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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. 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

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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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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

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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

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

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).

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