlike the
case of short range interactions, the length scale of magnetic correlation is set by
λF, not by r̄. This means that for λF ≫ r̄ the critical fluctuations of magnetization
will be strongly suppressed, an expectation corroborated by the virtual absence of
critical scattering of the carriers by the Mn spins in bulk Pb1−x−yMnxSnyTe [26],
where the free holes drive a ferromagnetic transition [2], and its presence in, e.g.,
EuS:Gd and EuO:Gd [27], in which a short range ferromagnetic interaction between
the Eu spins dominates [28]. At the same time, it is expected that the suppression
of the critical fluctuations will make it possible to probe directly the magnetic long
range order by means of inter- or intraband magnetospectroscopy.
5. Effects of disorder, carrier-carrier interactions, and magnet-
ic polarons
So far we have disregarded the influence of the potential scattering and the
Coulomb interactions among the carriers upon the magnitude of Θ. The former is
known [29] to introduce a random phase shift in the oscillatory functions of equa-
tion 1. This random phase shift has no effect on the magnitude of the second mo-
ment of the distribution of Jij [30] but, after averaging over the disorder, leads to
the dumping exp(−r/ℓ) of the first moment [29,30], where ℓ is the mean free path
for elastic collisions. The corresponding reduction factor of Θ to the lowest order
in 1/kFℓ is given by 1 − 1/12(kFℓ)
2, 1 − π/8kFℓ, and 1 − π/4kFℓ for d = 1, 2, and
3, respectively. Hence the effect of disorder becomes important on approaching the
strongly localized regime, kFℓ → 1. In this range, however, the magnetic susceptibil-
ity of the carriers becomes substantially enlarged by the disorder-modified electron-
electron interactions [31]. The resulting enhancement factor of Θ, originating from
the RKKY interactions between the spins located at r > ℓ [32], can be written in the
form: AF = 1+FLs/ℓ, 1 + 2F ln(Ls/ℓ)/πkFℓ, and 1+3F (1− ℓ/Ls)/(2kFℓ)
2, respec-
tively. Here F is the effective Coulomb amplitude, which becomes greater than 1 for
kFℓ → 1; Ls =
√
(Dts) ≫ ℓ is the spin diffusion length, where D = kFℓ/dm
∗ and
ts is the spin-disorder scattering time [31,33]. The effects of disorder and electron-
electron interactions can also be incorporated into the self-consistent formalism by
taking into account collision broadening of the density of states as well as by de-
termining H ∗ from gµBH
∗ = I∂Ω/∂∆ with the Gibbs free energy of the carriers, Ω
containing effects of the disorder-modified electron-electron interactions [31].
A crucial question which arises at this point concerns the actual magnitude of the
enhancement factor AF in the real materials. It turns out that AF is approximately
equal to 1 + g3, where g3 is a parameter that controls the spin-splitting-induced
499
T.Dietl
positive magnetoresistance in disordered systems [31]. For example, by taking g3
determined for bulk n-Cd0.95Mn0.05Se [34] and superlattices of n-Si/Si0.5Ge0.5 [35]
we obtain AF ≈ 2.3 and 2.5, respectively. Such a mechanism leads also to an en-
hancement of the spin-disorder scattering rate, as demonstrated by an analysis of
the linewidth of spin-flip Raman scattering (SRRS) in n-Cd0.95Mn0.05Se [36]. By
contrast, the interactions within the Fermi liquid do not renormalize the carrier
spin-splitting ∆(T,H), as given by the Stokes shift of the SFRS line [36]. The lack
of contribution to ∆(T,H) from the exchange interactions among the carriers stems
from the fact that the projection Sz of the total spin S of the carriers undergoes
a change in the process of SFRS, while S, and thus the interaction energy, remain
conserved.
A particularly strong can be a combined effect of disorder and on-site Hubbard
repulsion in the vicinity of the metal-to-insulator transition (MIT) in doped semi-
conductors. Here, a growing amount of evidences has been collected in favour of a
phenomenological two-fluid model of electronic states [37]. According to that model
the conversion of itinerant electrons into local moments occurs gradually, and begins
already on the metal side of the MIT, leading to the coexistence of the extended
and strongly localized states. In magnetic materials, the local moments can be po-
larized, via the s-d interaction, the neighbouring Mn spins (there are about 200 Mn
ions within the Bohr orbit in n-Cd0.95Mn0.05Se). Bound magnetic polarons formed in
this way not only impose a local ferromagnetic order above Tc, but also constitute
the centers of efficient spin-dependent scattering for itinerant electrons [34,36,38].
The corresponding scattering rate is proportional to the degree of polaron polariza-
tion. The latter is proportional to the magnetic susceptibility of the Mn spins, χ(T ),
and therefore increases steeply on approaching the ferromagnetic phase transition.
This mechanism accounts presumably for critical scattering of holes and negative
magnetoresistance detected recently in p-Ga1−xMnxAs near Tc [5]. By contrast, no
such effects are seen in p-Pb1−x−yMnxSnyTe [26], where a large dielectric constant
reduces the Hubbard repulsion, and thus precludes the formation of bound magnetic
polarons.
We conclude that, in fact, the disorder is expected to enlarge a characteristic
temperature of the ferromagnetic interactions in semiconductor structures. This en-
hancement may also be important in pure 1D systems, where the interaction-driven
separation of the charge and spin degrees of freedom modifies J(r) [39]. In the zero-
dimensional case such as quantum dots both correlation effects and the fluctuations
of magnetization associated with the finite volume visited by the carriers are of
the paramount importance. A variant of the theory developed for bound magnetic
polarons [40] should be applied for those systems.
6. Quantitative estimations
Figure 1 shows the magnitude of Θ(x) for p-type and n-type bulk Cd1−xMnxTe,
calculated by equation 5, with the values of x̃(x) and To(x), as determined by Gaj
et al. [17] at T = 1.7 K as well as by taking m∗ = 0.8 and 0.1mo as well as
500
Ferromagnetic transition
0.0 0.1 0.2 0.3 0.40.001
0.01
0.1
1
10
100
∼
×n = 1 1018 cm-3
×n = 2 1019 cm-3
×p = 2 1019 cm-3
Cu
rie
-W
eis
s te
mp
era
tur
e Θ
(K
)
Mn concentration x
T0
Cd1-xMnxTe, T > 1 K
Figure 1. Mean-field value of the Curie-Weiss temperature Θ calculated from
equation 5 for p- and n-type Cd1−xMnxTe, compared to antiferromagnetic tem-
perature To(x). Ferromagnetic phase transition occurs at Tc = Θ − To. Material
parameters as determined at 1.7 K together with the enhancement factor AF = 1
were adopted for the calculation (after Dietl et al. [13]).
INo ≡ βNo = −0.88 and INo ≡ αNo = 0.22 eV for the holes and the elec-
trons, respectively [17,41]. It has been noted that the spin-orbit interaction re-
duces the spin-splitting of the Γ8 heavy holes at the Fermi level according to [42]
∆(k) = I|M · k|/gµBk, which after angular averaging results in an effective spin
density of states ρ̃F = 1
2
ρF. We see in figure 1 that for sufficiently high hole concen-
trations so that the holes remain delocalized [43], Θ > To in a wide range of Mn
concentrations. Furthermore, since for the above parameters the Kondo temperature
TK ≈ εF exp[−1/(3|I|ρ̃F)] = 1.1 K, a crossover to the Kondo regime, TK > Θ, To
will take place at relatively low Mn concentrations, x < 1%. These considerations
suggest, therefore, that a ferromagnetic phase transition can occur above 1 K in p+
II–VI compounds. This is unlike the case of n-type doping, for which no ferromag-
netic phase transition is expected above 1 K, as shown by the two bottom curves
in figure 1. It is worth noting, however, that on lowering temperature To decreases,
especially for low Mn concentrations [44]. Indeed, for the highest value of χo ever
reported for any DMS [44], that is χo = 4.8 × 10−3 emu/g for Cd0.99Mn0.01Se at
15 mK, a ferromagnetic phase transition is predicted from equation 5 for material
parameters of n-Cd1−xMnxSe [40] provided that n > 1.5 × 1019 cm−3. Since I > 0,
there is no Kondo effect for the electrons in DMS.
Turning to the case of holes in DMS quantum wells we note that the confinement
501
T.Dietl
and possibly the biaxial strain lead to a splitting of the heavy and light hole bands
[45] as well as to a strong anisotropy of the spin-splitting [46]. Actually, the coupled
system of the 2D holes and the Mn spins is Ising-like as the spin-splitting of the
ground-state subband undergoes a maximum for the magnetization parallel to the
growth axis. By taking parameters suitable for the uppermost heavy-hole subband
in a quantum well of Cd0.9Mn0.1Te, m
∗
h = 0.25mo and |ϕo(z)|
2 = 1/LW, where
LW = 50 Å, we predict the ferromagnetic transition to occur at about 2 K for
AF = 1, independently of the hole area concentration p, since in our model the 2D
density of states does not vary with εF. At the same time, the saturation values of
∆ and M do depend directly on p according to ∆s(T ) = IM(T,H∗
s )/gµB, where
H∗
s = Ip/2gµBLW, a value of the order of 1 kOe for p = 2× 1011 cm−2.
Finally, we note that in the case of 1D structures Θ increases with decreasing
the carrier concentration. This, together with the correlation effects discussed above
demonstrate the outstanding properties of such systems.
7. Comparison with experimental results
The relevant experimental studies [14,15] have been carried out on 2D struc-
tures grown by molecular beam epitaxy (MBE) [11]. The nitrogen modulation-doped
structures consist of a single 8 nm quantum well (QW) of Cd1−xMnxTe embedded
in Cd0.66Mg0.27Zn0.07Te barriers grown coherently onto a (001) Cd0.88Zn0.12Te sub-
strate. Such a layout insures large confinement energies for the holes in the QW,
minimizing at the same time the effects of lattice mismatch. Nitrogen-doped region
in the front barrier is at the distance of 20 nm or 10 nm to the QW. Furthermore, in
order to reduce depleting effects, two additional nitrogen doped layers reside at the
distance of 100 nm from the QW on both sides. The nominal hole concentrations in
the doped structures, evaluated from a self consistent solution of the Poisson and
Schrödinger equations, are 2 × 1011 cm−2 and 3 × 1011 cm−2 for the two employed
values of the spacer width. In the same way we obtain L̃W = 6.3 nm as an effective
width of the hole layer, which is related to the hole ground state envelope function
ϕo(z) by L̃W = 1/
∫
dz|ϕo(z)|
4. For a control purpose, an undoped structure was
also grown and examined.
An examination of the photoluminescence (PL) and its excitation spectra (PLE)
was carried out as a function of temperature and the magnetic field [14,15]. The
magnetic field was parallel to the growth direction and the two circular polarizations
in the Farady geometry were employed. As shown in figure 2, the PL splitting
∆ = |E− − E+| under such conditions is not only exceptionally large, but increases
in a dramatic way on lowering temperature. Actually, in all doped samples, a colossal
value of ∂∆/∂H was observed above a characteristic temperature Tc, and a zero field
splitting below Tc, where Tc = 1.8 K for x = 0.024 [14] and 2.5 K for x = 0.037
[15]. The corresponding experimental data are summarized in figure 2. No such
effects were visible either in the undoped structure or in the presence of illumination
by white light that depletes the QW from the carriers. The appearance of zero-
field splitting was preceded by a critical increase of χ, as shown in figure 2b. The
502
Ferromagnetic transition
0 1 2 3 4 5
0.0
5.0
b
H 0
Temperature (K)
1/ χ (10
3 em
u)
#2
#2 under
illumination
#4 p = 0
0 200 400 600
1660
1665
a
4.2K
2.89K
2.14K
1.65K
1.36K
# 2
Magnetic field (Oe)
E
ne
rg
y
(m
eV
)
Figure 2. (a) Energy E± of photoluminescence maxima at σ+ and σ− circular
polarizations (full and empty symbols, respectively) as a function of the mag-
netic field H for selected temperatures in a modulation-doped p-type QW of
Cd0.976Mn0.024Te. Dotted lines are guides for the eye; solid lines denote the as-
sumed initial slope of E± at 2.14 K. (b) Inverse magnetic susceptibility calcu-
lated from d(E− −E+)/dH at H → 0, as given by data in (a) for p-type QW of
Cd0.976Mn0.024Te (full circles). Note the presence of a ferromagnetic transition.
Results for empty Cd0.976Mn0.024Te QWs, where antiferromagnetic interactions
dominate, are shown by empty symbols (after Haury et al. [14]).
susceptibility in the limit of vanishing fields was determined from
χ(T ) = (gµB/|α− β|)(∂∆/∂H). (8)
The transition temperature deduced from the extrapolation of the susceptibility
data shown in figure 2b agreed well with the temperature deduced from the appear-
ance of the spontaneous magnetization. The critical behaviour of χ(T ) is unlike the
gradual changes of χ(T ) associated with the formation of magnetic polarons [40,47],
for which the finite volume involved precludes the existence of any second order
phase transition [40,48].
These findings are interpreted as a ferromagnetic phase transition driven by
the free holes which by means of the RKKY mechanism mediate ferromagnetic
exchange interactions between the Mn spins. This conclusion is strongly supported
by the theoretical model discussed above. According to that model [13], the critical
fluctuations of magnetization are of minor importance, so that we may write χ(T ) =
C/(T + To − Θ), where the influence of the delocalized carriers is described by Θ.
According to the results of figure 2b, and in agreement with the model, the same
503
T.Dietl
values of the Curie constant C are observed in the presence and in the absence of
the holes. At the same time, again in accord with the model, Tc does not depend on
p (as long as holes are delocalized) but increases with x̃. From Tc and Θ = Tc − To
the enhancement factor AF is obtained to be 2.5± 0.5, an expected value in view of
the discussion presented in section 5.
In addition to enhancing Θ, static disorder and carrier-carrier interactions impose
a strong influence on the optical properties. In particular, they lead to a band-gap
narrowing [49] that amounts to 29 meV for all doped samples studied, as deduced
from the energy difference between the free exciton reflectivity for the empty QW and
the PL in the presence of the holes (7 meV at H = 0), and from an exciton binding
energy of 22 meV, according to an estimate based on the 1s − 2s excitonic energy
separation [50] in the undoped sample. Moreover, these renormalization effects are
expected to vary strongly with the degree of carrier polarization. Accordingly, while
the evaluated value of the splitting of the heavy hole subband corresponding to the
full hole polarization, ∆ ≈ 6 meV for p = 2×1011 cm−2, is not inconsistent with the
experimental results of figure 2, the achievement of the saturation with the lowering
of temperature proceeds slower than predicted by the model. A work is under way
aimed at determining the effect of the renormalization on the dependence ∆(T,H)
as well as at examining the domain structure in this novel ferromagnetic system.
8. Conclusions and outlook
The findings presented above demonstrate that p-type doping actually consti-
tutes the method for substantial enhancement of magnetic effects, and then magne-
tooptical effects in DMS, leading – in appropriately designed structures – to a fer-
romagnetic phase transformation. Additional enhancement of the tendency towards
the ferromagnetic ordering is possible by engineering such a microscopic distribution
of the magnetic ions, which would reduce To and/or increase Θ. Since the interac-
tions between localized spins mediated by the carriers are long range ones, a simple
mean-field approach gives a correct quantitative description of magnetic properties,
even in reduced dimensionality systems. At the same time, the data make it pos-
sible to evaluate the strength of many body effects for the case of two-dimensional
hole gas. Moreover, while our results demonstrate the possibility of changing the
magnetic phase by light, other means – such as gates – are expected to also provide
a high degree of control over magnetic properties in modulation-doped structures.
This opens new prospects for further studies of coupled carrier liquids and localized
spins in novel geometries and material systems.
Acknowledgements
I would like to thank my principal co-workers from Grenoble Yves Merle d’Aubig-
né, Joel Cibert, and André Wasiela for fruitful and enjoyable collaboration. The work
in Poland was supported by KBN Grant No. 2–P03B–6411, while the collaboration
by Polish-French cooperation project “Polonium” and Joseph Fourier University.
504
Ferromagnetic transition
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507
T.Dietl
Феромагнітний перехід у розведених магнітних
напівпровідниках
Т.Діетль
Інститут фізики і Коледж природничих наук Польської академії наук,
Польща, PL-02668 Варшава, ал. Льотнікув, 32/46,
Отримано 6 липня 1998 р.
Зроблено огляд недавніх теоретичних робіт про вплив взаємодії
Рудермана-Кіттеля-Касуї-Іоміди між локалізованими спінами у си-
стемах різної вимірності в розведених магнітних напівпровідниках
(РМН). Оскільки ця взаємодія далекосяжна, її вплив на температур-
ну і польову залежності намагніченості і спінове розщеплення зон
оцінюється в наближенні середнього поля, беручи до уваги змінену
безладом взаємодію носій-носій. Теоретичні оцінки показують, що
експериментально досяжні густини дірок достатньо високі для то-
го, щоб спричинити фазовий перехід парамагнетик-феромагнетик у
тривимірних модульованих структурах II–VI РМН. Показано, що ре-
зультати теоретичних обчислень підтверджуються недавніми маг-
нетооптичними дослідженнями вирощених методом НВЕ зразків
Cd1−xHnxTe/Cd1−4−7HgyZnz:N, що містять одну модульовану кванто-
ву яму глибиною 8 nm. Ці дослідження свідчать про наявність фе-
ромагнітного переходу, спричиненого двовимірним дірковим газом.
Перехід відбувається при температурі між 1.8 і 2.5 K, залежно від кон-
центрації марганцю x, що узгоджується з теоретичною моделлю.
Ключові слова: напівмагнітні напівпровідники, взаємодія RKKY,
квантові ями
PACS: 75.50.Rr, 75.30.Hx, 75.50.Dd, 78.55.Et
508
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