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Effects of Rashba and Dresselhaus spin–orbit interactions on the
ground state of two-
dimensional localized spins
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2014 J. Phys.: Condens. Matter 26 196005
(http://iopscience.iop.org/0953-8984/26/19/196005)
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1 © 2014 IOP Publishing Ltd Printed in the UK
1. Introduction
Recently, the topic of spin–orbit coupling in a thin
ferromag-netic layer has attracted much attention because of its
potential to permit the control and manipulation of electronic spin
and charge degrees of freedom in the field of spintronics. There
are two main types of spin–orbit interactions for this thin layer
system. One is the Rashba spin–orbit interaction, induced by a
structural inversion asymmetry originating from an electric field
perpendicular to the plane of the system [1]. The other is the
Dresselhaus spin–orbit interaction, induced by the lack of bulk
inversion symmetry in the crystalline structure [2]. The strength
of the Rashba spin–orbit interaction can be tuned by external gate
voltages or heterojunction materials. An oxi-dized Gd surface is
known to have a strong Rashba spin–orbit coupling of
∼0.01 eV·nm [3]. Experimentally reported torque magnitude in
ultrathin magnetic bilayers such as Pt/Co is con-sistent with
similar Rashba spin–orbit coupling magnitude
at the oxidized Gd surface [4]. Theoretical calculations on
ultrathin (
-
J H Oh et al
2
interaction is to convert a uniform ferromagnetic ground state
to spiral (in this paper we use the word ‘spiral’ for both helical
and cycloidal patterns) and skyrmion phases when it exceeds a
certain threshold. The skyrmion phase is the topological spin
texture where the spins point in all directions, as if wrap-ping a
sphere, and it is thought to be a good candidate as a further
building block in spin devices [12]. The formation of a skyrmion
crystal state was recently found in the metallic ferromagnet MnSi
[13], in the thin film of a non-centrosym-metric magnetic crystal
FeCoSi [14] and in a hexagonal Fe film on an Ir(1 1 1) surface
[15]. Many theoretical attempts to find the ground state of the
two-dimensional system have been made, usually with the
Ginzburg–Landau energy func-tional, namely the continuum model [12,
13, 16, 17]. Due to its simplicity, the continuum model was
successful in provid-ing good explanations of experimental results.
Now, however, in order to understand the details of the operating
principles and their enhancement of spin devices, it is necessary
to look more deeply into the origin of the interaction terms in the
con-tinuum model. For this, it would be beneficial to start with a
quantum mechanical model because variables such as cou-pling
strengths depend sensitively on basic structural param-eters, as
previously shown in the cases of a single atom or two atoms under
spin–orbital interaction [18–20].
Thus, it is timely to examine and understand the ground states
of a ferromagnetic layer, starting with spin–orbital cou-pling, in
order to fully assess the performance of spin devices. In this
work, we adopt the indirect exchange Hamiltonian for a
ferromagnetic state of a two-dimensional system which is assumed to
be influenced by both Rashba and Dresselhaus spin–orbit
interactions. We then derive the Hamiltonian for localized spins,
including the Dzyaloshinskii–Moriya inter-action as well as the
Heisenberg exchange coupling. The strengths of each interaction are
examined as a function of atomic distance. With calculated system
parameters, we dis-cuss the formation of various ground states as a
function of temperature and external magnetic field. By plotting
the mag-netic field–temperature (H-T) phase diagram, we present the
approximate phase boundaries between the spiral, skyrmion and
ferromagnetic phases.
2. Hamiltonian
In order to study the ground state of a ferromagnetic layer, we
consider the indirect exchange model in a two-dimensional spin
system [21, 22], where spins of localized d-orbitals are mediated
by itinerant s-orbital electrons. Additionally, the itinerant
electrons are assumed to be influenced by the Rashba and the
Dresselhaus spin–orbit interactions. The Hamiltonian is modeled as
H = H0 + Hsd, where
⎡⎣⎢
⃗ ⃗ ⃗ ⃗⎤⎦⎥
⃗
∫
∫
∑
∑
ψ α σ α σ ψ
ψ σ ψ
= +ℏ
ˆ× · +ℏ
·
= − ′ ·
∼′
′†
′′
†
′H
p
mz p p
H J a d
r r r
r r r S r
d ( )2
( ) ( ),
( ) ( ) ( ).
s ss
R D
s ss
sd sd
s ss s s s
0
2
02
(1)
with the momentum operator ⃗ = − ℏ∇p i . Here, H0 describes
non-interacting electrons under the Rashba and Dresselhaus
interactions with their coupling strengths, αR and αD,
respec-tively; ⃗σ σ σ= ( , )x y and σ σ σ= −∼ ( , )x y are vectors
of the Pauli spin matrix, and ψ†s (ψ) is a creation (annihilation)
field opera-tor with s = ↑, ↓. Note that σ∼ is
introduced to account for the Dresselhaus spin–orbit coupling,
which breaks the rota-tional invariance within the x − y
plane. We assume that the two-dimensional electron gas is confined
in the x-y plane and thus the electric field responsible for the
Rashba inter-action is along the z-direction. Hsd denotes the ‘s-d’
inter-action between the itinerant electrons and the localized spin
S(r) with a strength of Jsd. S(r) is a localized function at each
atomic site Rj as S(r) = ∑j Sj δ (r−Rj) and a0 is a
lattice con-stant of the atomic system.
The non-interacting Hamiltonian H0 is easily diagonalized in the
spatial coordinates by using the plane-wave basis of,
∑ψ = ·A
Cr( )1
es sk
k rk
i (2)
where A is the area of the system and Cks is an annihilation
operator at k = (kx, ky) and a spin state s. Then,
eigenfunctions are obtained by additional rotation in the spin
space;
⎜ ⎟⎛⎝⎜
⎞⎠⎟
⎛⎝
⎞⎠
⎛
⎝⎜⎜
⎞
⎠⎟⎟= −
∼∼
η η
η η↑
↓− −
−
+
CC
C
C
1
2e ee e
k
k
i i
i k i
k
k
k k
k (3)
with
ηα αα α
=++
tank k
k kkR x D y
R y D x (4)
and corresponding eigenvalues are given by [23],
α α α α= ℏ ± + +±E km
k k k2
( ) 4R D R D x yk2 2
2 2 2 (5)
with ⃗= ∣ ∣k k .The ground state of the spin system can be
obtained by
deriving an effective Hamiltonian for localized spins S(r). This
is fulfilled by integrating out the itinerant electron degrees of
freedom [24]. Firstly, we make a Schrieffer–Wolff transfor-mation
on the Hamiltonian, a sort of perturbation approach, to eliminate
terms which are linear in Jsd. The transformed Hamiltonian is given
by,
∫ τ τ= + ℏ − +∞
H Hi
H H J2
d [ ( ), ] ( )sd sd sd00
3 (6)
where Hsd(τ) denotes the Heisenberg operator of the inter-action
Hamiltonian; Hsd(τ) = eiτH0/ℏHsd e−iτH0/ℏ. At this
stage, we assume a weak exchange coupling Jsd between localized
spin and electrons so that terms of a higher order than Jsd2 can be
neglected, as is usual in indirect exchange models [19–21]. The
approximation is commonly applied to weak ferromagnetic systems,
for example, the rare earths with incomplete 4f shells, the
actinides, and the diluted magnetic alloys [22].
Below we will also consider the implications of the present
theory on strong ferromagnets, such as band magnetic materi-als of
Ni, Co and Fe. Strictly speaking, strong ferromagnets go beyond the
validity of the present approach. Nevertheless, such an application
may still provide some hints as to the
AQ3
J. Phys.: Condens. Matter 26 (2014) 196005
-
J H Oh et al
3
feasible modification of their ground state under spin–orbital
interactions.
By inserting a detailed form of Hsd into equation (6), the
transformed Hamiltonian reads as,
⎡⎣⎢
⎤⎦⎥∫∑ τ τ δ τχ= + ℏ · · + · × −′
∞′ ′ ′H H
J a
AS S S s s
1d ( )
1
2( ( ))sd
jjj j j j jj j j j0
204
20
(7)where the dyadic tensor χ is defined by,
τ τ κχ ≡− − ′ =κ κ κ κ′′
′′i
k x y zs s( )2
[ , ( )], , , ,j j j j
with ∑ ∑τ σ=κ τ κ τℏ + − ℏ′ ′ ′⃗C Cs ( ) e · e ej s sH
s s s sH
q
q Rk k q k
i i /,
†,
i /j 0 0 . (8)
3. Range functions
Next, we focus on the localized spin by averaging out the
elec-tron degree of freedom. Namely we write the Hamiltonian of the
localized spins as,
⎡⎣⎢
⎤⎦⎥∫∑ τ τ δ τχ= ℏ · · + · × −′
∞′ ′ ′H
J a
AS S S s s
1d ( )
1
2( )M sd
jjj j j j jj j j j
204
20
(9)where 〈···〉 denotes the expectation values of the electronic
operators over the Hamiltonian H0. In calculating the expectation
values, we rely on a non-equilibrium Green's function method
associated with the generating functional [25, 26]. According to
the calculations, the expectation value of an electronic spin 〈sj
(t)〉 is found to be zero; the second term of the above equation
also vanishes due to the time-reversal symmetry (simply seen from
the fact that the odd number of multiplication of spins turn its
direction oppositely by the application of the time-reversal
operation and that its expectation value should be zero if the
system has the time-reversal symmetry). By taking 〈sj
(t)〉 = 0 into account, the first term, 〈χ j′j (τ)〉, is
then equal to the expec-tation of the itinerant spin fluctuation.
Its expectation value can be obtained from the
fluctuation-dissipation theorem [27] or the second-order
derivatives of the generating func-tional with respect to
fictitious fields. The result is sum-marized by,
⃗ ⃗τ σ τ σ τχ· · = ℏ · · − −′ ′ > ′ ′ < ′{ }( )S S S G R S
G R( ) ImTr ( ) ( , )( ) ,j j j j j jj j jj2 (10)where the greater
(G>) and lesser (G <
∞> < > <
∞> <
mG G
mG G G G
mG G
R R R
R R R R R
R R R
( )2
Im d ( , ) ( , ),
( )2
Re d [ ( , ) ( , ) ( , ) ( , )] ,
( )2
Im d ( , ) ( , ).
A
S
0
5
2 00 0
5
2 00 1 1 0
5
22
01 1 (13)
Equation (11) represents the RKKY-type interactions modi-fied by
the Rashba and Dresselhaus spin–orbit interactions. In fact, the
result is similar to those of the previous works on the two
localized spin problems [19, 20], and thus can be the extension of
those to the two-dimensional spin lattice with both Rashba and
Dresselhaus spin–orbit interactions.
The first term in the parenthesis of equation (11) represents an
isotropic Heisenberg exchange interaction, implying that (negative)
positive values of strength Φ0 + Φs lower the sys-tem
energy by the (anti-) parallel arrangement of localized spins. The
second term adds anisotropy to the otherwise iso-tropic Heisenberg
interaction and prefers to orient two spins to the line that
connects them. This term is encountered in the direct interaction
problem of two dipole moments, where the interaction strengths are
proportional to R−3 for their dis-tance of R. In the present work,
however, because the interac-tion is mediated indirectly by
itinerant electrons, it is found that the range functions Φ are a
complicated function of the distance between localized spins. The
last term represents the Dzyaloshinskii–Moriya interaction with its
strength vec-tor ΦA djj′. This apparently originates from the
Rashba and Dresselhaus spin–orbit interactions. This term lowers
the system energy by rotating each localized spin to an angle of
90°-degree with respect to its adjacent one in the planes
per-pendicular to their connecting vector djj′. Thus this term
van-ishes if the two localized spins are parallel or anti-parallel.
The direction of the connecting vector djj′ depends on the type of
spin–orbit interaction. In the case of the Rashba interaction the
vector is perpendicular to the line connecting the positions of two
atoms, while its direction is position-dependent in the case of the
Dresselhaus interaction as expressed in equation (12). We find that
the Rashba and Dresselhaus spin–orbit interac-tions play equivalent
roles in contributing eigenenergies and in determining the range
functions because ΦA is nearly pro-portional to α α+D R2 2 in
equation (13). Their different roles are found only in a connecting
vector djj′, which leads us to different patterns of spin
arrangement depending on each spin–orbit strength. As inferred from
equations (11) and (12), in the Rashba interaction the rotation of
a localized spin rela-tive to its adjacent one occurs about the
axis that is perpen-dicular to both the connecting line and the
z-direction, while in the Dresselhaus interaction the rotation axis
depends on the position of two atoms. If the atoms are separated
along the x− or y−directions, the rotation axis is parallel to the
line
J. Phys.: Condens. Matter 26 (2014) 196005
-
J H Oh et al
4
connecting them; if they are aligned at 45 degrees between the
x− and y−directions, the axis becomes perpendicular to both the
connecting line and the z−direction.
For the numerical evaluation of the range functions, it is
convenient to transform equations (13) into the energy space. By
making use of the Green's functions in Appendix A, the range
function can be rewritten as,
∫∫∫
μ
μ
μ
Φ = − −
Φ = − Δ + Δ
Φ = − Δ Δ
−∞
∞
−∞
∞
−∞
∞
Ef E F E F E
Ef E F E F E F E F E
Ef E F E F E
R R R
R R R R R
R R R
( ) Re d ( ) ( , ) ( , ),
( ) Re d ( )[ ( , ) ( , ) ( , ) ( , ) ] ,
( ) Re d ( ) ( , ) ( , )
A
S
0
(14)
where f(E) is the Fermi–Dirac distribution function,
∑
∑
δ δ
α α α
δ δ
= ℏ · − + −
Δ = ℏ ∂∂
·+
− − −
+ −
+ −
[ ]
[ ]
F Em A
E E E E
F Em R A k k k
E E E E
R
R
( , )2
1e ( ) ( )
( , )2
1e
1
4 /
( ) ( )
x y R D
k
k Rk k
k
k R
k k
2i
2i
2 2
(15)
and F ΔF( ) are the Hilbert transformations of F (ΔF),
respec-tively, so that we have 1/(E − Ek−i 0+) instead of
delta func-tions in the above equations. It should be noted that,
when both the Rashba and Dresselhaus interactions are finite, ±Ek
in equation (5) becomes non-isotropic in the k-space. Thus, the
summation over k in the above equation leads us to the range
functions, depending on both the direction and size of R. This
non-isotropic nature is strong only at low energies of electrons
around k = 0. Accordingly, if the chemical potential μ is
much larger than the spin–orbit energy the range functions should
be almost isotropic. In section 5, we will demonstrate this point
via numerical simulations.
4. Continuum model
It is interesting to relate the result of equation (11) to the
con-tinuum model, which informs quantitative estimates of system
parameters in further applications to device simulations.
A standard process starts with the assumption that the localized
spins vary smoothly in the space and that the range functions are
short ranged, including only the nearest neigh-bor interaction
[28].
Then, a localized spin Sj′ at a site Rj′ may be expanded in a
Taylor series:
= + ′ ·∇ + ′ ·∇ ′ ·∇ + ⋯ |κ κ′ =S R R R S r(1
1
2( )( ) ) ( ) .j j j j j j j r Rj (16)
By inserting this into equation (11) and retaining the lowest
terms, we obtain the energy expression in terms of S(r):
⎜ ⎟⎡
⎣⎢⎢
⎛⎝
⎞⎠
⎛⎝⎜
⎞⎠⎟
⎤⎦⎥
∫= ∂∂ +∂∂
+ ∂∂
· ∂∂
+ · × ∂∂
+ · × ∂∂
E Jx
Jy
Jx y
x y
rS S S S
D SS
D SS
dM xx yy xy
x y
( )2
( )2
( )
( ) ( )
(17)
where
⎡⎣ ⎤⎦∑
∑
= Φ + Φ
= − Φ
κκ κ κ
κ κ
′′JN
aC R R R R
N
aC R RD d
2( ) ( )
( )
csd
jj S j j j
csd
jA j j j
( )
02 0
( )
02 0
(18)
with Nc, the number of atoms in a cell a02 and j, standing for
atomic neighbors of the origin atom R = 0. Here, we
neglect a constant energy term and the contribution from the second
term in equation (11) due to its smallness, as shown in the
following section.
It is heuristic to examine equation (17) in a square lattice
with j running for the nearest neighbors and Nc = 1. In
this case, J(xx) = J(yy) = J and
J(xy) = 0. Then, we obtain
̂= Φ
= ˆ−
= − ˆ + ˆ
J C a
D x D y
D y D x
D
D
( ),
,
sd
xD R
yD R
0 0
( )
( )
(19)
with
αα
αα
= Φ
= Φ
D Ca
a
D Ca
a
2( )
,
2( )
.
D sdD A
R sdR A
0
0
0
0 (20)
Figure 1. The range functions Φ0,S, A are plotted as a function
of kFR for various spin–orbit coupling strength α. The chemical
μ = 10 eV is used at T=100 Csd K.
Calculated results are nearly independent of temperature as long as
kBT ≪ μ.
0 2 4 6 8-80
-40
0
40
(a)α= 0.2 eV nmα= 0.1 eV nmα=0.05 eV nmα=0.01 eV nm
0 2 4 6 8-10
-5
0
5
10(b)
0 2 4 6 8kFR
-100
-50
0
50
ΦA
(meV
)Φ
S(m
eV)
Φ0(
meV
)
(c)
J. Phys.: Condens. Matter 26 (2014) 196005
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J H Oh et al
5
5. Results and discussion
5.1. Numerical results of range functions
Now we proceed to study the behavior of the range functions by
varying the spin–orbit coupling strength α. We choose the chemical
potential, μ = 10.0 eV. Other material parameters
such as Jsd and a0 do not affect the range functions because they
appear in the prefactor Csd of HM in equation (11) and give a
simple scaling effect. A Fermi wave vector associated with the
chemical potential, μ= ℏk m2 /F 2 , is 1.62 Å−1; and, if we
consider a lattice constant a0 = 2.506 Åfor cobalt,
the chemical potential means kFa0 = 4.061.
Figure 1 shows the range functions in terms of kFR for vari-ous
spin–orbit strengths, where R is the distance between two localized
spins. In the case of zero spin–orbit coupling (α = 0),
we find that Φ0(R) is well fitted to the form of sin (2kFR)ℏ2/(8π2
mR2), the conventional RKKY interaction in the two-dimensional
system [29, 30]. As the spin–orbit coupling con-stant α increases
from zero, the bands of conduction electrons split and each of the
wave vectors ±Ek in equation (5) associ-ated with the bands
contribute differently to oscillatory and decaying characters in
the range functions. The calculated results are somewhat
complicated. With an increased α, Φ0(R) is slightly modified.
Meanwhile, the range functions ΦS(R) and ΦA(R) start from zero and
grow in magnitude. In particu-lar, the function ΦA becomes
comparable with the function Φ0 across the whole range of the
distance when α ∼ 0.1 eV·nm.
In figure 2, for a given distance kFR = 4.06 we show
calcu-lated range functions as a function of the spin–orbit
coupling strength α (thick lines for αD = 0 and thin
lines for αD = αR). One can see that Φ0 decreases with
the increasing coupling strength α, while ΦA and ΦS become larger
in amplitude. Above approximately α = 0.1 eV·nm, the
system has compara-ble interaction strengths for Φ0 and ΦA. Thus,
with this amount
of spin–orbit strength, a different ground state rather than a
uniform spin arrangement may be expected. By increasing the
spin–orbit interaction further, we find that Φ0 and ΦA become
smaller, while ΦS are dominant around α = 0.4 eV·nm.
When both Rashba and Dresselhaus interactions are finite, then the
circular symmetry in eigenvalues in equation (5) is broken and,
correspondingly, the range functions are also expected to have
direction dependence. However, it is noted that the thin lines
(αD = αR) in the figure are very similar to the thick
lines (αD = 0) for each range function. This means that
the ratio between αR and αD is unimportant and a factor α α α= +D
R2 2 governs the interaction strength. In other words, the range
functions are largely circular symmetric within the interested
spin–orbit strength (about ≲ 0.2 eV·nm) and thus the
approxi-mation of =≶G 02 made in Appendix A is well justified.
5.2. Spin texture of ground states
We now examine a spin texture of the ground state for a
two-dimensional system under spin–orbit interaction. In order to
obtain a spin texture of the system, we adopt the mean field
approximation for the free energy under external magnetic fields
(details in Appendix B). In this approximation, each spin is
assumed to be in the effective magnetic field produced by its
surrounding spins and thus the problem is reduced to that of
independent spins under its own external magnetic field. The
effective magnetic field on each spin or on the ground states of HM
are determined to minimize the free energy of those independent
spin systems over all possible spin config-urations. The resulting
magnetic field is written in equation (B.9). Despite its simple
form, the magnetic field is calculated via coupled non-linear
equations depending on the directions of all spins. Due to this,
the ground states of HM are compli-cated, even in the mean-field
approximation, and so we adopt a variational approach to solve
equation (B.9), starting from a well-known spin configuration. We
consider three initial types of spin configurations on a squared
atomic lattice (the ferro-magnetic, spiral and skyrmion phases) and
thus the param-eters in equations (19) and (20) become available.
An external magnetic field is assumed to be along the z-direction
and to give rise to a Zeeman energy splitting only, as in Appendix
B.
Without the spin–orbit interaction and external magnetic fields,
the modeled two-dimensional system is expected to be a stable
ferromagnetic state below a certain temperature TC. To simulate the
situation, we find that the range functions of equation (11) should
be short ranged, and thus we confine our attention to the nearest
neighbor interaction for simplicity. This is one of the drawbacks
in the present theory because a full consideration of the
interaction range fails to predict a correct ferromagnetic phase.
Even though the exponentially decaying nature of the range
functions was addressed in the previous indirect exchange model,
taking impurity scatter-ing into account, the magnetic interaction
was still found to depend on a power-law due to the contributions
from the higher-order momentum of susceptibility [31, 32]. In the
cases of magnetic alloys, the effects of the band dispersion and
phase randomness from impurity scattering diminish the
Figure 2. The range function Φ0 (solid), ΦS (dashed) and ΦA
(dotted) are compared as a function of the spin–orbit strength α α
α= +R D2 2 with αD = 0 (thick lines) and
αD = αR (thin lines). We use μ = 10 eV and
the distance between two localized spins
R = 2.506 Å, the nearest neighbor distance of a
hexagonal cobalt, to be kFR = 4.06.
0.0 0.1 0.2 0.3 0.4α (eV.nm)
-20
-10
0
10
20Φ
(m
eV)
Φ0
ΦS
ΦA
kFR=4.06
J. Phys.: Condens. Matter 26 (2014) 196005
-
J H Oh et al
6
values of the range functions over a long distance, hence the
short-ranged approximation for the range functions can be used to
describe magnetic properties [33]. Correspondingly, within the
approximation, one can expect that spin textures in this work apply
well to the magnetic behavior of a two-dimensional system for
magnetic alloys, and that they do so qualitatively for band
magnetic materials.
We first examine a spiral state, which is known to be a sta-ble
state in small magnetic fields. Its magnetic unit vector is
described by:
ϕ ϕˆ = − ˆ × ˆ− ˆ · + ˆ ·( )m z zq q q R q Rcos sin sin cosj jH
H j (21)where a pitch vector q and an angle φH are variation
parameters. For minimizing the free energy of equation (B.4) over
q, we make use of a periodic unit cell with a size of Lx
× Ly = 200 a0 × 200a0 and calculate
the free energy at each reciprocal lattice point. It is found that
q0 giving the minimum free energy is nearly equal to that of a
continuum model. Namely, by expressing q in a polar coordinate of
q = ∣q∣(cos θq, sin θq), both angles of φH and θq are
determined to maximize the amplitude of a pitch vector,
θ ϕ ϕ∣ ∣ =
− +DJ
qcos (2 ) D sin ( )
2.
D q H R H (22)
According to the above equation, in the cases of
DD = 0 or DR = 0, q giving the minimum free
energy is independent of its angle and thus the free energy is
degenerate for the propa-gation direction θq. However, when both
spin–orbit terms are finite, the free energy is lifted in a
preferred direction. For example, in the case of DD = DR,
θq = π/4 is a propagation direction of a spiral state to
give the minimum free energy.
Figure 3 shows prototypical examples of spiral states for three
cases of the Rashba and the Dresselhaus spin–orbit couplings. When
either the Rashba or the Dresselhaus term is zero, the free energy
is degenerate for the direction of a pitch vector; figures 3(a) and
(b) show one of the ground states, say at θq = π/8. In
the case of the limited Rashba term, the direction θq is
independent of φH from equation (22), where spins always rotate in
the plane containing the pitch vector q. This is also the case for
figure 3(c), where both spin–orbit terms are equal. Alternatively,
in the case of the limited Dresselhaus term, the spin rotation
depends on the direction of the pitch vector as shown in figure
3(a); for θq = 0, the spin rotation is completely
different from figures 3(b) and (c), because it occurs in the plane
perpen-dicular to a q vector, while for θq = π/4 the spin
rotation is the same as in the other cases.
As another feasible ground state, we examine the skyrmion phase.
Prior to a two-dimensional skyrmion lattice, we first consider an
isolated skyrmion that has the spin pointing down at
r = 0 and up and far away r → ∞. A possible
solution can be inferred from equation (17) as
θ
θ
θ
ˆ = ++
ˆ = ++
ˆ =
mD x D y
r D D
mD x D y
r D D
m
sin (r),
sin (r),
cos (r)
xR D
R D
yD R
R D
2 2C
2 2C
z C (23)
where θC(r) denotes the tilting angle of a spin measured from
the z-axis smoothly interpolating from
θC(r = 0) = π to
θC(r = ∞) = 0. The above relations are
justified because, by inserting them into equation (17), the
corresponding energy functional depends only on r, θC(r) and
∂θC(r)/∂r variables. Namely, the relations hold well if ≪ +D D D D2
( )D R D R2 2 . Spin rotations of the isolated skyrmion in the
two-dimensional plane are now clear: in the case of the limited
Rashba, the spin direction is radial in the x − y plane
in all directions from the skyrmion core; in the case of the
limited Dresselhaus, its x− and y−components exhibit an anti-vortex
configuration. The skyrmion number defined by ∫dr (∂x m
× ∂ym)·m/4π is
− +D D D D( ) / ( )D R R D2 2 2 2 for equation (23). In fact,
because the skyrmion number should be an integer, the trial ground
state of equation (23) is the exact skyrmion phase only in the case
of DD = 0 or DR = 0.
In order to study the skyrmion lattice, an initial
configura-tion can be prepared by close-packing the individual
isolated skyrmions of equation (23), for example, as triangular and
square patterns.
Then, from the self-consistent solution of equations (B.5) and
(B.9) via successive iterations, the local minimum of the
Figure 3. Spiral states are shown for (αD,
αR) = (0.05, 0.0) (eV·nm) in (a), (0.0, 0.05) in (b) and
(0.035, 0.035) in (c). T = 100 Csd K,
μ0Hex = 0, and S = 1/2 are used. In the cases
of (a) and (b), θq defined in the text is π/8.
(a)
0 1 0 20 300
3
6
9
y (a
0)
x (a0)
y (a
0)y
(a0)
(b)
0 1 0 20 300
3
6
9
(c)
0 1 0 20 300
3
6
9
J. Phys.: Condens. Matter 26 (2014) 196005
-
J H Oh et al
7
free energy is obtained by varying the distance between
skyr-mions for the different arrangements.
It is found that the triangular arrangement has lower energy
than that of the square arrangement in the case of the limited
Rashba and the limited Dresselhaus. This fact is in agree-ment with
previous experimental results [13, 14]. When both Rashba and
Dresselhaus spin–orbit couplings are finite, the local minimum
point is barely to be found due to its com-plicated energy surface.
In other words, this may mean that the skyrmion phase is not likely
to be a ground state when both Rashba and Dresselhaus spin–orbit
couplings are finite, as indicated by the non-integer skyrmion
number.
We show prototypical examples of the skyrmion phases for the
limited Rashba and Dresselhaus spin–orbit interactions,
respectively, in figure 4. The calculated spin textures are found
to
be similar to those from the isolated skyrmions. However,
details are slightly modified because we lift the constant
spin-moment constraint via equation (B.5). That is, the magnitude
and direc-tion of each spin are changed to have a lower free energy
as well as a slightly enhanced elongated shape along an axis.
It is noted that the spin textures have topologically dif-ferent
shapes depending on the type of spin–orbit coupling. As seen from
figures 3 and 4, in the case of the limited Dresselhaus
interaction, spins rotate in planes depending on the q vector in
the spiral phase and skyrmion phase. In the case of the limited
Rashba interaction, spin rotations occur in the plane containing
the q vector for the spiral state and paral-lel to the radial
direction from cores in the skyrmion phase. If both types of
spin–orbit interaction are present, the com-plicated spin texture
is calculated, which looks like magnetic bubbles, as in figure
4(c).
By comparing the free energy of the three trial phases, we can
map a minimum one as a function of external magnetic field and
temperature. The result is summarized in figure 5. At a zero
magnetic field, it is found that the spiral state is the most
stable one as long as the temperature is less than
ν= +T J S S2 ( 1) / 3C (24)
with the number of the nearest neighbor ν. Then, by increas-ing
the magnetic field for a given temperature, the skyrmion state
becomes stable and eventually the spin system exhibits a
ferromagnetic state, and even a paramagnetic state with fur-ther
increases in the magnetic field.
After various calculations with changing system parame-ters, we
find that the boundaries between the magnetic phases are very
similar to a magnetic moment curve mz(T) of ferro-magnetism at
Hex = 0 (solid lines in figure 5). Thus, by fitting the
calculated results to the magnetic moment curve mz(T), the
boundaries, Bc1(T) between the spiral and skyrmion phases,
Figure 4. Skyrmion states are shown for (αD,
αR) = (0.05, 0.0) (eV·nm) in (a), (0.0, 0.05) in (b) and
(0.035, 0.035) in (c). T = 100 Csd K,
μ0Hex = 0.3B0 and S = 1/2 are used.
(a)
0 20 400
20
40y
(a0)
y (a
0)y
(a0)
x (a0)
(b)
0 20 400
20
40
(c)
0 20 400
20
40
Figure 5. Boundaries among spiral, skyrmion, ferromagnetic (FM)
and paramagnetic phases are shown in the H − T space.
Each symbol represents different sets of a parameter, such as (αD,
αR, S) = (0.08 eV·nm,0.0,1/2) (circles),
(0.08 eV·nm,0.0,3/2) (triangles), (0.05 eV·nm,0.0,1/2)
(boxes), and (0.05 eV·nm,0.0,3/2) (crosses). Normalized
magnetic moments of a ferromagnetic state (Hex = 0) are
also plotted with solid lines for comparison. The same results are
calculated for the limited Rashba interaction. In the case of both
a finite Dresselhaus and Rashba interaction, no boundaries were
found.
PM
FM
Skyrmion
Spiral
0.0 0.4 0.8 1.2T/Tc
0.0
0.3
0.6
0.9
µ 0H
ex/B
0
J. Phys.: Condens. Matter 26 (2014) 196005
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J H Oh et al
8
and Bc2(T) between the skyrmion and ferromagnetic phases, are
obtained as:
=
=
B T c Bm T
S
B T c Bm T
S
( ) ( )
,
( ) ( )
c
c
1 1 0z
2 2 0z (25)
where γ= +B S D D a J( ) /D R0 2 2 02 and fitting parameters
c1,2 are also a function of DD, R and J. Because we adopt the mean
field approximation, the calculated phase boundaries look like the
result of the Stoner model,
mz(T) ∝ (TC/T − 1)1/2 as a function of
temperature. However, this is somewhat different from the phase
boundaries measured in experiments [13, 14]. To produce more
realistic boundaries, it is necessary to use a sophisticated
approach such as a Monte Carlo method, rather than the mean field
approximation [14].
6. Summary
For the case where Rashba and Dresselhaus spin–orbit couplings
exist in a two-dimensional spin system, the Dzyaloshinskii–Moriya
interaction has been derived based on the indirect exchange
Hamiltonian, together with the RKKY interaction. Based on a Green's
function method, the strength of the Dzyaloshinskii–Moriya
interaction was examined as a function of atomic distance and
spin–orbital strength. With the calculated Dzyaloshinskii–Moriya
strengths, we discussed the formation of various magnetic ground
states as a func-tion of temperature and external magnetic field.
By plotting the H-T phase diagram, we presented approximate
boundaries between spiral, skyrmion and ferromagnetic phases.
Appendix A. Green's function
Bare lesser and greater Green's functions ±≶g are defined,
for
each branch of spin states, by
− = −ℏ
− ′ =ℏ
′
∼ ∼
∼ ∼
±< ′
±† ′
±
±>
± ±†
g t t C t C t
g t t C t C t
k
k
( , )1
i( ) ( )
( , )1
i( ) ( ) .
k k
k k (A.1)
Via the average over H0, these Green's functions are calculated
as,
τ μ
τ μ
=ℏ
− −
= −ℏ
−
μ τ
μ τ
±> ± − ℏ
±< ± − ℏ
±
±
[ ]g f E
g f E
k
k
( , )1
i1 ( ) e
( , )1
i( )e
E
E
k
k
( ) /i
( ) /i
k
k
(A.2)
where ±Ek are eigenvalues of H0 as shown in equation (5).The
Green's functions in equation (10) are defined by:
∑τ τ
τ τ τ η σ η σ
= ·
= + + − −
≶ ≶
≶+≶
−≶
+≶
−≶
A
g g g g
G R G
G k 1 k
( , )1
e ( )
( )1
2( )( , )
1
2( )( , )(cos sin )
k
k Rk k
k k k k
i,
,x y
Now we write,
τ α σ α σ θ= + × + +∼≶ ≶ ≶ ≶⃗ ( )G z G GG R 1 R( , ) [ ( ˆ ) ] ·
ˆ ˆR D R0 1 2
with
⎛⎝⎜
⎞⎠⎟ ⎡⎣ ⎤⎦
⎡⎣ ⎤⎦
∑
∑
·
·
τθ
τ τ
τ τ τ
= ∂∂
∂∂ −
−
= +
≶+ − +
≶−≶
≶+≶
−≶
GR R A E E
g g
GA
g g
R k k
R k k
( , ) ,1 1
ie
1( , ) ( , ) ,
( , )1
2e ( , ) ( , ) .
R k
k R
k k
k
k R
1,2i
0i
(A.3)In the case of αR≠0 and αD≠ 0, the energy dispersions
±Ek are not circular symmetric in the k-space. Due to this,
resulting Green's functions G0,1,2 are also not symmetric in the
R-space generally. However, this non-symmetric nature turns out to
be weak because it is mainly attributed to a small portion of the
k−space where energy is less than the spin–orbit coupling strength.
In the summation of equation (A.3), we neglect the azimuthal angle
dependence in the R-space by taking the average over the angle.
This leads to
τ τ= ∣ ∣≶ ≶G GR R( , ) ( , )0,1 0,1 and τ =≶G R( , ) 02 , though
these are exact in the case of αRαD = 0.
Appendix B. Mean field approach
Under the mean field approximation, the Helmholtz free energy of
the system HM is approximated by
ϕ≤ + − ≡H H .M Z0 0 (B.1)
Here, HZ is some Hamiltonian simple enough for the free energy 0
to be exactly evaluated. Thus, ϕ is a variational upper bound of
the Helmholtz free energy. Following Weiss's idea of a molecular
field, we choose
∑γ= − ·H B SZi
i ieff
(B.2)
with the gyromagnetic ratio γ. Here, we write an effective
magnetic field as the sum of an external field and a molecular
field on each atom i:
μ= +B H Bi ieff 0 ex mol (B.3)
This choice yields for 0 :
⎧⎨⎩
⎫⎬⎭
⎡⎣
⎤⎦
⎡⎣
⎤⎦
∑
∑
= −·
= −+
γ
γ
γ
∣ ∣
∣ ∣
k T e
k T
lnTr
lnsin h (S )
sin h
i
k T
i
k
k
B S
B
B
0 B
B
1
2 T
1
2 T
i ieff
B
ieff
B
ieff
B
(B.4)
where S is a maximum value of spin moment. Then, the average of
spin moment at a site i over the Hamiltonian HZ is given by:
γ
δδγ
= = −
=∣ ∣
∣ ∣
=⎜ ⎟⎛⎝
⎞⎠
⎛⎝⎜
⎞⎠⎟SB
k TS
m SB
B mm
1i i
Si i
i
B B0
0
eff
B
eff
(B.5)
where Bieff is assumed to be parallel to mi and BS(x) is the
Brillouine function:
⎡⎣⎢
⎤⎦⎥
⎡⎣⎢
⎤⎦⎥=
+ + −B x SS
S
Sx
S
S Sx( )
2 1coth
2 1
2
2coth
1
2.S (B.6)
J. Phys.: Condens. Matter 26 (2014) 196005
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J H Oh et al
9
By replacing spin operators Si with their classical expecta-tion
value mi, the free energy ϕ is approximated as:
∑ϕ γ= + ∣ → + ·H B m .Mi
i iS m0eff
i i (B.7)
Now we demand the free energy ϕ to be stationary over two
variational parameters, Bimol and mi as
δϕδ
δϕδ
= =B
m
0 and 0.i imol (B.8)
Then, we obtain
⎡⎣
⎤⎦
∑γ = Φ + Φ
− Φ · − Φ ×
{ }C R R
R R
B m
d d m d m
2 ( ) ( )
2 ( ) ( ) ( ) ) .
i sdj
ji S ji j
S ji ji ji i A ji ij j
mol0
(B.9)
Acknowledgments
This work was supported by the Pioneer Research Centre Program
(NRF-2011-0027906). H-W LEE acknowledges financial support from
MOTIE (no.10044723).
References
[1] Rashba E I 1960 Sov. Phys. Solid State 2 1109 [2]
Dresselhaus G 1955 Phys. Rev. 100 580 [3] Krupin O, Bihlmayer G,
Starke K, Gorovikov S, Prieto J E,
Döbrich K, Blügel S and Kaindl G 2005 Phys. Rev. B 71 201403
[4] Garello K, Miron I M, Avci C O, Freimuth F, Mokrousov Y,
Blügel S, Auffret S, Boulle O, Gaudin G and Gambardella P 2013
Nature Nanotechnol. 8 587
[5] Park J-H, Kim C H, Lee H-W and Han J H 2013 Phys. Rev. B 87
041301
[6] Winkler R 2003 Spin–Orbit Coupling Effects in Two-
Dimensional Electron and Hole Systems (Berlin: Springer)
[7] Sinitsyn N A, Hankiewicz E M, Teizer W and Sinova J 2004
Phys. Rev. B 70 081312
[8] Zhou B 2010 Phys. Rev. B 81 075318 [9] Chang R S, Chu C S
and Mal'shukov A G 2009 Phys. Rev. B
79 195314 [10] Dzyaloshinskii I E 1957 Sov. Phys.—JETP 5
1259[11] Moriya T 1960 Phys. Rev. 120 91 [12] Iwasaki J, Mochizuki
M and Nagaosa N 2013 Nature
Commun. 4 1463 [13] Mühlbauer S, Binz B, Joinetz F, Pfeiderer C,
Rosch A,
Neubauer A, Georgii R and Böni P 2009 Science 323 915
[14] Yu X Z, Onose Y, Kanazawa N, Park J H, Han J H, Matsui Y,
Nagaosa N and Tokura Y 2010 Nature 465 901
[15] Heinze S, von Bergmann K, Menzel M, Brede J, Kubetzka A,
Wiesendanger R, Bihlmayer G and Blügel S 2011 Nature Phys. 7
713
[16] Han J H, Zang J, Yang Z, Park J-H and Nagaosa N 2010 Phys.
Rev. B 82 094429
[17] Ho K Y, Kirkpatrick T R, Sang Y and Belitz D 2010 Phys.
Rev. B 82 134427
[18] Lounis S, Bringer A and Blügel S 2012 Phys. Rev. Lett. 108
207202
[19] Lyu P and Liu N 2007 J. Appl. Phys. 102 103910 [20] Imamura
H, Bruno P and Utsumi Y 2004 Phys. Rev. B
69 121303 [21] Kasuya T 1956 Prog. Theor. Phys. 16 45 [22]
Majlis N 2004 The quantum Theory of Magnetism (Singapore:
World Scientific) p 241[23] Mishchenko E G and Halperin B I 2003
Phys. Rev. B
68 045317 [24] Simon P, Braunecker B and Loss D 2008 Phys. Rev.
B
77 045108 [25] Rammer J 2007 Quantum Field Theory of
Non-Equilibrium
States (Cambridge: Cambridge University Press)[26] Oh J H, Ahn D
and Hwang S W 2005 Phys. Rev. B
72 165348 [27] Giuliani G F and Vignale G 2005 Quantum Theory of
the
Electron Liquid (Cambridge: Cambridge University Press)[28]
Kittel C 1949 Rev. Mod. Phys. 21 541 [29] Fischer B and Klein M W
1975 Phys. Rev. B
11 2025 [30] Litvinov V I and Dugaev V K 1998 Phys. Rev. B 58
3584 [31] Jagannathan A, Abrahams E and Stephen M J 1988 Phys.
Rev. B 37 436 [32] Chesi S and Loss D 2010 Phys. Rev. B 82
165303 [33] Fert A and Levy P M 1980 Phys. Rev. Lett. 9 1538
J. Phys.: Condens. Matter 26 (2014) 196005
http://dx.doi.org/10.1103/PhysRev.100.580http://dx.doi.org/10.1103/PhysRev.100.580http://dx.doi.org/10.1103/PhysRevB.71.201403http://dx.doi.org/10.1103/PhysRevB.71.201403http://dx.doi.org/10.1038/nnano.2013.145http://dx.doi.org/10.1038/nnano.2013.145http://dx.doi.org/10.1103/PhysRevB.87.041301http://dx.doi.org/10.1103/PhysRevB.87.041301http://dx.doi.org/10.1103/PhysRevB.70.081312http://dx.doi.org/10.1103/PhysRevB.70.081312http://dx.doi.org/10.1103/PhysRevB.81.075318http://dx.doi.org/10.1103/PhysRevB.81.075318http://dx.doi.org/10.1103/PhysRevB.79.195314http://dx.doi.org/10.1103/PhysRevB.79.195314http://dx.doi.org/10.1103/PhysRev.120.91http://dx.doi.org/10.1103/PhysRev.120.91http://dx.doi.org/10.1038/ncomms2442http://dx.doi.org/10.1038/ncomms2442http://dx.doi.org/10.1126/science.1166767http://dx.doi.org/10.1126/science.1166767http://dx.doi.org/10.1038/nature09124http://dx.doi.org/10.1038/nature09124http://dx.doi.org/10.1038/nphys2045http://dx.doi.org/10.1038/nphys2045http://dx.doi.org/10.1103/PhysRevB.82.094429http://dx.doi.org/10.1103/PhysRevB.82.094429http://dx.doi.org/10.1103/PhysRevB.82.134427http://dx.doi.org/10.1103/PhysRevB.82.134427http://dx.doi.org/10.1103/PhysRevLett.108.207202http://dx.doi.org/10.1103/PhysRevLett.108.207202http://dx.doi.org/10.1063/1.2817405http://dx.doi.org/10.1063/1.2817405http://dx.doi.org/10.1103/PhysRevB.69.121303http://dx.doi.org/10.1103/PhysRevB.69.121303http://dx.doi.org/10.1143/PTP.16.45http://dx.doi.org/10.1143/PTP.16.45http://dx.doi.org/10.1103/PhysRevB.68.045317http://dx.doi.org/10.1103/PhysRevB.68.045317http://dx.doi.org/10.1103/PhysRevB.77.045108http://dx.doi.org/10.1103/PhysRevB.77.045108http://dx.doi.org/10.1103/PhysRevB.72.165348http://dx.doi.org/10.1103/PhysRevB.72.165348http://dx.doi.org/10.1103/RevModPhys.21.541http://dx.doi.org/10.1103/RevModPhys.21.541http://dx.doi.org/10.1103/PhysRevB.11.2025http://dx.doi.org/10.1103/PhysRevB.11.2025http://dx.doi.org/10.1103/PhysRevB.58.3584http://dx.doi.org/10.1103/PhysRevB.58.3584http://dx.doi.org/10.1103/PhysRevB.37.436http://dx.doi.org/10.1103/PhysRevB.37.436http://dx.doi.org/10.1103/PhysRevB.82.165303http://dx.doi.org/10.1103/PhysRevB.82.165303http://dx.doi.org/10.1103/PhysRevLett.44.1538http://dx.doi.org/10.1103/PhysRevLett.44.1538