Scattering of high-energy magnons off a magnetic skyrmion
We discuss the scattering of high-energy magnons off a single magnetic skyrmion within the field-polarized ground state of a two-dimensional chiral magnet. For wavevectors larger than the inverse skyrmion radius,krs>>1 the magnon scattering is dominated by an emerging magnetic field whose flu...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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Schroeter, S. Garst, M. 2018-01-05T17:56:55Z 2018-01-05T17:56:55Z 2015 Scattering of high-energy magnons off a magnetic skyrmion / S. Schroeter and M. Garst // Физика низких температур. — 2015. — Т. 41, № 10. — С. 1043–1053. — Бібліогр.: 51 назв. — англ. 0132-6414 PACS: 75.10.-b, 05.45.Yv https://nasplib.isofts.kiev.ua/handle/123456789/128087 We discuss the scattering of high-energy magnons off a single magnetic skyrmion within the field-polarized ground state of a two-dimensional chiral magnet. For wavevectors larger than the inverse skyrmion radius,krs>>1 the magnon scattering is dominated by an emerging magnetic field whose flux density is essentially determined by the topological charge density of the skyrmion texture. This leads to skew and rainbow scattering characterized by an asymmetric and oscillating differential cross section. We demonstrate that the transversal momentum transfer to the skyrmion is universal due to the quantization of the total emerging flux while the longitudinal momentum transfer is negligible in the high-energy limit. This results in a magnon-driven skyrmion motion approximately antiparallel to the incoming magnon current and a universal relation between current and skyrmion-velocity. We acknowledge helpful discussions with M. Mostovoy and A. Rosch. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур К 80-летию уравнения Ландау–Лифшица Scattering of high-energy magnons off a magnetic skyrmion Article published earlier |
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Scattering of high-energy magnons off a magnetic skyrmion |
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Scattering of high-energy magnons off a magnetic skyrmion Schroeter, S. Garst, M. К 80-летию уравнения Ландау–Лифшица |
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Scattering of high-energy magnons off a magnetic skyrmion |
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Scattering of high-energy magnons off a magnetic skyrmion |
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Scattering of high-energy magnons off a magnetic skyrmion |
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Scattering of high-energy magnons off a magnetic skyrmion |
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scattering of high-energy magnons off a magnetic skyrmion |
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Schroeter, S. Garst, M. |
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Schroeter, S. Garst, M. |
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К 80-летию уравнения Ландау–Лифшица |
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Физика низких температур |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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We discuss the scattering of high-energy magnons off a single magnetic skyrmion within the field-polarized
ground state of a two-dimensional chiral magnet. For wavevectors larger than the inverse skyrmion radius,krs>>1 the magnon scattering is dominated by an emerging magnetic field whose flux density is essentially determined
by the topological charge density of the skyrmion texture. This leads to skew and rainbow scattering
characterized by an asymmetric and oscillating differential cross section. We demonstrate that the transversal
momentum transfer to the skyrmion is universal due to the quantization of the total emerging flux while the longitudinal
momentum transfer is negligible in the high-energy limit. This results in a magnon-driven skyrmion
motion approximately antiparallel to the incoming magnon current and a universal relation between current and
skyrmion-velocity.
|
| issn |
0132-6414 |
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https://nasplib.isofts.kiev.ua/handle/123456789/128087 |
| citation_txt |
Scattering of high-energy magnons off a magnetic skyrmion / S. Schroeter and M. Garst // Физика низких температур. — 2015. — Т. 41, № 10. — С. 1043–1053. — Бібліогр.: 51 назв. — англ. |
| work_keys_str_mv |
AT schroeters scatteringofhighenergymagnonsoffamagneticskyrmion AT garstm scatteringofhighenergymagnonsoffamagneticskyrmion |
| first_indexed |
2025-11-25T20:35:31Z |
| last_indexed |
2025-11-25T20:35:31Z |
| _version_ |
1850523804226289664 |
| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 10, pp. 1043–1053
Scattering of high-energy magnons off a magnetic
skyrmion
Sarah Schroeter and Markus Garst
Institut für Theoretische Physik, Universität zu Köln, Zülpicher Str. 77a, Köln, 50937, Germany
E-mail: mgarst@uni-koeln.de
Received March 30, 2015, published online August 25, 2015
We discuss the scattering of high-energy magnons off a single magnetic skyrmion within the field-polarized
ground state of a two-dimensional chiral magnet. For wavevectors larger than the inverse skyrmion radius,
1skr the magnon scattering is dominated by an emerging magnetic field whose flux density is essentially de-
termined by the topological charge density of the skyrmion texture. This leads to skew and rainbow scattering
characterized by an asymmetric and oscillating differential cross section. We demonstrate that the transversal
momentum transfer to the skyrmion is universal due to the quantization of the total emerging flux while the lon-
gitudinal momentum transfer is negligible in the high-energy limit. This results in a magnon-driven skyrmion
motion approximately antiparallel to the incoming magnon current and a universal relation between current and
skyrmion-velocity.
PACS: 75.10.-b General theory and models of magnetic ordering;
05.45.Yv Solitons.
Keywords: magnetic skyrmion, chiral magnet.
1. Introduction
The experimental discovery of skyrmions in chiral
magnets [1–7] and in magnetic monolayers [8–10] has
triggered an increasing interest in the interaction of spin
currents with topological magnetic textures [11–30]. It has
been demonstrated [13,16] that skyrmions can be manipu-
lated by ultralow electronic current densities of 106 A/m2,
which is five orders of magnitudes smaller than in conven-
tional spintronic applications using domain walls. The adi-
abatic spin-alignment of electrons moving across a
skyrmion texture results in an emergent electrodynamics
implying a topological [11,12,30] as well as a skyrmion-
flow Hall effect [17]. In insulators, the interplay of thermal
magnon currents and skyrmions is marked by a topological
magnon Hall effect and a magnon-driven skyrmion motion
[23–25]. The topological nature of the magnetic skyrmions
is responsible for a peculiar dynamics [31–35] that is also
at the origin of these novel spintronic and caloritronic phe-
nomena, which are at the focus of the fledgling field of
skyrmionics [22].
In two spatial dimensions, skyrmions are identified by
the topological charge density
. top
1 ˆ ˆ ˆ= ( ),
4 x yn n nρ ∂ ×∂
π
(1)
where n̂ is the orientation of the magnetization vector. For
a magnetization homogeneously polarized at the boundary,
the spatial integral top
2 =d r Wρ∫ is quantized, W Z∈ ,
and thus allows to count skyrmions within the sample. In
turn, a finite winding number W translates to a
Fig. 1. (Color online) (a) A chiral magnetic skyrmion texture of
linear size rs. (b) Illustration of a classical magnon trajectory
within the xy plane scattering off a skyrmion positioned at R
with impact parameter b and classical deflection angle Θ .
© Sarah Schroeter and Markus Garst, 2015
Sarah Schroeter and Markus Garst
gyrocoupling vector G in the Thiele equation of motion of
the skyrmion [36], and the resulting gyrotropic spin-
Magnus force governs its dynamics [37]. As a conse-
quence, in the presence of an applied electronic spin cur-
rent, the skyrmions will acquire a velocity [14,15,17] that
remains finite in the limit of adiabatic spin-transfer torques
and small Gilbert damping α, giving rise to a universal
current-velocity relation [18].
In order to address the interaction of magnon currents
with magnetic textures, a corresponding adiabatic approx-
imation has been recently invoked on the level of the Lan-
dau–Lifshitz–Gilbert equation by Kovalev and Tserkov-
nyak [38]. This approximation has been used in Refs. 23,
24 to derive an effective Thiele equation of motion for the
skyrmion coordinate R in the presence of a magnon cur-
rent density J,
eff eff= ,× − × +β +G R G v v
(2)
with = 0β in the adiabatic limit. The effective velocity
eff 0= /( )Bg mµv J is related to the current density via
the g-factor ,g the Bohr magneton > 0Bµ and the local
magnetization 0m . The gyrocoupling vector is given by
0ˆ= 4 /( )Bz m g− π µG with units of spin density corre-
sponding to a flux of 2− π per area of a spin – 1
2
in a two-
dimensional system with the unit normal vector ˆ.z The
dots in Eq. (2) represent further terms omitted for the pur-
pose of the following discussion, that is, in particular, a
damping force proportional to the Gilbert constant α . Ne-
glecting these additional terms, Eq. (2) predicts for = 0,β
similar to the skyrmion-driven motion by electronic cur-
rents, a universal current-velocity relation
eff 0= = /( )Bg m− − µ JR v
with a skyrmion velocity that is
antiparallel to J. Consequently, a magnon current generat-
ed by a thermal gradient will induce a skyrmion motion
towards the hot region of the sample, which was indeed
observed numerically [23,24,27]. Mochizuki et al. [25]
also used Eq. (2) with = 0β to account for the experi-
mental observation of a thermally induced rotation of a
skyrmion crystal.
However, the question arises as to when the adiabatic
limit of Eq. (2) is actually applicable and under what con-
ditions. The validity regime of the adiabatic approximation
for magnon-driven motion of magnetic textures has not
been explicitly discussed in Ref. 38. In fact, in order to
account quantitatively for their numerical experiment Lin
et al. [24] introduced the β parameter in Eq. (2) on phe-
nomenological grounds calling it a measure for non-
adiabaticity. Subsequently, Kovalev [28] argued that a fi-
nite β parameter arises due to dissipative processes.
In contrast, we have recently shown by considering the
magnon–skyrmion scattering problem [29] that a mono-
chromatic magnon current with energy ε will give rise to
a reactive momentum-transfer force in the Thiele equation
which reads in linear response
||ˆ= ( )( ) ( ) ,k z k⊥ ε ε× σ ε × + σ ε +G R J J
(3)
where the magnon dispersion is 2
gap mag= ( ) /(2 )k Mε ε +
with the magnon gap gapε and the magnon mass magM .
This force on the right-hand side of Eq. (3) is determined
by the two-dimensional transport scattering cross sections
|| ( ) 1 cos
=
sin( )
dd
d
π
⊥ −π
σ ε − χ σ
χ − χ χσ ε
∫ (4)
where /d dσ χ is the energy-dependent differential scatter-
ing cross section of the skyrmion. In the limit of low-
energies 1skr , where rs is the skyrmion radius, s-wave
scattering is found to dominate so that ( ) 0⊥σ ε → and, as
shown in Ref. 23, the force becomes longitudinal to εJ .
This, in turn, implies a skyrmion motion approximately
perpendicular to the magnon current, || ( )
ˆ ,
| |
k
z ε
σ ε
→ ×R J
G
thus maximally violating the predictions of the adiabatic
limit of Eq. (2). This implies that Eq. (2) is not valid for
low-energy magnons whose wavevector is comparable or
smaller than the inverse size of the texture.
It is one of the aims of this work to demonstrate explic-
itly that in the high-energy limit, 1skr , on the other
hand, the momentum-transfer force of Eq. (3) due to a
monochromatic magnon wave indeed reduces to the form
of Eq. (2). The effective velocity in this case, however, is
to be identified with 2
eff mag(| )| /A M=v k where A is
the amplitude of the incoming magnon wave. In the high-
energy limit the magnon-skyrmion interaction is dominat-
ed by a scattering vector potential, i.e., an emerging orbital
magnetic field whose flux is quantized and related to the
skyrmion topology. As a result, the transversal momentum
transfer assumes a universal value in the high-energy limit
( ) 4k ⊥σ ε → π as anticipated in Ref. 25. Moreover, the lon-
gitudinal momentum transfer yields a reactive contribution,
εβ , to the β parameter that, in this limit, is determined by
the square of the classical deflection function ( )bΘ inte-
grated over the impact parameter b , see Fig. 1(b),
2| |= ( ( )) .
8
G k db b
∞
ε
−∞
β Θ
π ∫ (5)
As the scattering is in forward direction at high energies,
( ) 1/b kΘ , the parameter vanishes as 1/kεβ ∝ so that it is
indeed small for large 1srκ >> .
The outline of the paper is as follows. In Sec. 2 we
shortly review the definition of the magnon–skyrmion scat-
tering problem and some of the main results of Ref. 29. In
Sec. 3 we examine the scattering properties of high-energy
magnons including the skew and rainbow effects, the total
and transport scattering cross sections, and the magnon
pressure on the skyrmion leading to Eq. (2). We finish with
a short discussion in Sec. 4.
1044 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 10
Scattering of high-energy magnons off a magnetic skyrmion
2. Skyrmionic soliton and its spin-wave excitations
This section closely follows Ref. 29 and reviews the
magnon–skyrmion scattering problem in a two-
dimensional chiral magnet. We start with the standard
model for a cubic chiral magnet restricted to a two-
dimensional plane that is described by the energy function-
al [39,40]
2 2 ˆˆ ˆ ˆ ˆ= [( ) 2 2 ]
2
s
j i j i jn Q n n nBα α α
ρ
∂ + ε ∂ − κ (6)
with spatial index {1,2} = { , }x yα∈ and , {1,2,3}i j∈ ,
i jα∈ is the totally antisymmetric tensor with 123= 1∈ , and
sρ is the stiffness. The two length scales are given by the
wavevectors Q and κ . The former determines the
strength of the spin-orbit Dzyaloshinskii–Moriya interac-
tion, that we chose to be positive, > 0Q . The latter,
> 0,κ measures the strength of the applied magnetic field,
that is applied perpendicular to the two-dimensional plane,
ˆ ˆ=B z . We neglect cubic anisotropies, dipolar interactions
as well as magnetic anisotropies for simplicity. The latter
can be easily included resulting in an additional length
scale.
2.1. Skyrmionic saddle-point solution
The theory (6) possesses a topological soliton solution,
i.e., a skyrmion, as first pointed out by Bogdanov and Hu-
bert [41,42]. With the standard parametrization of the unit
vector ˆ = (sin cos , sin sin , cos )T
sn θ ϕ θ ϕ θ , the skyrmion
obeys
= ( ), = ,
2
π
θ θ ρ ϕ χ + (7)
where ρ and χ are polar coordinates of the two-
dimensional spatial vector = (cos ,sin ).ρ χ χr The polar
angle θ obeys the differential equation
2
2
2
sin cos 2 sin sin = 0,Q′θ θ θ θ′′θ + − + − κ θ
ρ ρρ
(8)
with the boundary conditions (0) =θ π and
( ) = 0limρ→∞θ ρ . At large distances 1ρκ >> , the polar
angle obeys the asymptotics ( ) /e−κρθ ρ ρ . The result-
ing skyrmion texture is illustrated in Fig. 1(a). The associ-
ated topolpgical charge density
top
1 1 sinˆ ˆ ˆ= ( ) =
4 4
s
s x s y sn n n
′θ θ
ρ ∂ ×∂
π π ρ
(9)
integrates to 2
top = 1d ρ −∫ r identifying the solution as a
skyrmion. The skyrmion radius rs can be defined with the
help of the area
2 2ˆ(1 )/2 ,z sd r n r− = π∫
and it is found to approximately obey rs∼ 1/κ2.
The skyrmion is a large-amplitude excitation of the fully
polarized ground state as long as its energy is positive,
which is the case for cr>κ κ where 2 2
c 0.8 ,r Qκ ≈ which is
the regime we focus on. For smaller values of κ ,
skyrmions proliferate resulting in the formation of a
skyrmion crystal ground state.
2.2. Magnon-skyrmion scattering problem
Magnon wavefunction. The magnons correspond to
spin-wave excitations around the skyrmion solution ˆsn
that can be analyzed in the spirit of previous work by
Ivanov and collaborators. [43–46] We introduce the local
orthogonal frame ˆ ˆ =i j ije e δ with 1 2 3ˆ ˆ ˆ=e e e× , where
3ˆ ˆ( ) = ( )se nr r tracks the skyrmion profile. For the two
orthogonal vectors we use
1̂ = ( sin , cos , 0)Te − ϕ ϕ and
2ˆ = ( cos cos , sin sin , cos )Te − θ ϕ θ ϕ θ .
The excitations are parametrized in the standard fashion
2 *
3ˆ ˆ ˆ ˆ= 1 2 | | ,n e e e+ −− ψ + ψ + ψ (10)
where ψ is the magnon wavefunction and
1 2
1ˆ ˆ ˆ= ( ).
2
e e ie± ± For large distances, srρ >> , this
parametrization assumes the form
2 1ˆ ˆ ˆˆ 1 2 | | ( )( e ) c.c. .
2
in z x iy − χ
≈ − ψ + + − ψ +
(11)
It is important to note that the local frame îe corresponds
to a rotating frame even at large distances reflected in the
phase factor e i− χ− in the second term. For the discussion
of magnon scattering, it will be convenient to introduce a
wavefunction labψ with respect to a frame that reduces to
the laboratory frame at large distances, that is simply ob-
tained by the gauge transformation
lab ( , ) = e ( , ).it t− χψ − ψr r (12)
Magnon Hamiltonian. In order to derive an effective
Hamiltonian for ψ , we consider the Landau–Lifshitz
equation
effˆ ˆ= ,t n n∂ −γ ×B (13)
with = / ,Bgγ µ where the effective magnetic field
eff
0
1( , ) =
ˆ( , )
Et
m n t
δ
−
δ
B r
r
is determined by the functional
derivative of the integrated energy density =E dtd∫ r .
Expanding (13) in lowest order in ψ , one finds that the
spinor *= ( , )T ψ ψΨ is governed by a bosonic Bogoliu-
bov–deGennes (BdG) equation
= ,z
ti τ ∂ Ψ Ψ (14)
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 10 1045
Sarah Schroeter and Markus Garst
with the Hamiltonian
2 2
0
mag
( )=
2
z
x
x
i
M
− − τ
+ + τ
a
∇1
1 (15)
where = ( , )T
x y∂ ∂∇ , and xτ and zτ are Pauli matrices.
The potentials are given by
2gap
0 2 2
sin(2 )sin( ) =
22
Qε θ θ
ρ − − − ρκ ρ
(16)
2
2 22 cossin
2
Q Q
′θ′− θ + κ θ− θ −
,
22gap
2 2
sin(2 )sin( ) = .
2 22
x
Q Q
ε ′θ θ θ′ρ + − θ − ρκ ρ
(17)
The magnon energy gap is defined by
2 2 2
gap
0 mag
= = ,
2
B sg
m M
µ ρ κ κ
ε
(18)
which also identifies the magnon mass magM . The vector
potential reads ˆ= ( )aχ ρ χa with ˆ = ( sin ,cos )Tχ − χ χ and
cos= sin .a Qχ θ
− θ
ρ (19)
It obeys the Coulomb gauge = 0∇a . The polar angle in all
potentials is the soliton solution, = ( )θ θ ρ , and depends on
the distance ρ .
Effective magnetic flux. Far away from the skyrmion the
Hamiltonian simplifies 0→ for ρ→∞ with
2 2
0 gap
mag
1 ˆ( )
= .
2
zi
M
− − τ χ
ρ + ε
∇1
1 (20)
The remaining vector potential is attributed to the choice of
the rotating orthogonal frame in the definition of the
magnon wavefunction, see Eq. (11). It can be easily elimi-
nated by the gauge transformation (12),
( )
lab =
zie− τ χ+π→Ψ Ψ Ψ , (21)
lab
1 cos 1= = sin .a a a Qχχ χ θ −
→ − − θ
ρ ρ
(22)
With respect to this laboratory orthogonal frame, the vector
scattering potential lab lab ˆ= aχ χa vanishes exponentially for
large distances, srρ >> .
The associated flux lab ˆ= ( ) = z∇× a will play an
important role in the following discussion, where
lab( ) = ( ( )).aχρ∂ ρ ρ
ρ
r
According to Stokes’ theorem the
total flux 2 ( ) = 0d∫ r r vanishes as laba is exponentially
confined to the skyrmion radius. However, there is an in-
teresting spatial flux distribution,
reg( ) = 4 ( ) (| |),− π δ +r r r (23)
reg top( ) = 4 ( sin ) .
4
s Q
ρ
ρ π −ρ − ∂ ρ θ πρ
(24)
Since for small distances lab ( ) 2/ ,aχ ρ → − ρ there is a singu-
lar flux contribution at the skyrmion origin with quantized
strength 4− π . As it is quantized, this singular flux will
not contribute to the magnon scattering. The regular part of
the effective magnetic flux, reg , only depends on the
radius ρ and is spatially confined to the skyrmion area. Its
spatial distribution can be related with the help of Eq. (9)
to the topological charge density top
sρ of the skyrmion in
addition to a term proportional to Q . While top
s−ρ is al-
ways positive, the latter term can also be negative so that
reg as a function of distance ρ even changes sign for
lower values of 2κ , see Fig. 2. The spatial integral over
the second term of Eq. (24) however vanishes so that the
total regular flux
2 2
reg top( ) = 4 = 4sd dρ − π ρ π∫ ∫r r
is quantized and determined by the topological charge of
the skyrmion [25,47].
2.3. Magnon spectrum
In order to solve Eq. (14) for the magnon eigenvalues
and eigenfunctions, one uses the angular momentum basis
( , ) = exp ( / ) ( )mt i t im− ε + χ ρr Ψ η with positive energy
0ε ≥ . The angular momentum m turns out to be a good
quantum number and the wave equation (14) reduces to a
radial eigenvalue problem for ( )m ρη that can be solved
Fig. 2. (Color online) Regular part of the effective magnetic flux
density (24) for various values of 2 2/Qκ . For lower values of
2 2/Qκ the flux density close to the skyrmion center is sup-
pressed and even becomes negative for 2 2/ 1.3Qκ . As a result,
the effective local Lorentz force evaluated along a classical
magnon trajectory with = 0b changes sign resulting in a sup-
pression of the deflection angle.
1046 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 10
Scattering of high-energy magnons off a magnetic skyrmion
with the help of the shooting method [29]. In order to ob-
tain positive expectation values of the Hamiltonian, one
has to look for eigenfunctions with a positive norm,
†
0
( ) ( ) > 0.z
m md
∞
ρρ ρ τ ρ∫ η η (25)
The resulting spectrum is shown in Fig. 3 as a function of
the parameter 2 2/Qκ that measures the strength of the
magnetic field. The magnon continuum with the scattering
states are confined to energies larger than the magnon gap
2
gapε ∝ κ , which increases linearly with the field (black
solid line). In the field range shown, there are three subgap
states that correspond to bound magnon–skyrmion modes.
While the breathing mode with angular momentum = 0m
exists over the full field range, a quadrupolar mode with
= 2m − emerges for lower fields just before the field-
polarized state becomes globally unstable at 2 2
cr 0.8Qκ ≈
(dashed-dotted line). The eigenenergy of the latter finally
vanishes at 2 2
bimeron 0.56Qκ ≈ , indicating a local instability
of the theory with respect to quadrupolar deformations of the
skyrmion, i.e., the formation of a bimeron [48]. Further-
more, a sextupolar mode with = 3m − only exists within the
metastable regime. The corresponding eigenfunctions of
these modes do not possess any nodes. We have not yet
found bound modes with a single or more nodes, which
might however emerge for = 1m − at larger fields.
Apart from the modes shown in Fig. 3, the spectrum of
also contains a zero mode with angular momentum
= 1m − given by
zm
1
sin
1= .
sin8−
θ ′− θ ρ
θ ′+ θ ρ
η (26)
This zero mode is related to the translational invariance of
the theory (6) that is explicitly broken by the skyrmion
solution. The real and imaginary part of the amplitude of
the eigenfunction (26) correspond to translations of the
skyrmion within the two-dimensional plane.
3. High-energy scattering of magnons
The properties of the magnon scattering states for arbi-
trary energies, gap ,ε ≥ ε have been discussed in Ref. 29. In
the present work, we elaborate on the scattering of
magnons in the high-energy limit, gapε >> ε , which corre-
sponds to magnon wavevectors much larger than the in-
verse skyrmion radius, 1.skr In this limit, the treatment
of the scattering simplifies considerably allowing for a
transparent discussion of characteristic features.
In the high-energy limit the magnon-skyrmion interac-
tion is governed by the scattering vector potential
ˆ( ) = ( )aχ ρ χa r of Eq. (19) so that the scattering has a pure-
ly magnetic character. In particular, in this limit one can
neglect the anomalous potential x , and the BdG equation
(14) reduces to a Schrödinger equation for the magnon
wavefunction
2 2
gap
mag
( )= .
2t
ii
M
− −
∂ ψ + ε ψ
a
∇ (27)
Setting ( , ) = exp ( / ) exp ( ) ( )k mt i t imψ − ε χ η ρr
with the
dispersion
2 2
gap
mag
=
2k
k
M
ε ε +
and wavevector > 0k , one
obtains the radial wave equation for ( )mη ρ
2
2 2
2
( ( )) = 0.m
m a k
χ
ρ
ρ
∂ −ρ ρ
− ∂ + + − η
ρ ρ
(28)
For large distances ( ) 1aχρ ρ → , which identifies the angu-
lar momentum of the incoming wave to be = ( 1)zL m − .
3.1. Eikonal approximation
As we are interested in the high-energy limit, we can
treat this wave equation in the eikonal approximation.
However, in order to make contact with Ref. 29, we first
give the resulting phase shift within the WKB approxima-
tion that is obtained by following Langer [49,50]
Fig. 3. (Color online) Magnon spectrum in the presence of
a single skyrmion excitation as a function of 2 2/Qκ measuring
the strength of the magnetic field [29]. The magnon gap
2 2
gap = /DM Qε ε κ increases linearly with the field (black solid
line). The field-polarized state becomes unstable at 2 2
cr 0.8Qκ ≈
(dashed-dotted line) while the theory (14) becomes locally unsta-
ble at 2 2
bimeron 0.56Qκ ≈ . Apart from the zero mode (not
shown), there exist three subgap modes with angular momentum
m = 0, –2, –3.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 10 1047
Sarah Schroeter and Markus Garst
2
2
2
0
( ( ))=WKB
m
m ak k d
∞ χ
ρ
−ρ ρ δ − − ρ+
ρ
∫
0| 1 |
2
m kπ
+ − − ρ (29)
where 0ρ is the classical turning point. The eikonal ap-
proximation for the phase shift is then obtained by taking
the limit k →∞ while keeping the impact parameter
= /( )zb L k fixed, ( )WKB
m b∞δ → δ , yielding
lab lab
2 2 2
| | 1
( ) ( | |)
( ) = =
1b
a a s b
b b d b ds
b s
∞ χ ∞ χ
∞
ρ
δ ρ
ρ − −
∫ ∫ (30)
where we used lab ( ) = ( ) 1a aχ χρ ρ ρ ρ − , see Eq. (22), and in
the last equation we substituted = /| |s bρ . This phase shift
is odd with respect to b , i.e., ( ) = ( )b b∞ ∞δ −δ − . Note that
the scattering is non-perturbative even in the high-energy
limit in the sense that the phase shift ( )b∞δ covers the
entire interval ( , )−π π as a function of b , see Fig. 4. In
particular, in the limit of small impact parameter 0b → :
2
1
2/( | |)( ) = sgn ( ).
1
s bb b ds b
s
∞
∞
−
δ → −π
−
∫ (31)
For impact parameters larger than the skyrmion radius,
,sb r>> , the phase shift vanishes exponentially.
The deflection angle in the eikonal approximation is
given by the derivative of ( )b∞δ ,
r( ) 2 4( ) = 2 = ( ) = ( ) ( ).eg
z
bb b b b
L k k
∞
∞ ∞ ∞
∂δ π′Θ δ Θ − δ
∂
(32)
The step of ( )b∞δ for head-on collisions, see Eq. (31),
leads to the delta function ( )bδ . The classical deflection
function is given by the regular part, which reads
regreg
2
1
| | ( | |)2( ) =
1
s b s b
b ds
k s
∞
∞Θ =
−
∫
(33)
2 2
reg
1= ( ) ,b x dx
k
∞
−∞
+∫
(34)
where in the last equation we substituted 2=| | 1x b s −
and used that the integrand is an even function of x . It is
determined by the regular part of the flux density, reg ,
given in Eq. (24), integrated along a straight trajectory
shifted from the x-axis by the impact parameter b . Its be-
havior as a function of b is shown in Fig. 5 for various
values of 2 2/Qκ . The deflection angle is always positive
implying that, classically, the Lorentz force attributed to
reg always skew scatters the magnons to the right-hand
side from the perspective of the incoming wave even for
negative impact parameters, see Fig. 1(b). Note that the
deflection angle possesses a local minimum at = 0b for
2 21.6Qκ , that however gets filled and transitions into a
maximum for larger values of κ . This change of curvature
at = 0b is related to the change of curvature of the flux
density reg ( )′′ ρ at the origin = 0ρ , see Fig. 2, that happens
for a similar value of κ . As the total flux of reg is quan-
tized, the deflection angle integrated over the impact pa-
rameter is just given by the universal value
reg ( ) = 4 /db b k
∞
∞
−∞
Θ π∫ .
Fig. 4. (Color online) Scattering phase shift for high-energy
magnons (30) as a function of impact parameter b for different
values of 2 2/Qκ . The scattering is nonperturbative as the phase
shift assumes values within the entire interval ( , )−π π .
Fig. 5. (Color online) Classical deflection angle for scattering of
high-energy magnons (33) as a function of impact parameter b
for different values of 2 2/Qκ . In the high-energy limit, the scat-
tering is in the forward direction with a deflection angle decreas-
ing with increasing wavevector k as reg ( ) 1/b k∞Θ . The inset
focuses on the change of curvature at = 0b for 2 21.6Qκ ≈ with
the same units on the vertical axis.
1048 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 10
Scattering of high-energy magnons off a magnetic skyrmion
3.2. Differential cross section
In the following, we consider a magnon scattering setup
where an on-shell magnon plane wave with wavevector
ˆ= kxk along the x-direction and amplitude A defined
within the laboratory orthogonal frame, see Eq. (12), is im-
pinging on the skyrmion, see also Fig. 1(b). At large dis-
tances this wavefunction assumes the asymptotic behavior
/
lab
e( , ) = e e ( ) ,
ik
i ikrkt A f
ρ
− ε
ψ + χ ρ
r (35)
where the scattering amplitude is given by
/4
2( 1)
=
e( ) = e (e 1).
2
i
ii m m
m
f
k
∞− π
δ− χ
−∞
χ −
π
∑ (36)
Note that the additional phase factor e i− χ arises from the
gauge transformation (12). The differential cross section is
then obtained by 2= | ( ) |f∂σ
χ
∂χ
.
High-energy limit of the scattering amplitude. In the
high-energy limit, we can replace the sum over angular
momentum numbers by an integral over the impact param-
eter, = ( 1) /b m k− , so that the scattering amplitude reads
approximately
/4
2 ( )e( ) = e (e 1),
2
i
i bibkf k db
k
∞− π
δχ ∞
∞
−∞
χ −
π ∫ (37)
with ( )b∞δ defined in Eq. (30). The differential cross sec-
tion in this limit,
2
2=| ( ) | = ( / ),kf S k Q
Q
∞
∞
∂σ
χ χ
∂χ
(38)
is then determined by the dimensionless function S , which
is shown in Fig. 6.
The support of the differential cross section is approxi-
mately limited by the extremal values of the classical de-
flection angle of Eq. (33) and Fig. 5. Note that the angle χ
is defined in a mathematically positive sense so that a posi-
tive Θ translates to a negative value of χ. It is strongly
asymmetric with respect to forward scattering reflecting
the skew scattering arising from the Lorentz force of the
emerging magnetic field reg .
Rainbow scattering and Airy approximation. Moreover,
the differential cross section exhibits oscillations. These
can be attributed to an effect known as rainbow scattering.
As the function reg ( )b∞Θ is even in b , there exist for a given
classically allowed deflection angle Θ always at least one
pair clb± of impact parameters that solve reg
cl( ) =b∞Θ ± Θ.
For a given angle Θ the magnons might, therefore, either
pass the skyrmion on its right- or left-hand side; these clas-
sical trajectories interfere leading to the oscillations in
/d dσ χ.
First, consider values 2 21.6Qκ for which reg ( )b∞Θ
possesses only a single maximum at = 0b . The maximum
value reg (0)∞Θ is known as rainbow angle and for values of
χ close to reg (0)∞−Θ , the interference effect of classical
trajectories can be illustrated with the help of the Airy ap-
proximation for the scattering amplitude. For such values
of χ, the 1− in the integrand of Eq. (37) can be neglected
as it only contributes to forward scattering. Expanding the
exponent of the remaining integrand up to third order in b
one then obtains
A( ) | =iryf∞ χ
/4
reg reg 3e= exp[ ( (0)) (0) ]
62
i kk db ibk i b
k
∞− π
∞ ∞
−∞
′′χ +Θ + Θ =
π ∫
reg/4
reg 1/3 reg 1/3
( (0))2 e= Ai ,
[ | (0) | /2] [ | (0) | /2]
i kk
k k
− π
∞
∞ ∞
χ +Θπ
− ′′ ′′Θ Θ
(39)
where in the last equation we identified the integral repre-
sentation of the Airy function Ai using that reg (0) < 0∞′′Θ .
In the inset of Fig. 6, we compare the differential cross
section at 2 2= 2Qκ with the Airy approximation resulting
from Eq. (39). The latter reproduces the exponential de-
crease for large angles reg< (0)∞χ −Θ corresponding to the
dark side and also the oscillations on the bright side,
reg> (0)∞χ −Θ , of the rainbow angle. It of course fails close
to forward scattering and for positive angles > 0χ where
the classical deflection angle has lost its support.
Close to 2 21.6Qκ ≈ even the derivative reg (0)∞′′Θ van-
ishes, see inset of Fig. 5, giving rise to a cubic rainbow
effect [51]. Finally, for smaller values of 2κ there also ex-
ist two pairs of classical trajectories that interfere in the
differential cross section.
Fig. 6. (Color online) Differential cross section of high-energy
magnons (38) for various values of 2 2/Qκ . It is asymmetric with
respect to = 0χ due to skew scattering, and the oscillations are
attributed to rainbow scattering. The inset compares the curve for
2 2/ = 2Qκ with the Airy approximation (39) (green solid line)
with the same units on the vertical axis; the arrow indicates the
position of the corresponding rainbow angle reg (0)/k Q∞− Θ .
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 10 1049
Sarah Schroeter and Markus Garst
3.3. Total and transport scattering cross section
We continue with a discussion of the total,
tot = /d d d
π
−π
σ χ σ χ∫ , and the transport scattering cross sec-
tion defined in Eq. (4). In order to determine their high-
energy limit, one first expresses 2/ =| ( ) |d d fσ χ χ in terms
of the exact representation (36) for the scattering amplitude
( )f χ and evaluates the integral over χ. Afterwards one
takes the high-energy limit k →∞ with keeping the impact
parameter = ( 1) /b m k− fixed.
The total scattering cross section of the skyrmion then
reduces to
2
tot = 4 (sin ( )) .db b
∞
∞
∞
−∞
σ δ∫ (40)
It saturates to a finite value in the high-energy limit, and its
dependence on κ is shown in Fig. 7. It decreases with in-
creasing κ and thus decreasing skyrmion radius rs as ex-
pected. One might expect that tot ~ sr
∞σ which however
only holds approximately.
Using that ( )b∞δ is an odd function of b , we obtain for
the transport scattering cross section ( )⊥σ ε in the high-
energy limit
2
0
8( ) = ( )(sin ( )) =db b b
k
∞
∞
⊥ ∞ ∞′σ ε δ δ∫ (41)
0sin(2 )8 4= = .
2 4k k
∞ ∞
−π
δ δ π −
(42)
In the last line, we further used the boundary values of the
function ( )b∞δ . It vanishes ( ) 1/k∞
⊥σ ε , but with a univer-
sal prefactor that is independent of κ .
Finally, for the longitudinal transport scattering cross
section we obtain for 1skr >>
2 2
|| 2
0
4( ) = (2( ) (sin ) sin cos ).db
k
∞
∞
∞ ∞ ∞ ∞ ∞′ ′′σ ε δ δ − δ δ δ∫
(43)
After integrating by parts this simplifies to
2 reg 2
|| 2
0
4 1( ) = ( ( )) = ( ( )) .
2
db b db b
k
∞ ∞
∞
∞ ∞
−∞
′σ ε δ Θ∫ ∫ (44)
It is given by the square of the classical deflection angle
(33) integrated over the impact parameter b . It vanishes as
2
|| 1/k∞σ ∼ in the high-energy limit with a prefactor whose
κ dependence is shown in Fig. 8. On dimensional grounds
one might expect 2 ~ 1/ sk r∞σ
which again only holds ap-
proximately.
3.4. Magnon pressure in the high-energy limit
We have shown in Ref. 29 by considering the energy-
momentum tensor of the field theory that the monochro-
matic plane wave of (35) with wavevector ˆ= kxk leads to
a momentum-transfer force in the Thiele equation of mo-
tion of the form given in Eq. (3) with the magnon current
2 0
eff
mag
| |ˆ= | | = .
4B
m kx A
g Mε µ π
GJ v
(45)
In the second equation, we have introduced the effective
velocity 2
eff
mag
ˆ= | | kx A
M
v and 0| |= 4 /( )Bm gπ µG
with the purpose of comparing with Eq. (2).
This momentum transfer is illustrated in Fig. 9. In the
high-energy limit, the transversal and longitudinal forces
are given by
effˆ ˆ= ( )( ) = 4 ( ) = ,k z z∞
⊥ ⊥ ε εσ ε × π × − ×F J J G v (46)
Fig. 7. Total scattering cross section of the skyrmion in the high-
energy limit, Eq. (40), as a function of 2 2/Qκ . It decreases for
increasing external magnetic field strength, 2κ .
Fig. 8. The longitudinal transport scattering cross section, Eq.
(44), vanishes as 2
|| 1/k∞σ ∼ in the high-energy limit. The panel
shows the κ-dependence of the prefactor.
1050 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 10
Scattering of high-energy magnons off a magnetic skyrmion
reg 2
|| || eff
| |= ( ) = ( ( )) ,
8
k k db b
∞
∞
ε ∞
−∞
σ ε Θ
π ∫
GF J v (47)
where we used Eqs. (41) and (44) as well as ˆ= | | z−G G .
They are indeed of the form given in Eq. (2). The transver-
sal momentum-transfer force, ⊥F , is universal, and ||F is
determined by the β parameter of Eq. (5) after identifying
( )bΘ with the classical deflection angle reg ( )b∞Θ .
Is there an intuitive classical interpretation of these
momentum-transfer forces? From the classical limit of the
Schrödinger equation (27) follows the equation of motion
for the coordinate ( )tr of a classical magnon particle [25]
mag regˆ= ( (| |)),M z×r r r (48)
with the regular part of the effective magnetic flux distribu-
tion reg of Eq. (24). Note that we have chosen in Eq. (27)
the charge to be +1. Consider the change of momentum, δp,
of this magnon particle after scattering off the static
skyrmion by integrating the left-hand side of Eq. (48),
mag m( ) = ( ) = ( ( ) ( ))agb dtM t M
∞
−∞
δ ∞ − −∞ =∫p r r r
cos ( ) 1
= .
sin ( )
b
p
b
Θ −
− Θ
(49)
In the last equation, we have exploited that at large dis-
tances the magnitude of momentum mag | ( ) |=M p±∞r
remains unchanged due to energy conservation, while the
orientation of velocity is determined by the scattering an-
gle ( )bΘ , see Fig. 1(b), that depends on the impact parame-
ter b of the trajectory.
This momentum ( )bδp is transferred to the skyrmion.
The momentum-transfer force on the skyrmion due to a
current of classical magnon particles along x̂ with density
0 /( )Bm gµ and velocity eff eff= | |v v is then given by
|| 0
eff= = ( ),
B
F m
v db b
gF
∞
⊥ −∞
− δ µ
∫F p (50)
with ||/ ||/= | |F ⊥ ⊥F . In the high-energy limit, the scattering
is in forward direction so that we can expand Eq. (49) in
the deflection angle ( )bΘ and the force becomes with
=p k
2
0
eff
1 ( ( ))
= .2
( )B
bm
v k db
g b
∞
−∞
Θ
µ Θ
∫F (51)
Finally using that the integral ( ) = 4 /db b k
∞
−∞
Θ π∫ is quan-
tized in the high-energy limit, that we already know from
the discussion in the context of Eq. (33), we recover
Eqs. (46) and (47).
For the understanding of the universality of F⊥ , it is al-
so instructive to consider alternatively the right-hand side
of the classical equations of motion (48). By integrating
the right-hand side, one obtains for the transversal momen-
tum change
2 2
reg reg= ( ) (| |) ( ).yp dt x dx b x
∞ ∞
−∞ −∞
δ − ≈ − +∫ ∫r (52)
In the last equation we employed the high-energy approx-
imation by straightening the magnon trajectory. It follows
then for the transversal force
2 20
eff reg= ( )
B
m
F v db dx b x
g
∞ ∞
⊥
−∞ −∞
+ =
µ ∫ ∫ (53)
0
eff= 4 ,
B
m
v
g
π
µ
(54)
where its universality is now directly related to the quan-
tized total flux of reg .
4. Summary
The scattering of high-energy magnons with
wavevectors 1skr >> off a magnetic skyrmion of linear size
rs is governed by a vector scattering potential. The associ-
ated effective magnetic field is related to the topological
charge density of the skyrmion and is exponentially con-
fined to the skyrmion area. The total flux is determined by
the topological skyrmion number and is quantized.
Fig. 9. (Color online) An incoming monochromatic magnon cur-
rent εJ leads to a momentum-transfer force F that is deter-
mined by the transport scattering cross sections, see Eq. (3). The
image shows the magnon wavefunction in the WKB approxima-
tion with the skyrmion being represented by the circle with radius
rs [29]. For high-energy magnons with wavevector 1skr >> , the
transversal force dominates, || / 1/F F k⊥ ∼ , resulting in a skyrmion
motion t∂ R approximately antiparallel to εJ with a small
skyrmion Hall angle 1/kΦ .
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 10 1051
Sarah Schroeter and Markus Garst
When a magnon traverses the skyrmion, classically
speaking, it experiences the resulting Lorentz force and is
deflected to a preferred direction determined by the sign of
the emergent magnetic flux. This results in skew scattering
with a differential cross section that is asymmetric with
respect to forward scattering, see Fig. 6. As the flux distri-
bution is rotationally symmetric, the classical deflection
angle ( )bΘ as a function of the impact parameter b is even
in the high-energy limit, ( ) = ( )b bΘ Θ − . As a consequence,
for a given deflection angle Θ there exist corresponding
classical trajectories with positive as well as negative b ,
i.e., that pass the skyrmion on the left-hand as well as on
the right-hand side. These trajectories interfere which leads
to oscillations in the differential cross section, an effect
known as rainbow scattering.
Magnons hitting the skyrmion also transfer momentum
giving rise to a force in the Thiele equation of motion, see
Eq. (3). In the high-energy limit, this force can be inter-
preted classically and assumes the form of Eq. (2). While
the transversal momentum-transfer force, F⊥ is universal
and determined by the total emergent magnetic flux, the
longitudinal momentum-transfer force, ||F is obtained by
integrating 2( ( ))bΘ over the impact parameter b leading to
the parameter εβ of Eq. (5). Since for large energies the
classical deflection angle is small, ( ) 1/b kΘ , the momen-
tum transfer is mainly transversal, || / 1/F F k⊥ . This leads
to a skyrmion motion t∂ R approximately antiparallel to the
magnon current εJ with a small skyrmion Hall angle
= / | |εΦ β G defined in Fig. 9,
2
2
( ( ))
1 1= = ( ( )) ,
2 8
( )
b db
k b db
k
b db
∞
∞
−∞
∞
−∞
−∞
Θ
Φ Θ ∝
π
Θ
∫
∫
∫
(55)
where the integral ( ) = 4 /b db k
∞
−∞
Θ π∫ is universal in the
high-energy limit. Interestingly, the Hall angle Φ at high
energies increases with decreasing skyrmion radius rs,
which is shown in Fig. 8 identifying ||= ( )/4k ∞Φ σ ε π.
While the skyrmion Hall angle Φ is small at high ener-
gies 1skr >> , we note that it increases with decreasing en-
ergy and assumes the maximum value[29] = /2Φ π in the
low-energy limit 1skr where s-wave scattering prevails
and Eq. (2) ceases to be valid.
We acknowledge helpful discussions with M. Mostovoy
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Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 10 1053
1. Introduction
2. Skyrmionic soliton and its spin-wave excitations
2.1. Skyrmionic saddle-point solution
2.2. Magnon-skyrmion scattering problem
2.3. Magnon spectrum
3. High-energy scattering of magnons
3.1. Eikonal approximation
3.2. Differential cross section
3.3. Total and transport scattering cross section
3.4. Magnon pressure in the high-energy limit
4. Summary
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