Magnetic properties of the new diluted magnetic semiconductor Zn1-xMnxAs2: evidence of MnAs clusters

Article abstract of DOI:10.1088/0953-8984/11/2/018

The preparation and magnetic properties of the diluted magnetic semiconductor Zn_1-xMn_xAs₂, opening a novel group of II-V₂ compounds alloyed with transition metal elements, are reported for the first time. Single crystals of Zn_1-xMn_xAs₂ are obtained with a modified Bridgman method. For x ≤ 0.2 their structure is isomorphic to the ZnAs₂ parent compound. Magnetic properties of Zn_1-xMn_xAs₂ are investigated for x = 0.01, 0.05 and 0.1 between T = 2 and 500 K in fields up to 60 kG. All the samples show a steep decrease of magnetization well above 300 K. In low fields (2-200 G) the temperature dependence of the magnetic susceptibility is strongly irreversible below T_b ≈ 250 K. The magnetization displays nonlinear field dependence starting from B approx. 1-2 kG and reaching full saturation above 40 kG. Its temperature dependence is weak between 2 and 300 K. The magnetic properties are explained by the presence of MnAs clusters. The distribution of cluster sizes is described by three overlapping Gaussian functions with the maxima at 1.7, 2.4 and 3.3 nm. At 300 K the corresponding magnetic moments are found to be 1.5, 4.1 and 10.4 in units of 10³ μ_B.

Full text of DOI:10.1088/0953-8984/11/2/018

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J. Phys.: Condens. Matter 11 (1999) 555–568. Printed in the UK
PII: S0953-8984(99)88321-0
Magnetic properties of the new diluted magnetic
semiconductor Zn1 Mn As : evidence of MnAs clusters
−x
x
2
R Laiho†, K G Lisunov†‡, E L a¨ hderanta† and V S Zakhvalinskii†‡
†
‡
Wihuri Physical Laboratory, University of Turku, FIN-20014 Turku, Finland
Institute of Applied Physics, Academiei str. 5, MD-2028 Kishinev, Moldova
Received 10 October 1997, in final form 22 September 1998
Abstract. The preparation and magnetic properties of the diluted magnetic semiconductor
Zn1−xMnxAs2, opening a novel group of II–V2 compounds alloyed with transition metal elements,
are reported for the first time. Single crystals of Zn1−xMnxAs2 are obtained with a modified
Bridgman method. For x 6 0.2 their structure is isomorphic to the ZnAs2 parent compound.
Magnetic properties of Zn1−xMnxAs2 are investigated for x = 0.01, 0.05 and 0.1 between T = 2
and 500 K in fields up to 60 kG. All the samples show a steep decrease of magnetization well
above 300 K. In low fields (2–200 G) the temperature dependence of the magnetic susceptibility
is strongly irreversible below Tb ≈ 250 K. The magnetization displays nonlinear field dependence
starting from B ∼ 1–2 kG and reaching full saturation above 40 kG. Its temperature dependence
is weak between 2 and 300 K. The magnetic properties are explained by the presence of MnAs
clusters. The distribution of cluster sizes is described by three overlapping Gaussian functions with
the maxima at 1.7, 2.4 and 3.3 nm. At 300 K the corresponding magnetic moments are found to
3
be 1.5, 4.1 and 10.4 in units of 10 µB .
1. Introduction
A small amount of transition metal element (Mn, Fe or Co) incorporated in a non-magnetic
semiconductor matrix has usually a strong influence on its magnetic, optical and transport
properties [1]. Such materials, or diluted magnetic semiconductors (DMSs), have interactions
not present in the parent compounds. Among them is the s–d exchange interaction between
band carriers and localized magnetic moments of the transition metal ions and the d–d
interaction between the ions themselves. The best known DMSs belong to the II–VI and
IV–VI groups. Their properties are reviewed e.g. in [1] and [2]. For a long time these materials
have been regarded as typical representatives of DMSs, giving minor attention to other diluted
systems. On the other hand, in recent investigations of II3V2 ternary, (Cd1−xMnx)3As2 and
(Zn1−xMnx)3As2, and quaternary, (Cd1−x−yZnxMny)3As2, compound alloys [3–7] unusual
phenomena not observed earlier in the II–VI and IV–VI DMSs were discovered. Among them
are coexistence of two spin-freezing effects, one at a low (T < 4 K) [3, 4] and the other at a
high (T ∼ 200 K) [5–7] temperature in one and the same material, anomalous low-temperature
electronic properties [8] and strong dependence of the electron effective mass on the applied
magnetic field [9, 10].
However, the group of the II–V semiconductors is much larger than the group of the II3V2
compounds containing the materials mentioned above.The unusual properties of the latter
stimulate interest to find new DMSs among other II–V compounds. In this work we report our
effort to extend the group of DMSs to II–V2 semiconductors and communicate results of the
first investigations of the new alloys Zn1−xMnxAs2.
0
953-8984/99/020555+14$19.50 © 1999 IOP Publishing Ltd
555

5
2
2
56
R Laiho et al
. Preparation of the crystals
.1. Properties of the ZnAs2 host material
Zn1−xMnxAs2 can be obtained by substituting Mn for Zn in ZnAs2. The properties of the
host material are reviewed in [11]. In particular, the crystal structure of ZnAs2 belongs to
5
the β-ZnP2 type (space group P21/c, or C ) having eight formula units in the unit cell with
2
h
◦
the parameters a = 9.287 Å, b = 7.61 Å, c = 8.01 Å and β = 102.5 [12]. However,
the structure of ZnAs2 differs from that of β-ZnP2 by the absence of Zn–Zn bonds. Along
with Zn–As bonds also As–As bonds are present. The As atoms form chains along the c-axis
which are connected with Zn atoms [11]. Electronic properties of ZnAs2 are governed by
the relatively wide band gap (0.9 eV) and impurities or intrinsic defects. Undoped crystals
1
4
16
−3
are p-type semiconductors with hole concentration p = 10 –10 cm [13, 14] at 300 K.
By doping with Se or Te [14] the p-type conduction can be changed to n-type with electron
16
−3
concentration n = 8 × 10 cm [14]. The anisotropic structure of ZnAs2 leads to the
dependence of certain physical quantities on the crystal direction, e.g. the Hall coefficient
R[100]/R[001] = 5 and the resistivity ρ[100]/ρ[001] = 100 [15].
2.2. Preparation and characterization of Zn1−xMnxAs2
First we investigated the interval of forming Zn1−xMnxAs2 solid solutions within the system
of ZnAs2–MnAs2 alloys. The samples were synthesized in the interval of 0 6 x 6 0.5
using the modified Bridgman method (slow cooling of a melt in the presence of a temperature
gradient in a furnace). Stoichiometric amounts of Zn, Mn and As (99.999% purity) were
mixed and put into a quartz ampoule after covering its inner wall with a layer of carbon. The
◦
◦
synthesis was performed in vacuum at 1000 C (the melting point of ZnAs2 is 771 C) by
◦
◦
−1
◦
cooling the material down to 800 C at the speed of 50 C h and further to 700 C at the
speed of 10 C h . The polycrystalline ingots were analysed with x-ray powder diffraction
◦
−1
5
2
measurements. A monoclinic structure (space group P21/c, or C ) with z = 8 and lattice
h
◦
parameters a = 9.28 Å, b = 7.68 Å, c = 8.03 Å and β = 102.3 was identified unambiguously
for x = 0.3. The positions of the diffraction peaks correspond well with those of ZnAs2.
Incorporation of Mn in the lattice was revealed by the clearly observed x-ray fluorescence
background and by variations in the intensities of the x-ray diffraction lines. Generally, the
applied powder diffraction method allowed the phase analysis of the system ZnAs2–MnAs2 in
the interval of 0 6 x 6 0.5 establishing the upper boundary for existence of Zn1−xMnxAs2
solid solutions as x ≈ 0.2–0.3. No dependence of the lattice parameters on the composition
could be established.
Single crystals of Zn1−xMnxAs2 were grown in the interval of x = 0–0.2 by increasing
◦
−1
the amount of the starting materials, prolonging the time of the synthesis at 1000 C to 20 h
◦
and decreasing the cooling rate in the domain of crystallization down to 1.5–2 C h . X-ray
analysis showed good agreement with the results obtained for the polycrystalline ingots and
interplane distance in a satisfactory agreement with that of ZnAs2. This shows that substitution
of Zn for Mn in ZnAs2 practically does not distort the ZnAs2 type crystal structure up to
x = 0.2. No trace of a second phase could be detected within this interval of the composition.
The concentrations of Zn, Mn and As in the ingots were controlled by weighing the amounts
of the starting materials with accuracy of 0.001%. No material fractions or traces of volatile
components could be found inside the crucible containing the ingot. X-ray phase analysis
gave the same diffraction line patterns for specimens prepared from different parts of the
ingots having x 6 0.2. This brings the error in determination of x close to that of the x-ray
analysis, about 1.5%. At higher values of x weak peaks due to an impurity phase were found

Magnetic properties of Zn1−xMnxAs2
557
in polycrystalline samples of the system. Small single crystals of As were segregated from
this phase in some ingots with x between 0.3–0.5.
3. Magnetic measurements
Dc magnetic measurements were made between 2 and 300 K with a SQUID magnetometer
using unoriented Zn1−xMnxAs2 crystals with x = 0.01, 0.05 and 0.1, and an oriented specimen
with x = 0.05. The values of x 6 0.1 were chosen below and far enough from the upper
boundary of the interval of the solid solutions. The temperature of the sample was controlled
with flowing helium gas. Before every measurement the sample was annealed for 1.5–3 h at
◦
1
40 C to avoid the influence of possible remanent magnetization.
Thetemperaturedependenceofthemagnetization, M(T ), inweak(2–200G)andmoderate
(0.2–10 kG) fields was measured after cooling the sample in zero (B < 0.1 G) field (ZFC)
from 300 to 2 K or while cooling it in a field (FC). As can be seen from figure 1 the plots
of χZFC(T ) and χFC(T ) (χ = M/B) measured in low fields diverge clearly in the region
of T . 300 K. Additionally, χZFC(T ) has a broad maximum around Tb ≈ 250 K. The
difference between χZFC(T ) and χFC(T ) decreases strongly on increasing the field of the
measurement in the interval of 2–200 G (for convenience the plots in figure 1 are presented
so that their χFC(T ) parts coincide with the χFC(T ) measured at B = 2 G). The observed
irreversibility is suppressed as B is increased up to 0.5–0.7 kG and disappears completely
above 2 kG.
Figure 1. Temperature dependences of χZFC (open symbols) and χFC (closed symbols) for
Zn1−xMnxAs2 with x = 0.05, measured at B = 2 G (1), 5 G (2), 20 G (3), 50 G (4) and 200 G (5).
The scale of the χ-axis corresponds to χZFC and χFC for B = 2 G. The other plots are shifted so
that their FC parts coincide with that for B = 2 G. Inset: χZFC and χFC against T for samples
with x = 0.01 and 0.1, measured in the field of 10 G.

558
R Laiho et al
In figure 2 is displayed the magnetic field dependence of the magnetization, M(B), at
different temperatures for the sample with the Mn concentration x = 0.01, and in figure 3 for
that with x = 0.05. The magnetization increases at first very rapidly with increasing field but
starts to deviate from a linear growth at B ≈ 1–2 kG. In the sample with x = 0.01 it reaches
a broad maximum around Bmax ≈ 20–30 kG but this maximum is not observed when x is
increased to 0.05. The temperature dependence of the magnetization between 2 and 200 K is
quite weak. A measurable difference of the magnetization for the specimen with x = 0.05
and the field oriented perpendicular and parallel to the (100) plane is evident from figure 4.
Figure 2. Dependence of the magnetization of Zn1−xMnxAs2 on the magnetic field for x = 0.01
and T = 300 K (1), 200 K (2), 100 K (3), 20 K (4) and 2 K (5). Inset: magnetic field dependence
of the magnetization corrected by subtracting the diamagnetic contribution (Md ) for the same
∗
∗
∗
specimen and T = 300 K (1 ), 100 K (3 ) and 2 K (5 ).
To complete the picture above we have investigated the temperature dependence of the
susceptibility, χ, between 250 and 500 K for B = 20 G, 200 G, 2 kG and 60 kG. As can
be seen from figure 5 both χZFC(T ) and χFC(T ) decrease strongly in the temperature region
above 300 K suggesting a ferromagnetic (FM) transition in the specimen with x = 0.01. The
transition temperature, TC, increases when B is increased. Additionally, the measurements in
the field of B = 60 kG (squares) show that at T > 370 K the susceptibility is negative and
that above 410 K it becomes roughly independent of temperature, demonstrating the presence
of a diamagnetic contribution, Md, in the magnetization of this sample. Subtraction of Md
leads to disappearance of the maxima in the M(B) curves (see the inset to figure 2). For the
specimens with x = 0.05 and 0.1 a similar transition is observed (see figure 6), with TC being
close to that of the sample with x = 0.01.

Magnetic properties of Zn1−xMnxAs2
559
Figure 3. Dependence of the magnetization of Zn1−xMnxAs2 for x = 0.05 on the magnetic field
for x = 0.05 and T = 300 K (1), 200 K (2), 100 K (3), 20 K (4) and 2 K (5). Inset: ηM/σs against
B for T = 1.9 K (a) and T = 20 K, 50 K, 100 K, 200 K and 300 K (b).
Figure 4. Dependence of the magnetization of Zn1−xMnxAs2 (x = 0.05) on the magnetic field
∗
for B k (100) at T = 200 K (1), 50 K (2) and 5 K (3) and B ⊥ (100) at T = 200 K (1 ) and
∗
5
K (3 ). The solid lines are fits with equation (4).

560
R Laiho et al
Figure 5. Susceptibility against temperature in Zn1−xMnxAs2 for x = 0.01 and B = 20 G (1),
2
00 G (2), 2 kG (3) and 60 kG (4). The open and the closed symbols represent χZFC and χFC,
respectively. The dashed line indicates the Curie temperature of MnAs.
Figure 6. Susceptibility against temperature in Zn1−xMnxAs2 for x = 0.01 (1), 0.05 (2) and
0
.1 (3) and B = 200 G. The open and the closed symbols represent χZFC and χFC, respectively.
The dashed line exhibits the Curie temperature of MnAs. For convenience the data (2) and (3) are
shifted along the χ axis by the amounts shown above the corresponding curves.

Magnetic properties of Zn1−xMnxAs2
561
4. Discussion
The magnetic properties of Zn1−xMnxAs2 described above give strong evidence for presence
of small FM clusters of MnAs in all the samples investigated. Above TD = 394 K bulk MnAs
has hexagonal B81 (NiAs-type) structure and below TD the orthorhombic B31 (MnP-type)
structure. Both these phases are paramagnetic (PM) [16]. At TCc ≈ 306 K (on cooling in a
zero field) a first-order magnetostructural phase transition takes place whereupon the hexagonal
modification B81 is while spontaneous (FM) magnetization, σs(T ), appears. On heating at
B = 0 σs vanishes at TCh ≈ 317 K [16]. The average temperature of TCc and TCh for MnAs is
shown by the dashed line in figures 5 and 6. The values of TC in the investigated Zn1−xMnxAs2
specimens are close to this line. When B is increased from 0 to 60 kG TCh and TCc are shifted to
higher temperatures by about 20 and 30 K, respectively [16]. A similar shift of TC is observed
between 20 G and 60 kG in Zn1−xMnxAs2 with x = 0.01 (figure 5). It is worth mentioning
that the steep decrease of magnetization at TIC ≈ 310 K indicating the presence of MnAs
clusters was observed recently in the III–V DMS In0.82Mn0.18As [17].
From the difference between the magnetization of Zn1−xMnxAs2 at T . TC and T & TC
−3
and the value of σs = 630 emu cm (300 K) [18] the volume fraction, η, of MnAs was
estimated to be 0.023 and 0.035 for the samples with x = 0.05 and 0.10, respectively, and
less than 0.005 for that with x = 0.01. This corresponds to fractions β = 0.97 and 0.73 of
the total amount of Mn entering the MnAs clusters in Zn1−xMnxAs2 for y = 0.05 and 0.10,
respectively. The values of η lie definitely beyond the sensitivity of the x-ray analysis. Since
the overwhelming majority of Mn ions is located in the clusters no regular dependence of the
lattice parameters on composition is expected, in agreement with our observations.
For a detailed analysis over extended temperature regions on both sides of TC it is
convenient to choose the sample with x = 0.05, because the amount of free (i.e. not entering
the clusters) Mn ions in this sample is estimated to be much smaller than in the specimen with
x = 0.10 and the diamagnetic contribution to the net magnetization is negligible compared
with the specimen having x = 0.01. The latter is evident from (i) the absence of the maxima in
the M(B) curves (cf figures 3 and 4 and figure 2) in the sample with x = 0.05 and (ii) the fact
that the negative susceptibility (figure 5) is observed in our experiments only in the specimen
with x = 0.01.
In figure 7 is shown the saturation magnetization, Ms, of Zn1−xMnxAs2 (triangles) along
with the dependence of σs on T for MnAs [18] extrapolated to T = 0 and multiplied by
the factor of η = 0.023 (dashed line). The functions Ms(T ) and ησs(T ) coincide down to
T ≈ 50 K. Hence, the volume fraction of MnAs in Zn1−xMnxAs2 and, consequently, the
fraction of Mn ions in MnAs, β = 0.97, are the same with those obtained above from the data
in the vicinity of TC. Suggesting that the deviation of Ms(T ) from ησs(T ) below 50 K is due
to a PM contribution, Mp(T ), of the free Mn ions, we fit Ms(T ) between T = 2 and 300 K
∗
∗
with the sum of ησs(T ) + Mp(T ). Here Mp(T ) ≈ χp(T )B with B ≈ 40 kG being the field
where the contribution of the MnAs clusters to the net magnetization is practically saturated
while that of the free Mn ions is still sufficiently far from saturation and can be approximated
reasonably with a linear function. The susceptibility χp(T ) satisfies the Curie–Weiss law
χp(T ) = C1/(T − θ1).
(1)
2
eff
2
In equation (1) C1 = p µ Nf /3kB is the Curie constant, peff is the effective number
B
of Bohr magnetons per free Mn ion, Nf is their concentration and θ1 is the Curie–Weiss
temperature determined by the interaction between the free Mn ions. From the best fit with the
−
4
−3 −1
G
observed data of Ms(T ) we obtain C1 = 1.3 × 10 emu K cm
and θ1 = −4.2 K. The
negative sign of θ1 implies that the interaction between the free Mn ions in Zn1−xMnxAs2 is

562
R Laiho et al
Figure 7. Temperature dependence of the saturation magnetization of Zn1−xMnxAs2 for x =
0
.05 (triangles) compared with the reduced saturation magnetization of MnAs ησs (line 1) and the
sum of ησs and the PM contribution of free Mn ions (line 2). Inset: temperature dependence of the
inverse susceptibility above TC.
2
+
antiferromagnetic (AF) as in all known DMSs containing Mn. With peff = 35 for Mn
19
−3
we obtain Nf = 1.8 × 10 cm and the fraction of free Mn ions, 0.026, corresponding
to β = 0.974 which agrees very well with the value of β obtained above in the interval of
5
0–300 K and for T & TC. We conclude, that the temperature dependence of the saturation
magnetization in Zn1−xMnxAs2 is most probably determined by that of the MnAs clusters with
minor influence of the free Mn ions below 50 K.
At temperatures above TC, namely between 320 and 380 K, the susceptibility exhibits
the Curie–Weiss law again (see the inset to figure 7) but with other values of the Curie–Weiss
−
3
−3 −1
G
temperature, θ2 = 310±20 K, andtheCurieconstant, C2 = (3.3±0.3)×10 emuKcm
corresponding to peff = 4.8 ± 0.4 per Mn ion. These values coincide within the experimental
error with θ = 285 K and peff = 4.45 in the PM hexagonal phase of bulk MnAs [16] existing
above 394 K. In bulk MnAs the width of the orthorhombic phase is 80–90 K. As follows from
the shift of TC with B, about 30 K (figure 6), in the sample with x = 0.01 and from the value
of TC & 300 K and the onset of the Curie–Weiss law (∼320 K, figure 7) in the sample with
x = 0.05 it does not exceed ∼20–30 K in MnAs clusters of Zn1−xMnxAs2.
The above discussion gives sufficient evidence that the magnetic properties of
Zn1−xMnxAs2 are determined mostly by small ferromagnetic MnAs clusters. Generally,
small single domain FM particles incorporated in a solid matrix can exist in two different
states [19]. At T > Tb or above the blocking temperature thermal fluctuations can cause
a sort of Brownian rotation of the magnetic moment of the particles with the result that
an assembly of such particles exhibits superparamagnetic (SP) behaviour. In this case M,
corrected against the temperature dependence of the saturation magnetization of the clusters,
would be a Langevin-type function of the ratio B/T . At T < Tb the moments of the particles
are blocked, with their directions distributed at random over the sample volume violating
the B/T -scaling of M. There are two conceivable sources of the blocking barriers: (i) the

Magnetic properties of Zn1−xMnxAs2
563
anisotropy energy of individual particles and (ii) the dipolar interaction between the moments
of different particles.
In the case (i) the moment of each particle is stabilized independently when its anisotropy
energy, KV , becomes high enough to counteract the thermal excitations ∼kBT . Here K is
the density of the anisotropy energy and V is the volume of the particle. After removal of the
external field the moments of the particles relax towards an equilibrium state. The relaxation
time, τ, is given by 1/τ = f0 exp(−KV/kBT ), where f0 is a frequency factor of the order of
9
−1
2
1
0 s . When the value of τ = 10 s is used as the criterion for transition to stable behaviour
the energy barrier must be 25 kBT . Then the blocking temperature can be written as [19]
(
Tb
anis)
= KV/25 kB.
(2)
The temperature dependence of M is determined only by that of the saturation magnetization
of the FM particles, σs, so that the ratio M/σs would be temperature independent.
In case (ii) the blocking temperature and the magnetization will satisfy the equations (see
equation (A10) and (A11) in appendix A)
(
Tb
inter)
2
1/2
= µ I0 /3 kB
(3)
and
M = MsB/(B + ξMs)
(4)
respectively. In equation (4) Ms = µN is the saturation magnetization of the specimen, µ and
N are the moment and the concentration of the particles (neglecting the distribution of their
6
−1/3
is the mean distance between particles and
sizes), I0 = zI /r , zI = 11.6, r = 2(4πN/3)
ξ = 17.3.
The blocking transition (i.e. transition from the SP to stable behaviour) of the assembly of
single-domain particles can be identified by deviation of χZFC(T ) from χFC(T ). The blocking
temperature Tb is defined by the cusp of χZFC(T) meaning that for T > Tb the thermal energy
is large enough to enable the clusters to be oriented by an external magnetic field. Below
Tb clusters cannot overcome the blocking barriers (independent of their nature) [19] with the
help of thermal excitations. Blocking temperatures as high as the room temperature were
established e.g. in heterogeneous metallic Cu–Co alloy films containing nanometre scale FM
Co-rich clusters [20] and in III–V semiconductor GaAs with Fe3GaAs precipitates [21]. Lower
values of Tb were observed in some granular systems demonstrating SP response of the clusters
and related giant magnetoresistance (see [21] and references therein). Accordingly, the high-
temperature irreversible phenomena shown in figure 1 can be attributed to the presence and
blocking of the MnAs cluster moments in the samples. It is interesting to note that the blocking
transition is broad in Zn1−xMnxAs2 and overlaps with the transition from PM to FM state in
MnAs clusters (cf figures 1, 5 and 6). Consequently, the SP region corresponding to unstable
cluster moments is practically unobservable.
Asevidentfromfigure3theratioofM/σs isinallfieldsindependentofT withintheerrorof
2–3%, excluding the data at 2 K which are influenced by increasing PM contribution of the free
Mn ions when the temperature is lowered (figure 7). This is consistent with the assumption that
the anisotropy energy barriers are the main reason for stabilizing the cluster moments. From
the temperature dependence of χZFC(T ) in the lowest applied fields (figure 1) we calculate
the distribution function of the blocking temperatures (equation (B3) in appendix B),
ꢀ
ꢁ
1
d
T χZFC(T )
F(T ) =
− α.
(5)
2
γ dT
σ (T )
s
The distribution function, f (R), of the cluster radius, R (or more strictly the radius of the
sphere of equivalent volume) is calculated using the relation between the cluster volume and

564
R Laiho et al
Figure 8. The distribution function of the cluster radius of Zn1−xMnxAs2 for x = 0.01 (line 1),
0
.05 (line 2) and 0.1 (line 3). Inset: composition of the distribution function for the compound
with x = 0.05 (×) by three overlapping Gaussian distributions.
the blocking temperature given by equation (2) and the dependence of K on T [18]. The
constants γ and α are determined by normalizing f (R) to unity and using the condition
f (R) = 0 at R = 0. As evident from figure 8(a) the normalized distribution function
for the sample with x = 0.05 has a sharp peak at R1 ≈ 3.3 nm. Analysis of f (R)
with the multiple Gaussian function reveals two other peaks centred at R2 ≈ 2.4 nm and
R3 ≈ 1.7 nm. As follows from figure 8(b) the size distribution functions are similar for all
the samples. When x is increased the main peak is sharpened leading to some shift of the
cluster sizes towards R1. The calculated moments at 300 K corresponding to the values of
3
3
3
R1, R2 and R3 are µ1 ≈ 10.4 × 10 µB, µ2 ≈ 4.1 × 10 µB and µ3 ≈ 1.5 × 10 µB,
3
respectively. The average values hRi ≈ 3.1 nm and hµi ≈ 8.8 × 10 µB are evaluated
with the distribution function shown in figure 8(a). The concentration of clusters can be
calculated from the equation N ≈ Ms/σshV i where hV i is the mean cluster volume. The
17
−3
obtained value N ≈ 1.8 × 10 cm corresponds to the mean intercluster distance, hri =
−
1/3
2
(4πN/3)
≈ 22 nm. The cluster parameters found above permit us to compare the scales
6
−3
of the anisotropy energy of a mean cluster, Wa ≈ |K|hV i ≈ 0.5 eV (K ≈ −6 × 10 erg cm
2
3
at 300 K [18]), and of the energy of dipolar interaction, Wd ≈ zJ hµi /hri ≈ 0.013 eV.
The long-range character of the dipolar interaction is taken into account by introducing a
constant zJ ≈ 33 stemming from extension of the interaction beyond the nearest neighbours
(
see comments on equation (A2) in appendix A). Since Wd is less than 3% of Wa the dipolar
interaction can be neglected and the description of the magnetic properties of Zn1−xMnxAs2
given above is consistent. The difference between the magnetization measured for B k (100)
and B ⊥ (100) does not vanish at B ≈ 50 kG. On the other hand, the anisotropy field
of MnAs is three times lower, Ba = 18.3 kG at 300 K [18]. This suggests that the
observed anisotropy of the magnetization is connected with some non-sphericity of the cluster
shape.

Magnetic properties of Zn1−xMnxAs2
565
Finally, we discuss the magnetization data assuming that blocking of the cluster moments
is caused only by the dipolar interaction between the clusters, i.e. neglecting the influence of the
anisotropy energy Wa and the distribution of the cluster sizes. The dependence of M on B for
all the specimens can be fitted with equation (4) using the product of V N (since Ms = σsV N)
and ξ as adjustable parameters. An example is shown in figure 4, giving V N = 0.017±0.001
(
0.020±0.001) and ξ = 15±1 (7±1) for the cases of the parallel (perpendicular) orientation
of B with respect to the (100) plane. Hence, the value of ξ differs somewhat from 17.3,
predicted by the model of interacting clusters and, additionally, depends on the direction of the
magnetic field. The values of V and N can be found separately with equation (3), using those
5
of V N and Tb ≈ 250 K. This gives the values of R = 10.9 nm (10.4 nm), µ = 4.6 × 10 µB
5
15
−3
15
−3
(
4.0 × 10 µB), both at Tb, N = 3.2 × 10 cm (3.7 × 10 cm ) and r = 84 nm (80 nm).
With the values of µ and r above we evaluate Wa ≈ 19.3 eV (16.8 eV) and Wd ≈ 0.67 eV
(
0.55). The value of Wa ꢀ Wd contradicts with the assumption made in this paragraph that
our magnetization data could be explained by attributing the blocking of the cluster moments
to the dipolar interactions.
5. Conclusions
In this work the temperature and the field dependences of the magnetization of Zn1−xMnxAs2
are investigated for = 0.01, 0.05 and 0.1, opening a novel group II–V2 of DMSs. Analysis
of the data gives strong evidence for presence of small MnAs clusters in this compound.
The irreversible phenomena observed in Zn1−xMnxAs2 starting from x = 0.01 reflect the
blocking transition in dynamics of the magnetic moments at temperatures comparable with
the ferromagnetic Curie temperature of bulk MnAs. The dependence of M on B is consistent
with the behaviour of the magnetization below the blocking temperature. The analysis of
the temperature dependence of the saturation magnetization demonstrates that the majority of
Mn ions in Zn1−xMnxAs2 are located in the MnAs clusters. The distribution function of the
sizes, the magnetic moments and the concentration of the clusters are obtained. Comparing
the anisotropy energy of individual particles and the energy of intercluster dipolar interaction
it is concluded on basis of our magnetization data that the anisotropy energy of the clusters
is the reason for the blocking of their moments. A dependence of the magnetization on the
direction of the applied magnetic field suggests some non-sphericity of the cluster shape.
Appendix A
The blocking of the moments due to the dipolar interaction is a cooperative process similar to
freezing of spins in spin-glasses. Therefore, this phenomenon can be analysed in a similar way
using the mean-field theory. Following the method proposed in [22] we treat a disordered lattice
of interacting cluster moments, µ, in the dipolar approximation, neglecting the distribution of
∗
the cluster sizes. Then the equations defining the mean field B at the ith site may be written as
i
ꢂ
ꢃ
∗
X
µB
kT
∗
j
B = B +
Jij µL
(A1)
i
j
where B is the external field, L(y) = cotanh(y) − 1/y is the Langevin function and
2
3
ij
Jij = (1 − 3 cos ϕij )/r are random variables describing the dipolar interaction. Here ϕij is
∗
the angle between the direction of the field B at the site j and the vector distance rij between
j
the sites i and j. In equation (A1) the summation over the sites j is extended to infinity due
to the long-range character of the dipolar interaction. To describe the distribution of the fields

566
R Laiho et al
∗
∗
∗ 2
∗ 2
B we introduce the mean value, hB i, and the variation, h(B ) i − hB i , where the angular
i
parenthesis mean the averaging over the configurations. Assuming that the random variables
ϕij and Rij in different sites are uncorrelated, we obtain from equation (A1) in a standard way
[22] the following relations,
∗
∗
hB i = B + J0hµL(µB /kT )i
(A2)
(A3)
and
∗
2
∗ 2
2
2
∗
h(B ) i − hB i = I hµ L (µB /kT )i.
0
P
P
3
ij
3
In equations (A2) and (A3) J0 =
hJij i = −(1/2)
hr i = −zJ /r and I0 =
j
j
P
P
2
ij
2
6
ij
6
−1/3
is the mean distance
(
hJ i − hJij i ) = (11/8)
hr i = zI /r . Here r = 2(4πN/3)
j
j
between the nearest sites, and the values of the constants zJ ≈ 33 and zI ≈ 11.6 are obtained
by performing the numerical summation.
The averaging over the configurations in these equations can be made using the Gaussian
∗
∗
2
2
1/2
∗
distribution of fields, f (B ) = exp[−(B − B1) /2B ]/[(2π) B2], with B1 ≡ hB i and
2
2
2
∗ 2
∗ 2
B ≡ h(B ) i − hB i . This gives
Z
ꢀ
ꢁ ꢂ
ꢃ
∗
∗
2
∗
J0µ
dB
(B − B1)
µB
kT
B1 = B +
√
exp
−
L
(A4)
(A5)
2
2B
2
B2
2π
and
Z
ꢀ
ꢁ
ꢂ
ꢃ
2
∗
∗
2
∗
µB
I0µ
dB
(B − B1)
2
2
B =
√
exp
−
L
.
2
2
2B
2
B2
2π
kT
It is convenient to analyse the mean-field equations (A4) and (A5) by introducing an
analogue of the Edwards–Anderson order parameter, q, and local magnetization, m [22, 23],
defined as
∗
∗
q = hL2(µ /kT )i and m = µhL(µB /kT )i
(A6)
B
respectively. They are connected to the parameters B1 and B2 by the relations
2
2
2
B1 = B + J0m and B = I0µ q
(A7)
following from equations (A2), (A3) and (A6) and the definition of B1 and B2. Replacing
B1 and B2 in equations (A4) and (A5) with those in equations (A7), and substituting
∗
1/2
B = B + J0m + tµ(I0q) we obtain finally the following system of equations for q and m,
Z
ꢂ
ꢃ
ꢀ
ꢁ
2
p
dt
t
µ
2
q =
√
exp
−
L
(B + J0m + tµ I0q)
(A8)
2
π
2
kT
and
Z
ꢂ
ꢃ ꢀ
ꢁ
2
p
dt
t
µ
m = µ
√
exp
−
L
(B + J0m + tµ I0q) .
(A9)
2
π
2
kT
The macroscopic magnetization is then M = mN.
The state of stable moments is defined by q
state corresponds to the interval of non-zero solutions of equation (A8) in zero field given by
inequality T 6 Tb
6= 0 and m = 0 at B = 0. Therefore, this
(
inter)
where
(
Tb
inter)
2
1/2
= µ I0 /3 kB.
(A10)
2
Since L(y) is a finite function (L(y) → 1 as y → ∞ and exp(−t /2) decays rapidly
with increasing t) the expression under the integral in the right-hand side of equation (A9)
is determined by small values of t. Therefore, for high fields the third term in the argument
of the Langevin function can be neglected, resulting in m ≈ L[µ(B + J0m)/kBT ]. This

Magnetic properties of Zn1−xMnxAs2
567
equation is solved in the limits of µB/kBT ꢀ 1 and B ꢀ |J0|m taking into account the
asymptotic form of the Langevin function, L(y) ≈ 1 − 1/y at y ꢀ 1. The solution is given
by m ≈ (µB + kBT )/(B + |J0|µ). For large cluster moments we have then µB ꢀ kBT
resulting in a weak temperature dependence of the magnetization. Finally, the macroscopic
magnetization M = mN can be expressed in the form
MsB
M ≈
(A11)
B + ξMs
where Ms = µN and ξ = πzJ /6 ≈ 17.3.
Appendix B
The distribution function of the blocking temperatures can be obtained with the following
method [24]. The ZFC susceptibility of a system of identical clusters satisfies the conditions
χZFC = χ0 at T < Tb and χZFC = C/T at T > T0, where C is the Curie constant for clusters.
The low-field contribution to the susceptibility from the stable clusters is small and depends
on T only by σs. Then the susceptibility of a system of arbitrary clusters can be written as
Z
T
C
∗
∗
χZFC(T ) = χ0 +
F(T ) dT
(B1)
T
0
where the first and the second terms on the right-hand side represent the contributions of the
stable and the SP clusters, respectively, and F(T ) is the distribution function of the blocking
2
temperatures. It is convenient to represent the parameters χ0 and C as χ0 = γ ασ and
s
2
C = γ σ (where α and γ are constants) depending on T only by the temperature dependence
s
of σs. Then equation (B1) can be written in the form
ꢀ
Z
ꢁ
T
1
2
∗
∗
χZFC(T ) = γ σs α + T
F(T ) dT
.
(B2)
0
Finally, F(T ) can be given in the form
ꢂ
ꢃ
1
d
χZFCT
F(T ) =
− α
(B3)
2
γ dT
σs
obtained by differentiation of equation (B2) with respect to T . The values of the constants γ
and α can be determined by normalizing F(T ) to unity and using the condition F(T ) = 0 at
T = 0, respectively.
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