Exchange interactions, spin waves, and Curie temperature in zincblende half-metallic sp-electron ferromagnets: the case of CaZ (Z = N, P, As, Sb).

Abstract

Using first-principle calculations in conjunction with the frozen-magnon technique we have calculated the exchange interactions and spin-wave dispersions in the series of the zincblende half-metallic II-V (CaZ, Z = N, P, As, Sb) ferromagnets. The calculated exchange constants are used to estimate the Curie temperature within the random phase approximation. The large Stoner gap in these alloys gives rise to well-defined undamped spin waves throughout the Brillouin zone. Moreover we show that the spin-wave stiffness constants for the considered systems are among the largest available for local moment ferromagnets. The predicted Curie temperature of half-metallic CaN is noticeably higher than the room temperature with respect to the other compounds, and thus we propose CaN as a promising candidate for future applications in spintronic devices.

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IOP PUBLISHING
JOURNAL OF PHYSICS: CONDENSED MATTER
J. Phys.: Condens. Matter 23 (2011) 296001 (6pp)
Exchange interactions, spin waves, and
Curie temperature in zincblende
half-metallic sp-electron ferromagnets: the
case of CaZ (Z = N, P, As, Sb)
A Laref1,2, E S ¸as ¸ıo˜ glu3,4and I Galanakis5
doi:10.1088/0953-8984/23/29/296001
1Department of Physics, Pohang University of Science and Technology, Pohang 790-784,
Korea
2Max-Planck-Institut f¨ ur Mikrostrukturphysik, Weinberg 2, D-06120 Halle, Germany
3Peter Gr¨ unberg Institut and Institute for Advanced Simulation, Forschungszentrum J¨ ulich
and JARA, 52425 J¨ ulich, Germany
4Department of Physics, Fatih University, 34500, B¨ uy¨ ukc ¸ekmece,˙Istanbul, Turkey
5Department of Materials Science, School of Natural Sciences, University of Patras,
Patras 26504, Greece
E-mail: larefamel@yahoo.com, e.sasioglu@fz-juelich.de and galanakis@upatras.gr
Received 6 April 2011, in final form 8 June 2011
Published 27 June 2011
Online at stacks.iop.org/JPhysCM/23/296001
Abstract
Using first-principle calculations in conjunction with the frozen-magnon technique we have
calculated the exchange interactions and spin-wave dispersions in the series of the zincblende
half-metallic II–V (CaZ, Z = N, P, As, Sb) ferromagnets. The calculated exchange constants
are used to estimate the Curie temperature within the random phase approximation. The large
Stoner gap in these alloys gives rise to well-defined undamped spin waves throughout the
Brillouin zone. Moreover we show that the spin-wave stiffness constants for the considered
systems are among the largest available for local moment ferromagnets. The predicted Curie
temperature of half-metallic CaN is noticeably higher than the room temperature with respect to
the other compounds, and thus we propose CaN as a promising candidate for future applications
in spintronic devices.
(Some figures in this article are in colour only in the electronic version)
1. Introduction
Since the discovery of half-metallic ferromagnetism in CrAs
when grown as thin film in the metastable zincblende structure
using the molecular beam epitaxy technique by Akinaga et al
[1], a lot of interest has been focused on the study of similar
pnictides and chalcogenides (for a review on these alloys
see [2] and for a wider review on half-metals see [3]) due
to their potential applications in spintronic devices and their
coherentgrowthontopofbinarysemiconductorswiththesame
lattice structure.Although the interest has mainly focused
on the compounds containing transition metal atoms, several
theoretical studies have been also devoted to the so-called sp-
electron ferromagnets. Unfortunately, contrary to transition
metal pnictides, the sp-electron ferromagnets have not yet been
synthesized experimentally. In 2003 Geshi et al were the first
topredict usingfirst-principles calculations thatCaP, CaAs and
CaSb compounds in the zincblende structure are half-metallic
ferromagnets, i.e. one spin band is metallic while the other
is semiconducting [4]. In the following year Kusakabe et al
focused on the CaAs alloy and found that ferromagnetism is
due to a flat band created by the hybridization of the localized
Ca d and As p-states [5]; the p orbitals transform using the
same symmetry group with the t2g d orbitals and thus can
hybridize. Followingthese twopioneering papers a lotofeffort
has been devoted to the study of sp-electron ferromagnets in
several lattice structures. Sieberer and collaborators studied
all possible II–V combinations in the zincblende (zb) structure
0953-8984/11/296001+06$33.00
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J. Phys.: Condens. Matter 23 (2011) 296001 A Laref et al
Table 1. Lattice parameter a in˚ A used in the calculations (CaN is taken from [6] and the rest from [4]), atom miand total mcellspin magnetic
moments in μB, Stoner gap ?S in eV and spin-wave stiffness constant D in meV˚ A
2for the CaZ (Z = N, P, As, Sb) compounds.
mcell
?SDamCa
mZ
mVoid1
mVoid2
CaN
CaP
CaAs
CaSb
5.47
6.55
6.75
7.22
0.04
0.12
0.15
0.19
0.98
0.85
0.81
0.77
0.01
0.04
0.05
0.03
−0.03
−0.01
−0.01
0.01
1.00
1.00
1.00
1.00
0.76
0.54
0.44
0.30
790
1268
1420
1554
and found that ferromagnetism exists for large unit cell
volumes [6]; Yao and collaborators expanded this study to
cover also the case where the V-column element is Bi [7].
Volnianska and Boguslawski have also considered the case of
rocksalt, NiAs and Zn3P2structures, and have shown that the
rocksalt structure is the most stable while spin polarization is
more stable for nitrides [8]. Similar work by Gao et al has
confirmed that only nitrides retain the half-metallic character
in the rocksalt structure [9].
studied the case of II–VI zincblende compounds where the
VI-column element is C, Si or Ge [10–12] while a study of
SrC also in the wurtzite structure exists in the literature [13].
Li and Yu in 2008 have shown that the inclusion of spin–
orbit coupling destroys the half-metallic character in the case
of zincblende Ca(As, Sb, Bi) alloys but not for the lighter
CaN and CaP compounds [14].
CaC [15], and the properties of surfaces and interfaces with
binary semiconductors of the Ca and Sr nitrides [16] have also
been studied. Finally the case of I–V compounds (Li, Na, K)
(N, P, As) has been studied and nitrides have also been found
to be half-metallic ferromagnets [17].
Although the electronic properties and the spin magnetic
moments of the half-metallic sp-electron ferromagnets have
been studied in great detail, the stability of ferromagnetism
via the calculation of exchange constants has been explored
only in the case of carbonides.
constants for rocksalt SrC and BaC have been calculated
using the frozen-magnon approximation, and in [10] similar
calculations have been performed for the zb CaC, SrC and BaC
alloys. The leading interactions are between the carbon atoms
which mainly carry the magnetic moments. Estimated Curie
temperatures were found to be higher for the compounds in the
zb structure and CaC exhibited the highest one, reaching 735 K
within the random phase approximation (RPA). The scope of
our present study is to explore the magnetic properties such as
exchange constants, Curie temperatures and spin waves for the
zb CaZ (Z = N, P, As, Sb) alloys, which are the most studied
ones, and since, as mentioned above, the CaN and CaP retain
their half-metallic character even when spin–orbit coupling is
taken into account [14]. These properties are directly related to
the robustness of ferromagnetism and thus are relevant to any
potential applications of these alloys. To perform this study
we use the same methodology as for the zb transition metal
pnictides and chalcogenides used in [18]. In section 2 we will
provide details about the method which we adopt, in section 3
we will analyze our results and finally in section 4 we will
summarize our results and present our conclusions.
Gao and collaborators also
The surface properties of
In [12] the exchange
2. Method
The ground-state electronic structure calculations are carried
out using the augmented spherical waves method (ASW) [19]
within the atomic-sphere approximation (ASA) [20]. In the zb
lattice two empty spheres are used to account for the empty
sites. The exchange–correlation potential is chosen in the
generalized gradient approximation [21]. A dense Brillouin
zone sampling 30 × 30 × 30 is used. The radii of all atomic
spheresarechosenequal. Forourcalculationswehaveusedthe
optimized lattice constants from [4] which are almost identical
to the equilibrium lattice constants calculated in [6] (for CaN
we used the value from [6]), and we present them in table 1.
Before presenting the key points of our method to
calculate the exchange constants we should briefly first discuss
excitations in these alloys.
excitations: (i) Stoner type and (ii) magnons.
excitations involve excitations from the occupied majority-
spin states to the unoccupied minority-spin states and thus are
affected by the exchange splitting of the bands [22]. Even in
usual metals they correspond to high excitation energies and
in systems like the ones studied here, where a gap exists in
the minority-spin gap and exchange splitting is very strong,
they can be neglected even when the Curie temperature is
calculated, as shown by Edwards and Katsnelson in the case
of sp-impurity bands in CaB6 [23].
energy position of these excitations is given by the so-called
Stoner gap ?S which is the energy difference between the
highestoccupied majority-spinstateandthelowestunoccupied
minority-spinstate, and thuscorresponds tothe lowestpossible
Stoner excitation energy [22].
values from our density of states (DOS) data in table 1 for
all four compounds under study and, as shown, the values
vary between 0.30 eV for CaSb and 0.76 eV for CaN and
thus Stoner excitations can be neglected in the discussion of
thermodynamic properties [23].
Thesecondkindofexcitationsaretheso-called‘magnons’
which involve collective excitations of the spin magnetic
moments. In a simplified picture we can assume that the
magnitude of the spin magnetic moments of the atoms does
not change with respect to the 0 K value calculated using ab
initio electronic structure methods. However, as we raise the
temperature atomic spin moments change their orientation in
such a way that the azimuthal angle can be described by a
propagating plane wave characterized by a vector q belonging
in the Brillouin zone.This is the so-called spin wave or
magnon.If the magnitude of q is small we can assume
that the energy for the creation of the spin wave is given by
the relation E(q) = D|q|2and thus depends only on the
There two kind of magnetic
The Stoner
An estimation of the
We present the calculated
2

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J. Phys.: Condens. Matter 23 (2011) 296001 A Laref et al
magnitude of the wavevector, and not its orientation in the
Brillouin zone; this usually occurs around the ? point as we
will show later on when presenting the energy dispersion of
the spin waves. The constant D is referred to as the ‘spin-
wave stiffness constant’ and we present its calculated value in
table 1. In transition metal ferromagnets like Fe, Co and Ni the
value of D is about 300–600 meV˚ A2(see table 2 in [24]) and
the largest known values are 715 meV ˚ A2for Co2FeSi [25]
and 800 meV ˚ A2for Fe53Co47[26]. But as shown in table 1
and discussed in section 3, for the compounds under study it
surpasses these values ranging from 790 meV ˚ A2for CaN up
to 1554 meV ˚ A2for CaSb. Magnons dominate the lower part
of the excitation spectra and especially for the systems under
study here and, as shown by the spin-wave dispersion energies
discussed in section 3, they are separated by a gap from the
Stoner excitations. Thus the consideration only of magnons is
justified for the calculation of the thermodynamic properties of
calcium pnictides.
Electronic structure results are used to estimate the
exchange constants within the frozen-magnon approxima-
tion [27].The method of the calculation of the exchange
constantswithinthe frozen-magnon approximationhas already
been presented elsewhere [28].
reasonably self-contained, a brief overview is given. Notice
that since the magnetism is almost exclusively concentrated on
theZ = N, P, As orSb atomsand theCa–Caintrasublatticeand
Ca–Z intersublattice interactions are negligible, as confirmed
also from our calculations, the equations presented below are
for systems with only one magnetic sublattice for reasons
of simplicity. It should be noted that the frozen-magnon
technique does not allow us to access spin-wave lifetimes since
it is a static approximation. The lifetimes can be obtained
from dynamical spin susceptibility calculations within time-
dependent density functional theory (TDDFT) [29] or many-
body perturbation theory [30]. In magnetic materials with a
Stoner gap larger than the spin-wave energies such as half-
metals or semiconductors the spin waves exist throughout the
Brillouin zone and are not damped. Thus, the frozen-magnon
method and TDDFT give practically the same results for the
spin-wave dispersion as shown in [29] for the case of NiMnSb,
which has also a large Stoner gap.
To calculate the interatomic exchange interactions we
use the frozen-magnon technique [27] and map the results
of the calculation of the total energy of the helical magnetic
configurations
Here, to make the paper
sn= (cos(qRn)sinθ,sin(qRn)sinθ,cosθ)
onto a classical Heisenberg Hamiltonian
(1)
Heff= −
?
i?=j
Jijsisj
(2)
where Jijis an exchange interaction between two Z sites and
si is the unit vector pointing in the direction of the magnetic
moment at site i. Rnare the lattice vectors, q is the wavevector
of the helix, and θ the polar angle giving the deviation of the
momentsfrom the z axis. Withinthe Heisenberg model (2), the
energy of frozen-magnon configurations can be represented in
the form
E(θ,q) = E0(θ) − sin2θJ(q)
where E0 does not depend on q and J(q) is the Fourier
transform of the parameters of exchange interaction between
pairs of N (P, As or Sb) atoms:
?
Calculating E(θ,q) for a regular q-mesh in the Brillouin zone
of the crystal and performing back Fourier transformation, one
obtains exchange parameters J0Rbetween pairs of Z = N, P,
As or Sb atoms.
Regarding the Curie temperature, we employ the RPA and
the Curie temperature is given by the relation [24]
(3)
J(q) =
R
J0Rexp(iq ·R).
(4)
1
kBTRPA
C
=6μB
M
1
N
?
q
1
ω(q),
(5)
where ω(q) =
excitations, μB is the Bohr magneton, M is the magnitude
of the atomic spin magnetic moments and the sum runs over
the N values of the q-wavevector considered in the Brillouin
zone. RPA formalism takes into account only transverse
spin fluctuations (magnons) and the single-particle spin-flip
excitations (Stoner excitations) are neglected. Thus its use
is well grounded for the alloys under study here since, as
we discussed above, the latter ones play no practical role in
the thermodynamics of the systems of interest in our study.
We have not employed the mean-field approximation (MFA)
since the latter corresponds to an equal weighting of the
low- and high-energy spin-wave excitations, leading to an
overestimation of the experimental Curie temperature contrary
to RPA where the lower-energy excitations make a larger
contribution to the Curie temperature leading to more realistic
values [24, 31, 32].
4
M[J(0) − J(q)] is the energy of spin-wave
3. Results and discussion
We will start our discussion from the electronic properties of
the compoundsunder study. As mentionedabove we have used
the equilibrium lattice constants from [6] for CaN and from [4]
for the other three compounds. In table 1 we gather the values
of the lattice constants which vary from 5.47˚ A for CaN up to
7.22˚ A for CaSb. We should note here that contrary to the other
three compounds, CaN’s lattice constant is close to the values
of several zb binary semiconductors (GaP has a lattice constant
of about 5.45˚ A and GaAs of 5.65˚ A) and thus coherent growth
should be more likely to occur for this alloy due to the smaller
strains imposed by the substrate. The reason for the large
equilibrium lattice constants is the high ionicity of the calcium
pnictides. As shown by Volnianska and Boguslawski due to
the very high ionicity presented by these alloys they prefer
to crystallize in structures with higher coordination number,
like the rocksalt one, to minimize electrostatic energy and
only for very large unit cell volumes does the zincblende
structure become stable [8]. But as shown by Kusakabe et al
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J. Phys.: Condens. Matter 23 (2011) 296001A Laref et al
in the zb lattice structure, the magnetism comes from a flat
band mechanism [5] and not from mechanisms like the p–d
hybridization and the double-exchange usually occurring in
ferromagnets containing transition metal atoms. Thus these
compounds can be considered as prototype systems to study
the occurrence of this mechanism.
In figure 1 we present the total density of states (DOS)
together with the anionic projected one.
gap occurs in the majority-spin band contrary to the usual
half-metallic pnictides containing transition metal atoms like
CrAs etc [2]. There is also a gap in the minority-spin band
and the Fermi level falls within a peak of the minority-spin
DOS. Notice also that the states below the gaps in both spin
directions are almost exclusively of anionic character (N, P, As
or Sb) while the bands above the gaps are of cationic character
(Ca). Also, the width of the gaps is unusually large and the
tendency as we move along the Vth column from N to Sb is
the decrease of the gap width followed by smaller band widths.
This behavior can be traced to the band structure [5]. Each Ca
atom provides two electrons which occupy the 4s-states in the
free atom, while each anion provides in total five electrons:
two occupying the valence s-states and three the valence p-
states in the free atom. Thus in total we have seven valence
electrons. The anions create an s-band deep in energy, not
shown here, which accommodates two valence electrons. The
p-states of the anions can hybridize exclusively with the t2gd-
states of Ca, which transform following the same symmetry
group. However, these states are high in energy above the
gaps for both spin directions, and thus the weight of the bands
below the gaps is mainly of anionic p-character. The small
Ca d-admixture in the occupied states is due to the fact that
anionic p-states are extended in space and, if we express the
wavefunctions within a sphere centered at the cationic center
and use as point of origin the cationic site, then t2gd-states
appear in the expression of the wavefunctions since they are
the only ones compatible with respect to their symmetry. This
d-admixture plays a crucial role since it leads to an almost
dispersionless flat p-band exactly at the top of the other two
p bands, as shown by Kusakabe et al [5]. The splitting of
the flat bands of different spin character leads to the complete
occupation of the one in the majority-spin band and to an
empty flat band in the minority-spin band. Thus the majority-
spin p bands contain three electrons and the minority-ones
two electrons. The Ca atoms also play a second decisive role
in the appearance of half-metallicity since they provide two
valence electrons which, due to the high-energy position of
the Ca 4s-states in the crystal, occupy the p-states offered by
the anionic atom. The difference in the width of the gaps and
the bands should be mainly attributed to the large variation of
the lattice constants as we move from CaN to CaSb. Large
unit cell volumes mean less hybridization and more atomic-
like behavior of the p-states and thus more narrow bands and
smaller gaps.
We should also discuss the spin magnetic moments
presented in table 1. As discussed above, we have also used
two empty sites (Void1 and Void2) to account correctly for
the empty space in the zb structure.
also included the spin magnetic moments at these two sites,
The half-metallic
In table 1 we have
Figure 1. Total density of states (DOS) and its projection on the
anionic atom (N, P, As, Sb) for the CaZ compounds. We have set the
Fermi level as the zero of the energy axis. Note that positive values
of DOS refer to the majority-spin electrons and negative values to the
minority-spin electrons.
but since the charge in the two voids is very small the spin
moments are also negligible. As expected from the discussion
on the DOSs, the magnetic moments are mainly carried by the
anions. Especially for CaN, where N is the lighter element,
the contribution of Ca to the total spin moment can be safely
neglected. As we move from CaN to CaSb the spin moment
at the anionic sites decreases and it is counterbalanced by an
increase of the spin moment at the Ca site which reaches a
value of 0.19 μBfor CaSb. The observed trend is expected
since the spin splitting for the valence 5p-states of the heavy
Sb atom is more difficult to achieve than for the valence 2p
states of the light N atom. In all cases the total spin magnetic
moment is exactly 1 μB. In the zincblende alloys like the
ones under study, irrespective of whether or not they contain
transition metal atoms, we have exactly four states below the
half-metallic gap: one s-like and three p–d ones (in the sp-
electron ferromagnets under study the p-character dominates
with respect to the transition metal ones [33]). Thus the total
spinmagneticmomentinμBfollowsa Slater–Pauling behavior
and should be |8− Zt|, where Ztis the total number of valence
electrons in the unit cell. Here Ca contributes two valence
electrons and the anionic atoms five, and thus in total we have
seven valence electrons and from the just-mentioned rule the
total spin magnetic moment should be 1 μB, as confirmed by
our electronic structure calculations.
Since the magnetic moments are mainly concentrated at
the anionic atoms we expect that their interaction is the most
relevant for the discussion of the stability of ferromagnetism.
Indeed our calculations show that the Ca–Ca intrasublattice
and Ca–N (P, As, Sb) intersublattice interactions have a
negligible contribution to the exchange constants, and thus
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J. Phys.: Condens. Matter 23 (2011) 296001A Laref et al
Figure 2. The effective Z–Z (Z = N, P, As, Sb) Ji(i = 1–4)
exchange interaction constants and the RPA-calculated Curie
temperature as a function of their corresponding lattice spacing. In
parenthesis is the number of neighbors within the ith coordination
shell. The Ca–Ca intrasublattice and Ca–Z intersublattice constants
are almost vanishing and thus are not presented.
we do not present them here. Moreover as we discussed in
section 2 and as deduced from the values of the Stoner gaps
(minimum energy required for such an excitation) presented in
table 1 the Stoner excitations, which involve excitation from
the occupied majority-spin states to the unoccupied minority-
spin states, occur at the high-energy region of the excitation
spectra and can be neglected [23]. The Z–Z intrasublattice
exchange constant within the frozen-magnon approximation
discussed in section 2 is presented in figure 2. The index
refers to the Z–Z coordination shell. We present results up
to the fourth coordination shell.
only the interactions between nearest anionic Z atoms J1play
an important role since contributions from other coordination
shells are almost vanishing. Each Z atom has 12 nearest Z
atoms which in reality are second-neighbors (each Z atom has
as nearest-neighbors eight Ca atoms).
ranging from about 6 meV for CaN down to 2.5 meV for
CaSb. The value for CaN is high enough to lead to stable
ferromagnetic order and a Curie temperature well above the
room temperature as we will also discuss later. The value
of J1is much larger for the lighter N atom due to the larger
anionic spin magnetic moment in CaN and the much smaller
N–N spacing inCaN with respect tothe other three compounds
under study. Here we should note that, for zb half-metallic
CaC studied in [10] the value of J1 characterizing the C–
C interactions was 0.906 mRyd, which equals 12.33 meV,
double the N–N values due to the much larger C spin magnetic
moment; C has one electron less than N and due to the Slater–
Pauling rule CaC has a total spin magnetic moment of 2 μB
carried mainly by the carbon atoms. Finally we should note
that in the case of transition metal zb-pnictides like CrAs the
Cr–Cr J1value is even larger reaching 18 meV, triple the N–N
value, due to the even larger Cr spinmagnetic momentof about
3 μB[18]. Thus the flat band mechanism can lead to stable
ferromagnetism as for CaN and the values of the exchange
constants depend mainly on the values of the spin magnetic
The first remark is that
The J1 has a value
Figure 3. The spin-wave dispersion along the high-symmetry lines in
the Brillouin zone for the CaZ compounds where Z is N, P, As and
Sb.
moments and not on the underlying mechanism responsible for
the appearance of ferromagnetism.
We will conclude our study with a discussion about the
Curie temperatures, TC, in the CaZ alloys. As we mentioned
in section 2 we employ the random phase approximation
(RPA) in which low-energy excitations have a larger weight on
the estimated TCwith respect to the high-energy excitations.
Moreover the stronger the ferromagnetism the larger should be
the value of the estimated TC. As we will show, the values
which we have calculated are compatible with this picture. In
figure 2 we present the RPA-calculated values of the Curie
temperature. As expected, trends follow the trend of the J1
exchange constants. For CaN TCis 430 K well above room
temperature, for CaP and CaAs it is close to room temperature
(350 K and 290 K respectively) and for CaSb it is 280 K,
being below room temperature.
RPA estimated TCof 735 K [10] and the RPA TCof zb CrAs
exceeds the 1000 K [18] following the trends of J1discussed
in the previous paragraph. Thus for realistic applications only
CaN seems to be appealing. The curve followed by the Curie
temperature in figure 2 does not exactly have the same gradient
as the curve for the J1since also further neighbor exchange
constants contribute to TCand as we can see in the figure J2,3,4
have an almost zero value for CaN but small positive values
for the other three alloys (although their values stay extremely
small compared to J1). Finally the higher TCvalue for CaN
with respect to the other three alloys can be also deduced
using as the starting point the energies of the spin waves. As
shown in equation (5), the
sum over the whole Brillouin zone of the
the spin-wave dispersion energy; we present these for all four
compounds under study in figure 3 along the high-symmetry
lines in the Brillouin zone. We can see that the dispersion
curve for CaN is for all q-points well above the curves for
the other three compounds especially along the X–W–K lines
where itmakesalsoa high-energyplateau. Thusthe sumwhich
we just mentioned is much smaller for CaN and the Curie
temperature which is the inverse of the sum is much larger.
As we move along the Vth column in the periodic table from
Note that zb CaC has an
1
TCis directly proportional to the
1
ω(q)where ω(q) is
5