Static and dynamic properties of [hkl]
low-symmetry trilayers
K. R. Celino1, C. H. Costa1,2,, C. G. Bezerra*,1,and C. Chesman1
1Departamento de F´isica Te´orica e Experimental, Universidade Federal do Rio Grande do Norte, Natal – RN 59078-900, Brazil 2Universidade Federal do Cear´a, Campus Avanc¸ado de Russas, Russas – CE 62900-000, Brazil
Received 15 September 2015, revised 14 December 2015, accepted 5 January 2016 Published online 28 January 2016
Keywords magnetic anisotropy, magnetic interlayer coupling, magnetization, phenomenological modeling, symmetry
∗Corresponding author: e-mail[email protected], Phone:+55 84 3215-3793, Fax: +55 84 3211-9217
We present a theoretical study about the influence of magneto-crystalline anisotropies on the static and dynamic magnetic properties of trilayers coupled via bilinear and biquadratic ex-change fields for situations in which the systems are grown in unusual [hkl] low-symmetry directions. We apply a real-istic phenomenological model, with a total free magnetic en-ergy that includes Zeeman interaction and magneto-crystalline anisotropies as well as exchange energy terms. We consider pa-rameters from the literature to illustrate our results for Fe/Cr/Fe systems. In particular, a total of six different magnetic scenar-ios for the [211] and [321] low-symmetry growth orientations
and three sets of exchange fields were analyzed, and the associ-ated magnetization, magnetoresistance, and spin-wave frequen-cies were calculated. Our results show that the combination of magneto-crystalline symmetries and exchange fields leads to various interesting properties, including different values of the saturation field for the magnetization and magnetoresistance curves. Regarding the spin-wave modes, we observed the pres-ence of Goldstone modes, associated with second-order phase transitions, resulting from the competition between the Zeeman and biquadratic energies.
© 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1 Introduction Research on magnetic multilayers has
been a field of intense activity in physics over the past two decades. The discovery of the antiferromagnetic bilinear cou-pling in Fe/Cr/Fe trilayers [1] and the observation of giant magnetoresistance (GMR) [2] were two major breakthroughs in this exciting research area [3]. Important subsequent find-ings include oscillations (from ferromagnetic to antiferro-magnetic) in the interlayer exchange coupling as a function of the Cr layer thickness [4] and the experimental observation of biquadratic exchange coupling [5]. Recent publications have focused on developing an understanding of the static proper-ties of quasiperiodic magnetic multilayers coupled via bilin-ear and biquadratic exchange fields [6, 7] and the effects of magneto-crystalline anisotropies [8]. For example, new fea-tures of the static and dynamic magnetic properties associated with the Fibonacci quasiperiodicity in multilayers grown in the [100] and [110] directions, including the discovery of an anomalous magnetoresistance, have very recently presented new possibilities for information storage technologies and applications in logical devices [9, 10]. For a recent review of metallic magnetic multilayers and GMR, see Refs. [11–14].
From an experimental perspective, advances in exper-imental growth techniques have enabled the synthesis of high-quality magnetic multilayers grown in different crys-tallographic orientations through either sputtering [15–17] or molecular beam epitaxy (MBE) [18, 19]. Quasiperiodic Fe/Cr multilayers presenting biquadratic and bilinear cou-plings have been experimentally synthesized, thereby exem-plifying these technical improvements [20]. The magnetic properties of these materials strongly depend on the stacking pattern of the layers, which can now be arranged in an un-usual fashion if necessary. Moreover, it is possible to grow magnetic multilayers to achieve properties that are subject to precise design and control through variations of either the thickness or composition of the individual layers. Con-sequently, magnetic multilayers with very specific proper-ties can be synthesized. Nevertheless, research in this area has often remained restricted to structures grown in con-ventional crystallographic orientations such as [100] and [110]. Moreover, fundamental studies of magnetic multi-layers with unusual crystallographic orientations are limited [21–25].
H y’ z’ x’ M2 θ2 θ1 M1 S M1 θH (b) z [001] x [100] y [010] y’ x’ z’ 1/2 1 1 [211] Plane y’ z’ x’ z [001] y [010] x [100] 1/3 1/2 1 [321] Plane
Figure 1 Schematic representations of (a) the trilayer structure and (b) the coordinate systems used for the [211] and [321] growth orientations.
The aim of this work is to push a little bit more the understanding of magnetic properties of ferromagnetic thin films with unusual crystallographic orientations, e.g., sys-tems structured into [hkl] low-symmetry directions. In par-ticular, we have investigated the properties of [211] and [321] magnetic trilayers using a phenomenological model in order to address static and dynamic properties of these systems, which have attracted attention in the last few years [25–27]. We focus on the effects of magneto-crystalline anisotropies, cubic and uniaxial, on the static and dynamic magnetic prop-erties of trilayers like Fe/Cr/Fe systems, which exhibit both bilinear and biquadratic exchange couplings [28]. This work is organized as follows. In Section 2, we present the physical model and the free magnetic energy terms for a single layer considering both the [211] and [321] growth directions. In Section 3, we extend the free magnetic energy of a single film to the trilayer case by including bilinear and biquadratic exchange terms. The static and dynamic properties of such structures are described in Sections 4 and 5, respectively. Finally, our findings are summarized in Section 6.
2 Single-layer model The general theory applied in
this work concerns a magnetic single-crystal layer of thick-ness d with a cubic lattice structure. The coordinate system is chosen such that the film surface is perpendicular to the
growth direction, namely, y, as depicted in Fig. 1a. By
con-sidering the geometry, the coordinate system (xyz) and its
relation to the crystalline axes xyz, we investigate the situa-tion in which a static magnetic field H is maintained within
the plane of the film at an arbitrary angle θHfrom the zaxis.
The equilibrium direction of the magnetization M also lies within the plane of the film and is characterized by the angle θ. The static properties of the system are governed by the com-petition among the various magnetic terms in the total free magnetic energy. Generally, each individual term attempts to
Thus, our first task is to determine the equilibrium values of
θby finding the minimum total free magnetic energy ET as
a function of the applied magnetic field H.
We consider three contributions to the free magnetic
en-ergy ET of a single magnetic layer at room temperature: the
Zeeman energy EZand the terms Ecaand Eua, which are the
cubic and uniaxial magneto-crystalline anisotropy energies, respectively. Moreover, we treat the ferromagnetic layers as
monodomains with no dynamic excitations. ETis of the form
ET = EZ+ Eua+ Eca. (1)
The magneto-crystalline anisotropies arise from atomic con-figurations in the crystal lattice. On the one hand, the intrin-sic atomic configurations are related to quantum electronic interactions. On the other hand, the extrinsic atomic con-figurations are related to the methods used during sample preparation. The free magnetic energy can be written as [16]
ET = −dM · H − dKua M2 (M· u) 2 +dKca M4 (M 2 xM 2 y + M 2 xM 2 z + M 2 yM 2 z). (2)
Here, Kcaand Kuarepresent the cubic and uniaxial anisotropy
constants, respectively, and Mx, My, and Mz represent the
moduli of the crystallographic projections of the
magnetiza-tion. After some calculation, ETtakes the form
ET = −dMH cos(θ − θH)− dKuacos2(θ− θu)+ Eca,
(3)
where θ is the orientation of the magnetization, θu is the
uniaxial anisotropy direction, and θH is the direction of the
applied magnetic field.
The cubic anisotropy energy may depend on the direc-tion of the film growth. In this case, this energy term can be rewritten as a function of the magnetization components with
respect to the xyz axis system (see, for example, Fig. 1b
for [211] and [321] directions). It is important to notice that whenever two of the Miller indices [hkl] are equal, we can choose a plane, for instance, [kk¯l], in which the expression for the cubic anisotropy energy simplifies, resulting in terms
of sin4θand cos4θ(see Refs. [24, 29]). A general expression
for the cubic anisotropy energy for an arbitrary direction [hkl] is presented in Appendix A.
In Fig. 2, we show the dimensionless cubic anisotropy
energy, Eca/dKca, for the [211] and [321] growth directions,
as a function of the planar angle θ. The main symmetry
axes for the [211] direction are θe= 0◦and 80◦(easy axes),
θi−h= 40◦ (intermediate axis), and θh= 130◦ (hard axis).
For the [321] direction, the main symmetry axes are θe= 0◦
(easy axis), θi−h= 50◦ (intermediate–hard axis), θi−e= 85◦
0 30 60 90 120 150 180 0.00 0.02 0.04 0.06 0.08 0.10 Dimensionless C ubic Anisotropy, Eca /dK ca Planar Angle, θ (º) [211]: θe= 0º ; θi-h= 40º ; θe= 80º ; θh= 130º [321]: θe= 0º ; θi-h= 50º ; θi-e= 85º ; θh= 125º
Figure 2 The dimensionless cubic anisotropic energy Eca/dkca
versus the planar angle θ for both the [211] (red solid line) and [321] (blue dashed line) growth directions.
As a matter of convention, we write the total free mag-netic energy in terms of experimental parameters or effective fields, i.e., Hca= 2Kca MS , Hua= 2Kua MS . (4)
Here, Hca is the effective cubic anisotropy field, Hua is the
effective uniaxial anisotropy field, and MS is the saturation
magnetization. Once the values of θ that minimize the free magnetic energy are found, we can obtain the normalized magnetization component along the field direction for a sin-gle film as follows:
M(H )
MS = cos (θ − θH). (5)
Figure 3a shows the magnetization versus the applied magnetic field for a [211] single film for four values of θH,
namely, θH= 0◦, 40◦, 80◦, and 130◦, corresponding to the
main symmetry axes of the [211] cubic anisotropy. In our numerical calculations for an ideal single film (Fe), we
con-sider Hua= Hca= 0.50 kOe and θu= 90◦(see, for example,
Ref. [7]). The canted and aligned phases are present when the
magnetic field is applied at θH = 0◦. At zero magnetic field,
the magnetization lies along θ∼ 86◦, and as the magnetic
field amplitude increases, it rotates toward the field direction
until the critical field H∼ 0.35 kOe is reached. At this point,
a first-order phase transition occurs, and the aligned state
emerges. For θH = 40◦, the canted and aligned phases are
again present. However, in this case, the system passes from the canted state into the aligned one through a second-order phase transition, i.e., the magnetization of the film continu-ously rotates toward the field direction with increased
mag-netic field. Similar behavior is observed when θH = 130◦.
In both situations, the magnetization saturates at high fields,
namely, at approximately H∼ 2.0 kOe. However, when the
Figure 3 Normalized magnetization curves for (a) [211] and (b) [321] single-crystal films (ideal Fe layers).
magnetic field is applied at θH = 80◦, only the aligned phase
is present. At this point, we should note the symmetry shift that arises because of the combination of cubic and uniaxial anisotropies. For the [211] single film, the major consequence of this symmetry shift is a change in the characteristics of two
of the film’s main axes, i.e., θ= 0◦ and θ= 40◦ present an
intermediate and hard signature, respectively.
For the case of a [321] single film, we also calculate the magnetization versus the external applied magnetic field
for four values of θH, namely, θH = 0◦, 50◦, 85◦, and 125◦,
which correspond to the main symmetry axes of the [321] cubic anisotropy. From Fig. 3b, we observe that the mag-netic phases for the [321] and [211] single films are simi-lar, with equivalent phase transition behavior. In particusimi-lar,
for θ= 50◦and θ= 125◦, the magnetization curves are
the aligned one, and the saturation of the magnetization
oc-curs at high fields of approximately H ∼ 2.0 kOe. When the
external magnetic field is applied along θH= 0◦, the
mag-netization rotates toward the field direction from θ∼ 88◦,
which is characteristic of a canted magnetic phase. As the
field increases, a first-order phase transition occurs at H ∼
0.28 kOe, and the aligned phase emerges. Finally, when the
field is applied at θH = 85◦, only the aligned state is present.
The combination of cubic and uniaxial anisotropies in the [321] single film not only changes the characteristic of three
of its main axes, i.e., causes θ= 0◦, θ= 50◦, and θ= 85◦to
present intermediate, hard, and easy signatures, respectively,
but also generates a perfect equivalence between θ= 50◦and
125◦, leading to indistinguishable behavior of the
magnetiza-tion when the magnetic field is applied along those direcmagnetiza-tions (degenerate magnetic states in relation to θH).
From the magnetization curves presented in Fig. 3, we note that the combination of cubic and uniaxial anisotropies strongly influences the magnetic behavior of the single films considered here. This combination is responsible for the sym-metry shift and, therefore, the magnetic states and corre-sponding phase transitions observed in [211] and [321] single films.
3 Trilayer model We now investigate a situation
ge-ometrically similar to the considered single-layer case, for which two magnetic films, also presenting cubic structure, are coupled. The so-called trilayer system is composed of two ferromagnetic single-crystal layers of the same
thick-ness, i.e., d= d1= d2, separated by a nonmagnetic spacer
layer of thickness S (Fig. 1a). Again considering the situ-ation in which a static magnetic field H is applied in the
plane of the film at an angle θH from the z axis, the
equi-librium directions of the magnetizations M1and M2 can be
determined by minimizing the total free magnetic energy of
the system. However, ET now consists of the terms listed in
Eq. (1) plus two additional contributions from the exchange
energies associated with the bilinear (Ebl) and biquadratic
(Ebq) couplings between the two individual magnetic layers.
Therefore, the total free magnetic energy takes the form
ET = EZ+ Eua+ Eca+ Ebl+ Ebq, (6)
which can be generalized to yield
ET = − 2 i=1 diMi· H − 2 i=1 diKua M2 i (Mi· u)2 + 2 i=1 diKca M4 i (M2 ixM 2 iy+ M 2 ixM 2 iz+ M 2 iyM 2 iz) − Jbl M1· M2 M1M2 + Jbq (M1· M2)2 M2 1M 2 2 . (7)
couplings, respectively. The exchange couplings may be
de-scribed by their exchange coupling fields defined as Hbl=
Jbl/dMS, which favors antiferromagnetic (when negative)
or ferromagnetic (when positive) alignment, and as Hbq=
Jbq/dMS, which favors a noncollinear (90◦) alignment
be-tween two adjacent magnetizations.
The equilibrium configuration is now characterized by
the planar angles of the magnetizations, namely, θ1 and θ2.
The equilibrium values of these angles must be determined as a function of the external applied field H; therefore, the nor-malized component of the magnetization in the direction of the applied field [28] and the normalized magnetoresistance [30] are obtained from
M(H ) MS = 2 i=1 Micos(θi− θH) 2 i=1 Mi (8) and MR(H ) MR(0) = [1− cos(θ1− θ2)] 2 , (9)
respectively. Here, MR(0) is the magnetoresistance at zero field.
4 Static properties The numerical results obtained
for the magnetization and magnetoresistance curves of the [211] and [321] trilayers are presented in this section. A total of three experimental parameter sets are considered in the
nu-merical calculations: (i) Hbl= −1.0 kOe and Hbq = 0.1 kOe
(|Hbl| Hbq), (ii) Hbl= −0.15 kOe and Hbq= 0.05 kOe
(Hbq∼ 13|Hbl|), and (iii) Hbl= −0.05 kOe with Hbq= |Hbl|.
These exchange fields are associated with specific values of the nonmagnetic spacer layer (Cr) thickness S of coupled Fe/Cr/Fe trilayers [28, 31]. Moreover, as in the case of the single films, we consider the effective fields of the cubic and uniaxial anisotropies to have the same intensities, namely,
Hua = Hca= 0.50 kOe, and we assume θu= 90◦. The
mag-netic field is applied along the symmetry axes previously considered for the [211] and [321] single films (see Section 2). Because the transition magnetic fields are the same for both magnetization and magnetoresistance, we henceforth focus our discussion on the magnetization curves.
The normalized magnetoresistance (top) and magneti-zation (bottom) curves for the [211] trilayer, considering the three specified sets of exchange fields, are depicted in Fig. 4.
The results for |Hbl| Hbq (left panel) show that when
θH = 0◦, the magnetizations remain in an antiferromagnetic
configuration in the low-field regime and rotate symmet-rically as the field increases until saturation is reached
at H ∼ 2.61 kOe. A similar nearly antiferromagnetic
configuration in the low-field regime is observed in both
situations when θH = 40◦ and θH = 130◦. However, the
Figure 4 Normalized magnetoresistance (top) and magnetization (bottom) curves for the [211] trilayer as a function of the applied magnetic field, considering the three sets of bilinear and biquadratic exchange fields. The intervals associated with the coexistence of magnetic phases are labeled in gray as 1, 2, 3, and 4. The vertical dashed lines represent critical fields.
until a second-order phase transition occurs at H∼ 0.85 kOe
(θH= 40◦) and H ∼ 0.87 kOe (θH = 130◦). These
transi-tions result in a spin-flop state that persists until fields of
amplitude H∼ 2.65 kOe (θH= 40◦) and H ∼ 2.53 kOe
(θH= 130◦), at which point the magnetizations become
aligned with each other along the direction of the applied
magnetic field. For θH = 80◦, because of the strong bilinear
coupling, the magnetizations remain antiparallel up through
a field of H∼ 0.83 kOe, where a first-order phase transition
occurs and a symmetric state emerges, with saturation at
H∼ 1.63 kOe.
In the central panel of Fig. 4, we present the numerical results for the second set of bilinear and biquadratic cou-plings for the [211] trilayer. Once the relation between the
exchange fields is such that Hbq∼ 13|Hbl|, a nearly
antiferro-magnetic state arises in the low-field regime when θH = 0◦.
As the field increases, the magnetizations continuously ro-tate toward the field direction until the saturated phase that
emerges at H∼ 0.71 kOe. For θH = 40◦, the magnetizations
rotate asymmetrically toward the field direction until the field
reaches H ∼ 0.22 kOe, where a first-order phase transition
occurs, with M1and M2becoming aligned at θ1 = θ2∼ 76◦
prior to saturation. As the field increases, the magnetizations rotate while remaining aligned until saturation is reached at
H∼ 1.92 kOe. A similar interpretation is valid for the case
in which a magnetic field is applied at θH = 130◦. In that
case, a first-order phase transition occurs at H ∼ 0.21 kOe,
with the magnetizations becoming aligned at θ1= θ2∼ 95◦
prior to saturation; saturation is reached at H ∼ 1.90 kOe.
For θH = 80◦, the antiferromagnetic state persists in the
low-field regime until the low-field reaches H ∼ 0.15 kOe, at which
point a first-order phase transition brings the magnetization into the saturated state.
The results for the [211] trilayer with the third set of exchange fields are shown in the right panel of Fig. 4. Again, the strength of the biquadratic field compared to the bilinear one is responsible for the nearly antiparallel
configura-tion in the low-field regime for θH = 0◦. An asymmetric
state emerges via a second-order phase transition at
H∼ 0.35 kOe. This asymmetric configuration persists up
through a field of H ∼ 0.50 kOe, where a first-order phase
transition occurs and saturation is reached. For θH= 40◦
and θH = 130◦, the magnetizations continuously rotate
until fields of H ∼ 75 Oe and H ∼ 70 Oe, respectively,
are reached. As in the previous set of exchange couplings,
the magnetizations become aligned at θ1 = θ2∼ 83◦ (for
θH= 40◦) or θ1= θ2∼ 90◦(for θH = 130◦) prior to saturation
through a first-order phase transition. In both cases, saturation
occurs at H ∼ 1.40 kOe. Finally, as expected, for θH = 80◦,
a first-order phase transition at H ∼ 0.05 kOe converts M1
and M2from an antiferromagnetic state to saturation.
The static properties of the [321] trilayer for the first set of
exchange fields (|Hbl| Hbq) are shown in the left panel of
Fig. 5. When θH= 0◦, the antiferromagnetic configuration
in the low-field regime evolves symmetrically as the field
Figure 5 Same as Fig. 4 but for the [321] trilayer system.
the single-layer case, the magnetic behavior for θH = 50◦and
θH = 125◦ is indistinguishable and completely equivalent.
In both situations, M1and M2rotate asymmetrically from a
nearly antiferromagnetic configuration until a second-order
phase transition occurs at H ∼ 0.79 kOe. The resulting
spin-flop states persist until a field of H ∼ 2.45 kOe is reached, at
which point the saturated phase emerges. For θH = 85◦, the
strong bilinear coupling causes the magnetizations to remain
antiparallel up through a field of H ∼ 0.80 kOe, for which
a symmetric state emerges via a first-order phase transition
and saturation is reached at H ∼ 1.66 kOe.
The ratio between the exchange fields (Hbq∼
1
3|Hbl|)
de-termines the nearly antiferromagnetic state that forms when
θH = 0◦ (see the central panel of Fig. 5). The continuous
rotation of the magnetizations toward the field direction
re-sults in a saturated state at H ∼ 0.61 kOe. For θH = 50◦and
θH = 125◦, the identical magnetic behavior observed in both
cases shows that M1 and M2 become aligned prior to
sat-uration at H ∼ 0.19 kOe via a first-order phase transition,
with θ1 = θ2 ∼ 80◦ (for θH= 50◦) or θ1 = θ2∼ 96◦ (for
θH = 125◦). In both cases, magnetization saturation occurs
at H ∼ 1.88 kOe. For θH = 85◦, the evolution from the
an-tiparallel configuration to the saturated state occurs through
a first-order phase transition at H ∼ 0.15 kOe.
Finally, the right panel of Fig. 5 shows the magnetization and magnetoresistance of the [321] trilayer for the third set of exchange fields, for which the biquadratic field has the
same magnitude as the bilinear field. For θH= 0◦, a
first-order phase transition at H ∼ 0.35 kOe takes M1 and M2
from the antiferromagnetic phase into the spin-flop phase as a result of the strong relative intensity of the biquadratic field. This spin-flop configuration persists up through a field
of H∼ 0.43 kOe, upon which another first-order phase
tran-sition occurs and the saturated state is reached. For θH = 50◦
and θH= 125◦, the magnetizations rotate asymmetrically
to-ward the field direction until a field of H∼ 65 Oe is reached,
at which point a first-order phase transition occurs, with M1
and M2becoming aligned at θ1= θ2∼ 85◦(for θH = 50◦) or
θ1= θ2 ∼ 91◦(for θH = 125◦) prior to saturation. As the field
increases, the coupled magnetizations rotate together until
saturation is reached at H ∼ 1.90 kOe. Finally, for θH = 85◦,
only the antiferromagnetic and saturated states, separated by
a first-order phase transition at H∼ 50 Oe, are present.
It is interesting to note the mismatch between the magnetoresistance and magnetization saturation fields when the biquadratic coupling is comparable to the bilinear one (second and third sets of exchange fields). This is illustrated in the respective panels of Fig. 4 for the [211] case and in Fig. 5 for the [321] case. In both scenarios, the magne-toresistance and magnetization present different saturation fields when the magnetic field is applied along a hard axis,
namely, for θH = 40◦and θH = 130◦(for [211] trilayers) and
for θH= 50◦ and θH = 125◦ (for [321] trilayers). We can
interpret this mismatch by considering that the magnetore-sistance varies linearly with cos(θ) in magnetic metallic multilayers (see Eq. (9)). Therefore, if the magnetizations become parallel before becoming aligned with the applied magnetic field, the magnetoresistance and magnetization
z’
y’
x’
z
1x
1z
2x
2y
y
M
1M
2H
θ
1θ
2θ
HFigure 6 Relation between the local axes system and the axes sys-tem of the plane of the film, where the zidirection lies along the
equilibrium orientation of the magnetization for the i-th film.
will present different saturation fields. A similar result has been observed in Fe/Cr/Fe [110] trilayers [8].
5 Dynamic properties To describe the dynamic
properties of the trilayer system, we must include two addi-tional energy terms in Eq. (6): dipolar and surface anisotropy energies. Therefore, the total magnetic energy can now be written as
ET = EZ+ Eua+ Eca+ Ebl+ Ebq+ Esa+ Edip. (10)
Here, Esa and Edip are the surface anisotropy and dipolar
energies, respectively, which are given by [32]
Esa = − 2 i=1 Ki sa M2 i Mi· ˆy 2 , (11) and Edip= 2π 2 i=1 diMi· ˆy 2 . (12) Here, Ki
sais the surface anisotropy constant of the i-th
mag-netic film. Henceforth, we follow the approach of Refs. [28, 32]. A self-consistent correlation between the static con-figuration and the dynamic response of the system is char-acterized by the ferromagnetic resonance (FMR) dispersion relations considered below.
The equation of motion for the magnetization of film i is written as
dMi
dt = −γMi× Heff, (13)
where γ= gμB/ is the gyromagnetic ratio, g is the Land´e
factor, μB is the Bohr magneton, and Heff is the effective
field acting on Mi. For each magnetic film, we construct a Cartesian coordinate system called the local axes system (see
Fig. 6), wherein zi-axis coincides with the equilibrium direc-tion of the magnetizadirec-tion. For the i-th film, the magnetizadirec-tion is then given by
Mi= mixixˆi+ miyˆy+ Mizizˆi, (14)
and we assume that mixi, miy Mizi. Note that the y-axis of
the local axes system coincides with the y-axis of the plane
of the film, i.e., y≡ y (see Fig. 6). The transformation that
relates the components of the magnetization in the two axes systems is given by
Mix = Mizisin θi+ mixicos θi, (15)
Miy = miy, (16)
and
Miz = Mizicos θi− mixisin θi. (17)
Likewise, the effective magnetic field is expressed as
Heff = hixixiˆ + hiyˆy+ Hizizi,ˆ (18)
with the relation between the effective field and the total energy defined by
Heff = −∇MET. (19)
By substituting the magnetization (14) and the effective field (18) into (13) and retaining only first-order terms, we obtain
dmixi
dt = γ(Mizihiy− miyHizi) (20)
and dmiy
dt = γ(mixiHizi− Mizihixi). (21)
After explicitly writing the equations of motion for the tri-layer structures, the following system of two equations must be solved: 1 γ dmixi dt = miy ∂ET ∂Mizi − Mizi ∂ET ∂miy (22) and 1 γ dmiy dt = Mizi ∂ET ∂mixi − mixi ∂ET ∂Mizi (23) Now, considering solutions of the type
mixi = m 0
miy= m
0
iyexp (−iωt) , (25)
we obtain, after some algebra, the matrix equation ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ − iω γ + HcaC1 H1 0 H2 −H3 − iω γ − HcaC1 H4 0 0 G2 − iω γ + HcaC2 G1 G4 0 −G3 − iω γ − HcaC2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ m0 1x1 m0 1y m0 2x2 m0 2y ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = 0, (26)
where we have assumed d1 = d2= 1 and M1z1 = M2z2≈
MS. From the resulting 4× 4 determinant condition, we
ob-tain (the explicit form of the elements of the matrix is in Appendix B) ω γ 4 + α0 ω γ 2 + α1= 0. (27)
Thus, the dispersion relation for the trilayer system is ω γ 2 = −α0 2 ± α 0 2 2 − α1, (28) with α0 = Hca2 C2 1+ C 2 2 + 2H2H4− G1G3− H1H3 (29) and α1 = H 2 ca (HcaC1C2) 2+ 2C 1C2H2H4− C 2 1G1G3 − C2 2H1H3 + (H2H4)2− H42G1H1 − H2 2G3H3+ G1G3H1H3. (30)
The spin-wave modes of the trilayer system consid-ered here consist of one high-frequency mode and one low-frequency mode. For the high-low-frequency mode, the dynamic
components of the magnetization, mixi and miy, are of
oppo-site signs for the individual adjacent magnetic layers, whereas for the low-frequency mode, they are of the same sign. In ex-act analogy to phonons, the low-frequency mode is related to an in-phase magnetization precession (acoustic mode), and the high-frequency mode is related to an out-of-phase mag-netization precession (optical mode). Figures 7 and 8 show the acoustic and optical modes of the spin-wave dispersion
relation for the three sets of exchange fields, Hbland Hbq, for
the [211] and [321] trilayers, respectively. As before, we
con-sider Hua = Hca= 0.5 kOe and θu= 90◦as well as values of
Hsa = 2 kOe and 4πMS = 20 kOe [28, 32].
rium static configurations of vectors M1and M2in a
particu-lar magnetic state, the critical fields at which phase transitions occur in these curves are consistent with those evidenced in
the magnetization results. The overall similarity of the disper-sion relations, for a particular set of exchange fields, between situations in which the magnetic field H is applied along an
equivalent axis of symmetry, for example, θH = 40◦ in the
[211] case (central panel of Fig. 7) and θH= 50◦in the [321]
case (central panel of Fig. 8), is an indication of the dominant
uniaxial effect that applies through the ratio Hua/Hca= 1.
Moreover, apart from certain minor details, the spin-wave dispersion relations are quite similar for the trilayers grown in the [211] and [321] directions that we have considered in this work.
Another interesting feature is the inversion of frequencies between the acoustic and optical modes [28, 25], which is observed for almost all directions of the applied magnetic
field θH. The exceptions in which no such frequency inversion
is present, given|Hbl| = Hbq, occur in the cases such that
θH = 40◦(50◦), 80◦(85◦), and 130◦(125◦) in the [211] ([321])
trilayer case. By contrast, when θH= 0◦, for both the [211]
and [321] trilayer cases, the frequency inversion occurs twice. Before concluding, let us discuss the presence of Gold-stone modes in Figs. 7 and 8. We again consider the analogy with the phonon case. For phonons, the excitation energy that is required for the presence of a Goldstone mode is zero. In our results, we can observe similar excitations for specific values of the magnetic field H and magnetic field orientation
θH. Such modes are present in the [211] trilayer case when
the magnetic field is applied along θH= 40◦ and θH = 80◦
for|Hbl| Hbq(left panel of Fig. 7) as well as when θH= 0◦
for Hbq∼ 13|Hbl| (central panel of Fig. 7). For the [321]
tri-layer case, the Goldstone modes are observed only for the
first set of exchange fields (|Hbl| Hbq) when θH = 50◦and
θH = 85◦ (left panel of Fig. 8). The common characteristic
of these configurations is the presence of second-order phase transitions, associated with the Zeeman and biquadratic en-ergies, that convert the system into the saturated phase. On the one hand, the Goldstone theorem predicts the existence of Goldstone modes in ferromagnetic materials in the presence of spontaneous symmetry breaking. On the other hand, as discussed by Zivieri et al. [25], in the presence of a second-order phase transition spontaneous symmetry breaking may
Figure 7 Spin-wave frequencies as a function of the external applied magnetic field H for the [211] trilayer for each of the three sets of bilinear and biquadratic exchange couplings: (a)|Hbl| Hbq, (b)|Hbl| > Hbq, and (c)|Hbl| = Hbq. The results show critical fields in
perfect agreement with magnetization curves. Notice that Goldstone modes (f −→ 0) are present for some very particular situations.
fore, the presence of Goldstone modes in our results may be associated with the second-order phase transitions due to the competition between the Zeeman and biquadratic energies on these systems.
6 Conclusions In summary, we present a theoretical
study in which we investigate the effects of cubic and uni-axial magneto-crystalline anisotropies on the static and dy-namic magnetic properties of single films and trilayers grown in [hkl] low-symmetry orientations. In particular, our phe-nomenological model considers a trilayer system composed of two magnetic single-crystal films, structured in a cubic lattice, separated by a nonmagnetic spacer. The contribu-tions to the free magnetic energy include Zeeman, cubic and uniaxial anisotropies, as well as bilinear and biquadratic ex-change energies. To illustrate our analysis for Fe/Cr/Fe sys-tems, magnetization and magnetoresistance curves as well as spin-wave frequencies were numerically calculated for three experimentally feasible sets of exchange fields that are known from the literature.
With regard to the static properties, the results for indi-vidual single-crystal layers (considering ideal Fe films) show that the magnetization undergoes two phase transitions in both [211] and [321] growth scenarios for all orientations θH of the applied magnetic field considered in this work. These transitions are either of first order (characterized by discon-tinuous jumps in the magnetization) or of second order (char-acterized by continuous rotation of the magnetization). The main result obtained from the magnetization curves of the [211] and [321] single films is the symmetry shift caused by a combination of cubic and uniaxial anisotropies. In par-ticular, two of the main symmetry axes of the [211] single
film undergo a change in symmetry, resulting in the θH = 0◦
and θH = 40◦ axes presenting an intermediate and a hard
signature, respectively. For the [321] single film, our results
show that when θH= 50◦and θH= 125◦, the magnetization
curves are perfectly superposed, i.e., the magnetic states for this particular situation are degenerate in θH. Therefore, cu-bic and uniaxial anisotropies in the [321] single film
im-pose a symmetry such that θH = 0◦, θH = 50◦, and θH = 85◦
present intermediate, hard, and easy signatures, respectively, with an indistinguishable equivalence between the directions
θH = 50◦and θH = 125◦.
For the trilayer system, both the magnetization and mag-netoresistance curves in both the [211] and [321] scenar-ios yield a rich magnetic phase diagram. As expected for a
regime in which|Hbl| Hbq, both the [211] and [321]
tri-layers exhibit magnetization curves in which the majority of the phase transitions proceed without discontinuous jumps (second-order phase transitions). In this case, because of the strength of the bilinear coupling, the magnetizations rotate continuously as the magnetic field increases, either symmet-rically or asymmetsymmet-rically. As the strengths of the bilinear and biquadratic fields become comparable, for example, in the second and third sets of exchange fields, the number of first-order phase transitions increases. This is a consequence
by the biquadratic term. For the [211] trilayer case, we
ob-serve that for θH = 40◦and θH = 130◦, M1and M2become
aligned prior to saturation, thereby causing a mismatch be-tween the magnetization and magnetoresistance saturation fields. For the [321] trilayer case, the same situation occurs
when θH= 50◦ and θH = 125◦. In particular, when the
bi-linear and biquadratic energies are comparable, our results show that the magnetizations of the films tend to align prior to saturation along the direction of the applied magnetic field. This effect has also been observed in [110] trilayers [8] and is attributable to the superposition of the cubic and uniaxial anisotropies when the bilinear and biquadratic energies are comparable.
With regard to the dynamic properties, as evidenced by the results for the spin-wave dispersion relations, the criti-cal fields at which phase transitions occur perfectly coincide with those for the magnetization curves. We observe the well-established inversion of the spin-wave frequencies between the acoustic and optical modes for certain directions of the applied magnetic field θH. Furthermore, Goldstone modes are observed in both the [211] and [321] trilayer systems. The presence of these Goldstone modes may be associated with the second-order phase transitions associated with the competition between the Zeeman and biquadratic energies, in which spontaneous symmetry breaking occurs. The mag-netic trilayers considered in this work can certainly be exper-imentally realized, and we hope that experimentalists will be encouraged to investigate them.
Appendix
A [hkl] cubic anisotropy energy We present a
gen-eral expression for the cubic anisotropy energy of a thin film grown with a generic [hkl] orientation. The cubic anisotropy can be written, in relation to the xyz crystalline axis, as
Eca= diKi ca |Mi|4 Mx2iMy2i+ Mx2iM2zi+ My2iMz2i. (A1)
Let{αi} and {αi} be the sets of coefficients of the
magnetiza-tion vector associated with the crystalline axis xyz and with
the plane of the film axes xyz, respectively (see Fig. 1). Let
us rewrite Eq. (A1), which is a function of{αi}, in terms of
{αi}. We must find an expression that relates these two sets
of coefficients. To do so, we may write ˆM, the unit vector
in the magnetization direction, as ˆM= α1ˆx+ α2ˆy+ α3ˆz or
ˆ
M= α1xˆ+ α2yˆ+ α3zˆ. Thus, we can write{αi} as a
func-tion of{αi} as follows: α1= ˆx · ˆM = ˆx · α1xˆ+ α2yˆ+ α3zˆ, (A2) α2= ˆy · ˆM = ˆy · α1xˆ+ α 2yˆ+ α3zˆ , (A3) and α3= ˆz · ˆM = ˆz · α1xˆ+ α2yˆ+ α3zˆ. (A4)
For a generic [hkl] plane, the unit vectors ˆx, ˆy, and ˆz are
given in terms of ˆx, ˆy, and ˆz as follows: ˆ x= √ 1 h2+ k2(−kˆx + hˆy) , (A5) ˆ y= √ 1 h2+ k2+ l2(hˆx+ kˆy + lˆz) , (A6) and ˆ z= √ 1 h2+ k2√h2+k2+l2 hlˆx+klˆy −k2+ h2 ˆz. (A7) Now, by substituting Eqs. (A5), (A6), and (A7) into Eqs. (A2), (A3), and (A4), we obtain
α1 = − k √ h2+ k2α 1+ h √ h2+ k2+ l2α 2 +√ hl h2+ k2√h2+ k2+ l2α 3, (A8) α2 = h √ h2+ k2α 1+ k √ h2+ k2+ l2α 2 +√ kl h2+ k2√h2+ k2+ l2α 3, (A9) and α3= l √ h2+ k2+ l2α 2− k2+ h2 √ h2+ k2√h2+ k2+ l2α 3. (A10) Finally, from Eq. (A1), we can obtain a final expression for
the cubic energy anisotropy Eca of a thin film grown with a
generic [hkl] orientation: E[hkl]ca = diKica h2k2h2+ k2+ l22h2+ 2k2+ l2 (h2+ k2)2(h2+ k2+ l2)2 sin 4 θ +2hkl −h6− h4k2+ h2k4+ k6+ 4hkl3−h4+ k4+ 2hkl5−h2+ k2 (h2+ k2)2 (h2+ k2+ l2)2 sin 3θ cos θ + h2+ k24+ 2l2h6+ k6+ l4h4− 4h2k2+ k4 (h2+ k2)2 (h2+ k2+ l2)2 sin 2 θcos2θ +2hkl 3h4− k4+ 2hkl5h2− k2 (h2+ k2)2(h2+ k2+ l2)2 sin θ cos 3 θ +h2k2l2 3h2+ 3k2+ l2+ l2h6+ k6 (h2+ k2)2 (h2+ k2+ l2)2 cos 4θ . (A11)
Here, we have used the relations α1= sin θ cos φ, α2 =
sin θ sin φ, and α3= cos θ. We have also assumed that φ = 0
because of the very strong demagnetization field, which sup-presses any tendency for the magnetization to tilt out of the
plane of the film. For the particular case in which h= k = m
and l= −n, we find, for example,
E[mm ¯n] ca = diKica 4 (2m2+ n2)2 ×−12m4+ 8m2n2+ 4n4 cos4θ +8m4− 4m2 n2− 4n4cos2 θ +4m4+ 4m2n2+ n4. (A12)
B Elements of the matrix The explicit form of the
elements of the matrix in Eq. (26) is
H1 = H cos (θ1− θH)+ Hca(B1− D1) + Huasin 2 (θ1)+ Hblcos (θ1− θ2) − 2Hbqcos2(θ1− θ2) + 4πMS− Hsa, (B1) H2 = −Hbl+ 2Hbqcos (θ1− θ2), (B2) H3 = H cos (θ1− θH)+ Hca(A1− D1) − Huacos (2θ1)+ Hblcos (θ1− θ2) − 2Hbqcos [2 (θ1− θ2)], (B3) and H4 = Hblcos (θ1− θ2)− 2Hbqcos [2 (θ1− θ2)]. (B4)
The Gicoefficients can be obtained from the Hicoefficients
and are as follows: (i) for the [211] direction,
A[211]
i =
1 200
−161 sin2(2θi)−14√6 sin (4θi)+168
, (B5) B[211] i = 1 20
2√6 sin (2θi)+ cos (2θi)+ 5
, (B6) C[211] i = √ 5 50
sin θi42 cos2θi− 8+√
6 cos θi ×9 sin2θi− 1, (B7) and D[211]i = 1 150
2√6 sin (2θi)1− 7 sin2
θi
+ cos2
θi161 sin2
θi− 5+ 48; (B8)
(ii) for the [321] direction, A[321]i = 1 66248 −62433 sin2 (2θi) − 1350√14 sin (4θi)+ 60, 200 , (B9) B[321] i = 1 33124
1170√14 sin (2θi)+ 7 cos (2θi)
−8281] , (B10) C[321] i = 9√13 16562
14 sin θi61 cos2θi− 12
+ 5√14 cos θi17 sin2
θi− 1, (B11) and D[321] i = 1 16562
30√14 sin (2θi)1− 15 sin2θi
+ cos2θ i 20811 sin2θ i− 4823 + 7056. (B12)
Acknowledgements We would like to thank two anony-mous referees for the comments that lead us to a substantial im-provement of this manuscript. We thank L.F.C. Pereira for a critical reading of the manuscript as well F. Bohn and M.A. Correa for fruit-ful discussions. We also would like to thank the Brazilian Research Agencies CNPq, CAPES, FAPERN, and INCT of Space Studies for
arship 07/54885-2. This research was performed with the aid of the Computer System of High Performance of the International Institute of Physics (IIP) – UFRN, Natal, Brazil.
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