![FIG. 4. Evolution of the spin noncollinearity in the Fe6 cluster. (a) Plot of θ = θ (t) − θ (0) at site 1 for the BXC [or S(t)] direction (black curve) and the Bkin direction (red dashed curve). (b) Same quantities as in (a) but calculated at atomic site 6. (c) Plot of √ Ā21+Ā22, where Ā =√∑3 i=1 A2i and Ai is introduced in Eq. (23), compared to Ā3 at atomic site 1. (d) Same quantities as in (c) but calculated at atomic site 6. The fields are measured within a sphere of radius R = 0.8 Å centered on the atom center.](/figures/fig-4-evolution-of-the-spin-noncollinearity-in-the-fe6-eibftb6g.webp)
PHYSICAL REVIEW B 95, 024412 (2017)
Ultrafast demagnetizing fields from first principles
Jacopo Simoni,
*
Maria Stamenova, and Stefano Sanvito
School of Physics, AMBER and CRANN Institute, Trinity College, Dublin 2, Ireland
(Received 25 April 2016; revised manuscript received 17 November 2016; published 12 January 2017)
We examine the ultrafast demagnetization process of iron-based materials, namely, Fe
6
clusters and bulk bcc
Fe, with time-dependent spin-density functional theory (TDSDFT). The magnetization continuity equation is
reformulated and the torque due to the spin-current divergence is written in terms of an effective time-dependent
kinetic magnetic field, an object already introduced in the literature. Its time evolution, as extracted from the
TDSDFT simulations, is identified as one of the main sources of the local out-of-equilibrium spin dynamics
and it plays a major role in the demagnetization process in combination with the spin orbit interaction. Such
demagnetization is particularly strong in hot spots where the kinetic torque is maximized. Finally, we find the
rate of demagnetization in Fe
6
to be strongly dependent on the direction of polarization of t he exciting electric
field and this can be linked to the out-of-equilibrium distribution of the kinetic field in two comparative cases.
DOI: 10.1103/PhysRevB.95.024412
I. INTRODUCTION
The search for practical solutions for increasing the speed
of manipulation of magnetic bits is essential for the progress
of modern information and communication technology. It has
been shown that there is an upper limit to the speed of the
magnetization switching process when this is driven by a
magnetic field [1,2]. An increase in power absorption beyond
this limit and for higher magnetic-field amplitudes push a spin
system out of equilibrium into a chaotic behavior and the
switching speed decreases. For this reason the discovery made
by Beaurepaire et al. [3] that a ferromagnetic Ni film could
be demagnetized by a 60-fs optical laser pulse has attracted
a great deal of interest and was the seed of a new field, now
called femtomagnetism.
In a standard pump-probe experiment the system is initially
excited by an optical pulse (pump) and then the magneti-
zation dynamics is monitored by analyzing a second signal
(probe) [4,5]. Depending on the minimal delay between the
pump and the probe, one can analyze the demagnetization
process at different time scales and thus observe the dissipation
mechanisms active at that particular time. The interpretation
of the results is, however, a complicated matter. In general,
for demagnetization processes observed on a time scale
ranging from nanoseconds to 100 ps, one considers an
empirical three-temperature model [6], where electrons, spins,
and phonons define three energy baths, all interacting with
each other. In contrast, ultrafast spin dynamics, taking place
within a few hundred femtoseconds, is yet not described in
terms of a single unified scheme and various models for the
demagnetization process have been advanced. These include
fully relativistic direct transfer of angular momentum from
the light to the spins [ 7,8], dynamical exchange splitting [9],
electron-magnon spin-flip scattering [10], electron-electron
spin-flip scattering [11], and laser-generated superdiffusive
spin currents [12].
Given the complexity of the problem, ab initio methods,
resolved in the time domain, provide a valuable tool to probe
the microscopic aspects of the ultrafast spin dynamics of real
*
simonij@tcd.ie
magnetic materials by means of time-dependent simulations.
In this work we apply time-dependent spin-density functional
theory (TDSDFT) [13,14] in its semirelativistic, noncollinear,
spin-polarized version to analyze the ultrafast laser-induced
demagnetization of two ferromagnetic transition metal sys-
tems: a Fe
6
cluster (see Fig. 1) and bulk bcc Fe. Recently,
within a similar theoretical description, it has been demon-
strated that the spin-orbit (SO) interaction plays a central
role in the demagnetization process [15–17]. Furthermore, it
wasshowedbyus[18] that the laser-induced spin dynamics
can be understood as the result of the interplay between the
SO-coupling potential and an effective magnetic field. The
so-called kinetic magnetic field [19,20] B
kin
(r,t) originates
from the presence of nonuniform spin currents in the system.
In this work we focus on the anatomy of B
kin
(r,t) and we
analyze in detail its role in the highly nonequilibrium process
of ultrafast demagnetization.
The first formulation of the spin dynamics problem in
transition metal systems was given in Refs. [19,20] by Katsnel-
son and Antropov, who laid down the foundation of density
functional theory (DFT) -based spin dynamics, by deriving
a set of equations of motion for the local magnetization
vector. In those seminal works the magnetization dynamics
was analyzed at the level of the adiabatic local spin-density
approximation (ALSDA), but actual applications to real out-
of-equilibrium systems were not described. Our purpose is to
clarify and quantify, through TDSDFT simulations at the level
of the noncollinear ALSDA, the role played by B
kin
(r,t)in
the laser-induced ultrafast spin dynamics of transition metal
ferromagnets.
The paper is divided into four main sections. In Sec. II we
define the various fields that couple t o the spins by isolating in
the continuity equation only the terms that play a major role
in the dynamical process. In Sec. III we present the results
of the calculations for Fe
6
clusters and show that hot spots
for demagnetization are associated with larger misalignment
of the kinetic magnetic field and the local spin density. This
becomes more clear through evaluation of material derivatives.
A demonstration of the effect of the polarization of the electric
field on the rate of demagnetization of Fe
6
is discussed in
Sec. IV. In Sec. V we show that previous observations for Fe
6
are valid for bulk bcc Fe as well. We summarize in Sec. VI.
2469-9950/2017/95(2)/024412(11) 024412-1 ©2017 American Physical Society
JACOPO SIMONI, MARIA STAMENOVA, AND STEFANO SANVITO PHYSICAL REVIEW B 95, 024412 (2017)
FIG. 1. (a) Typical electric-field pulse used to excite the Fe
6
cluster with the black arrow indicating the direction of the field.
The fluence of this pulse is 580 mJ/cm
2
. (b) Time evolution of the z
component of B
kin
and B
X
(exchange component of the field), with
respect to their values at t = 0 integrated over the system volume
μ
B
B
tot
(t) = μ
B
I
B
I
(t). (c) Time evolution of the variation of the
total magnetization S
tot
z
(t) =
I
S
I
z
(t) with respect to its initial
value. (d) Time evolution on atomic site 6 of the magnetization
variation along z and of the electron density variation with respect to
its value at t = 0 integrated inside a sphere of radius R = 0.9
˚
A.
In the Appendix we present a detailed derivation of the spin
continuity equation.
II. THEORY
We consider the TDSDFT problem within the ALSDA for a
spin-polarized system excited by an electric-field pulse. If one
neglects second-order contributions arising from the solution
of the coupled Maxwell-Schr
¨
odinger system of equations, the
dynamics will be governed by the usual set of time-dependent
Kohn-Sham (KS) equations
i
d
dt
ψ
KS
j
(r,t) = H
KS
(r,t)ψ
KS
j
(r,t). (1)
In Eq. (1) ψ
KS
j
(r,t) are the KS orbitals and the KS Hamiltonian
H
KS
(r,t) can be expressed in the velocity gauge formulation
and the minimal coupling substitution as
H
KS
(r,t) =
1
2m
−i∇ −
q
c
A
ext
(t)
2
− μ
B
ˆ
σ · B
s
[n,m](r,t) + v
s
[n](r,t), (2)
where
v
s
[n](r,t)=
d
3
r
n(r
)
|r − r
|
+ v
ALSDA
XC
[n](r,t)
+
I
V
I
PP
(|r − R
I
|)(3)
and
B
s
[n,m](r,t) = B
ALSDA
XC
[n,m](r,t) + B
ext
(r,t). (4)
Here v
s
(r,t) r epresents the usual noninteracting KS potential
and the full noninteracting magnetic field B
s
(r,t) consists of
the external one B
ext
(r,t) and the exchange-correlation (XC)
magnetic field B
ALSDA
XC
(r,t). In the equations above m is the
electron mass, q the electron charge, c the speed of light, A
ext
(t)
the vector potential associated with the external magnetic field,
ˆ
σ the spin operator, μ
B
the Bohr magneton, n the electron
density, and m the magnetization density. Then v
s
(r,t)is
decomposed into a Hartree contribution, an XC correlation one
v
ALSDA
XC
[n](r,t), and an ionic pseudopotential V
I
PP
(|r − R
I
|).
For a fully relativistic, norm-conserving pseudopotential the
SO coupling enters the KS equations in the form [21]
V
I
PP
(|r − R
I
|) =
l
¯
V
I
l
(r) +
1
4
V
I,SO
l
(r)
+
l
m=−l
V
I,SO
l
(r)
ˆ
L
I
·
ˆ
S |I,l,mI,l,m|
.
(5)
In Eq. (5) the orbital momentum operator associated with the
I th atomic center is
ˆ
L
I
, while the vectors {|I,l,m} are the
associated set of spherical harmonics centered on that given
atomic position. In Eq. (5) V
I,SO
l
(r) defines a generalized
space-dependent SO coupling parameter providing a measure
of the SO interaction strength close to the atomic site, while
¯
V
I
l
(r) includes all the ionic r elativistic corrections such as
the Darwin and the mass correction terms. Within the ALSDA
v
XC
(r,t) and B
XC
(r,t) are local functions in time of the electron
density and magnetization, which in turn are written in terms
of the time-dependent KS orbitals
n(r,t) =
j∈O
σ
ψ
KS
jσ
(r,t)
∗
ψ
KS
jσ
(r,t), (6)
m(r,t) =
j∈O
α,β
ψ
KS
jα
(r,t)
∗
σ
α,β
ψ
KS
jβ
(r,t), (7)
where O denotes occupied states. Starting from the set of
time-dependent KS equations in (1), it is possible to derive an
equation of motion for the magnetization, or a spin-continuity
equation, in terms of the noninteracting KS observables. This
reads
d
dt
m(r,t) =−∇ ·J
KS
(r,t) + μ
B
m(r,t) × B
s
(r,t)
+ T
SO
(r,t), (8)
where J
KS
(r,t) represents the noninteracting KS spin-current
rank-2 tensor
J
KS
(r,t) =
2mi
j∈O
ψ
KS†
j
ˆ
σ ∇ψ
KS
j
− H.c.
(9)
and the SO torque contribution reads
T
SO
(r,t) =
I
l,m
1
,m
2
O
j,α,β
V
SO
l
(|r − R
I
|)
×
ψ
KS
jα
l,m
1
,I
l,m
1
,I |L
I
|l,m
2
,I ×σ
αβ
×
l,m
2
,I
ψ
KS
jβ
. (10)
024412-2
ULTRAFAST DEMAGNETIZING FIELDS FROM FIRST . . . PHYSICAL REVIEW B 95, 024412 (2017)
The KS magnetic field B
s
(r,t) is taken as in Eq. (4), which in
the absence of an external magnetic field reduces to B
XC
(r,t).
In DFT there is a set of zero-force theorems stating that
the interaction between the particles cannot generate a net
force [22]. In the case of the exchange-correlation magnetic
field we have the exact condition
d
3
r m(r,t ) × B
XC
(r,t) =
0, which is satisfied by the ALSDA. Combining this equality
with the assumption that the currents at the system boundary
are negligible allows us to conclude that the only source of
global spin loss is the SO coupling torque T
SO
and that the
spin lost during the temporal evolution is transferred to the
orbital momentum of the system, which in turn is partially
damped into the lattice (we consider frozen ions). Hence we
have the relation
d
dt
d
3
r m(r,t ) =
d
3
r T
SO
(r,t), (11)
where the integration extends over the entire volume .
Within the ALSDA, the exchange-correlation functional
satisfies also a local variant of the zero-torque theorem [23],
which is not a property of the exact DFT functional [24–26].
According to this condition, m(r,t) × B
XC
(r,t) = 0 and there-
fore the exchange-correlation magnetic field cannot contribute,
even locally, to the magnetization dynamics. This leads us to
conclude that the local magnetization dynamics is solely the
result of the interplay between the spin-polarized currents and
the SO torque (in reality B
XC
can still contribute indirectly
to the spin dynamics through a dynamical modification of the
gap between up- and down-spin-polarized bands, which in
turn may give rise to an enhancement of the spin dissipation
via the spin-orbit-coupling channel). In order to elucidate this
view further we make use of the hydrodynamical formalism
applied to spin systems, which has already been introduced in
Refs. [27,28]. This approach needs to be slightly modified
in view of the fact that we are considering an effective
Kohn-Sham system and not a s et of independent spin particles.
In fact, as it was already pointed out in Refs. [19,20], Eq. (8)
can be written in a different form (the details of the derivation
are shown in the Appendix)
D
Dt
m(r,t) +
j∈O
∇ ·v
j
(r,t)m
j
(r,t) =−∇ ·D(r,t)
+ μ
B
m(r,t) × B
eff
(r,t) + T
SO
(r,t), (12)
where a couple of new terms appears. In the equation
D
Dt
=
d
dt
+ v · ∇ is a material derivative, v
j
(r,t) represents a
single Kohn-Sham state velocity field (see the Appendix), and
m
j
(r,t) = ψ
KS†
j
ˆ
σ ψ
KS
j
. On the right-hand side of Eq. (12), in
addition to the spin-orbit coupling torque T
SO
(r,t)wehavea
new term −∇ · D(r,t) that describes the spin dissipation in the
system due to the internal motion of the spin currents. It can
be interpreted as an effective spin-current divergence object
involving only transitions among different Kohn-Sham states
(interband transitions) [see Eq. (A16)]. Finally, the effective
field B
eff
isgivenbythesumoftwotermsB
eff
= B
XC
+ B
kin
,
with B
XC
the exchange-correlation field and B
kin
defined as
[see Eq. (A25)]
B
kin
(r,t) =
1
¯
Fe
∇n · ∇s
n
+∇
2
s
, (13)
with spin vector field s(r,t) =
m(r,t)
n(r,t)
.
Such a B
kin
(r,t) field has only an instrumental role in
the equations of motion for the spin density; a very similar
expression was already introduced in some previous work. In
Ref. [19] it is expressed in the form ∂
k
1
n
(m × ∂
k
m), while in
Ref. [20] it appears as
∇n∇m
n
. The interpretation of B
kin
may
look quite obscure at first, however, in Refs. [29,30]itwas
identified as a possible source of spin-wave excitations in the
form of a spin-spin interaction potential.
In order to clarify this point, let us consider the Heisenberg
interaction between two spins centered on atoms placed at a
distance d =|d|. We can assume naively, but reasonably, that
the spin-spin interaction between the two spin distributions,
computed at an arbitrary point r in space, may be expressed in
the form
H
eff
(r) s(r − d/2) · s(r + d/2), (14)
where it is more convenient for us to employ a spin field s(r),
which describes the spin distribution in space, instead of an
atom localized spin vector. Hence, H
eff
defines an effective
single-particle Hamiltonian. By averaging over the number of
electrons in the entire space we obtain
S
1
· S
2
d
3
rn(r)s(r − d/2) · s(r + d/2). (15)
Then, by expanding the spin density in a Taylor series up to
second order over the distance d and then neglecting the zeroth-
order contribution (we focus our attention on the nonlocal
term appearing in the expansion), after some straightforward
rearrangement we arrive at
S
1
· S
2
−
d
2
4
d
3
rn(r)∇s(r) · ∇s(r), (16)
which in turn becomes
S
1
· S
2
d
2
4
d
3
r
−∇ ·[n(r)s(r) · ∇s(r)]
+ m( r) ·
∇n(r) · ∇s(r)
n(r)
+∇
2
s(r)
. (17)
Finally, by considering a sufficiently large integration volume,
the use of the divergence theorem allows us to neglect all the
boundary terms with the consequent final expression
S
1
· S
2
d
2
4
d
3
r m(r) ·
∇n(r) · ∇s(r)
n(r)
+∇
2
s(r)
, (18)
which remarkably resembles the result in Eq. (13)forthe
kinetic magnetic field. We can therefore tentatively interpret
B
kin
(r,t) as an effective mean-field internal magnetic field,
which plays a role in coupling the spins at different loca-
tions in the system analogously to the Heisenberg spin-spin
interaction.
024412-3
JACOPO SIMONI, MARIA STAMENOVA, AND STEFANO SANVITO PHYSICAL REVIEW B 95, 024412 (2017)
III. ANALYZING SPIN DYNAMICS FROM TDSDFT
SIMULATIONS IN THE Fe
6
CLUSTER
Here we present the results of TDSDFT calculations,
performed with the
OCTOPUS code [31], where we simulate the
ultrafast demagnetization process in iron-based ferromagnetic
systems. In all those, at time t = 0 the system is in its
ground state. Then we apply an intense electric-field pulse
with a duration of less than 10 fs, which initiates the
dynamics. The pseudopotentials for Fe used in the calculations
are fully relativistic, norm conserving, and generated using
a multireference-pseudopotential scheme [32]atthelevel
implemented in
APE [33,34], which takes directly into account
the semicore states. For the XC functional we employ the
ALSDA with parametrization from Perdew and Wang [35].
Our simulations then consist in evolving in time the KS wave
functions, i.e., in solving numerically the set of equations (1).
The results are then interpreted through the magnetization
continuity equation (12).
In Fig. 1 the extracted magnetization dynamics of a Fe
6
magnetic cluster is presented. We use the LSDA ground-state
geometry of Fe
6
as extracted from Refs. [36,37]forwhich
we reproduce the reported therein spin state S = 20/2. The
nuclei are kept stationary during the dynamics. In Fig. 1(c)
we observe that the total loss of the z component of the
total magnetization S
tot
z
(t) is exactly equal to the variation
in value of its module |S
tot
| since the global noncollinear
contribution is negligible. This indicates that the spin is not
exchanged globally between the different components of the
magnetization vector, but, according to Eq. (11), it is at least
partially transferred into the orbital momentum of the system.
We note that, due to the electrostatic interactions with the
nuclei and due to the interaction with the laser field, the
rotational invariance of the electronic system is broken and
the total angular momentum is not conserved.
In Fig. 1(b) we observe that the average kinetic magnetic
field (over the entire simulation box, for
¯
F = 1) is comparable
in magnitude to the exchange component. At the same time,
B
tot
kin,z
shows a much more oscillatory behavior compared
to B
tot
X,z
. In particular, while B
tot
X,z
evolves smoothly in time
following the action of the optical excitation, B
tot
kin,z
presents an
abrupt variation at the onset of the electrical pulse. This is due
to the fact that the laser pulse directly excites currents, through
the term −∇ · D(r,t )inEq.(12), which in turn produces
a modification of the gradients of the charge/spin density,
even on a global scale since they are not conserved. Thus
we observe large variations of B
tot
kin,z
. In addition, B
X,z
can also
oscillate very strongly locally, following the temporal variation
of the densities, but in the evaluation of B
tot
X,z
these oscillations
are averaged out given that the densities are approximately
conserved over the entire simulation box. During the action of
the pulse we see a tendency of t he two fields to compensate
each other, an effect strongly resembling the Lenz law. After
the pulse, B
kin
continues to oscillate dramatically with its aver-
age value, only slowly increasing. In contrast, B
tot
x,z
decreases
(in absolute value) due the net dissipation of spin angular
momentum.
Moving from an analysis of global quantities to probing
locally the spin dynamics, in Fig. 1(d) we compare the
magnetization and the electron density around atomic site 6 at
the tip of the cluster [see inset of Fig. 1(a) for the numbering
labels of all the cluster atoms]. We define local magneti-
zation and charge associated with the particular atomic site
Ias
S
I
(t) =
1
|S
I
R
|
S
I
R
d
3
r m(r,t ),Q
I
(t) =
1
|S
I
R
|
S
I
R
d
3
rn(r,t) ,
(19)
where the integration volume S
I
R
is a sphere of radius R
centered at site I. Our results show that the loss of S
6
z
is not
taking place just during the action of the external pulse, but it is
rather distributed over the entire time evolution. This suggests
that the spin-sink mechanism is not directly related to the
coupling of the system to the laser field, but is rather intrinsic
to the electron dynamics following the pulse. Furthermore,
close to the atomic site, the temporal variation of the charge
Q
6
is much smaller in magnitude and smoother than that of
S
6
z
. In addition, for long times Q
6
settles close to an average
value, while S
6
z
continues to decrease. Hence the long-term
spin dynamics is not the result of a net charge displacement
from the region close to the ions to the interstitial space. These
observations are valid for all the atomic sites in the cluster.
If we now consider the continuity equation for the electron
density (see the Appendix for further explanation)
D
Dt
n(r,t) =−n(r,t)∇ ·v(r,t), (20)
where
D
Dt
n(r,t) is the material derivative of the electron density
D
Dt
n(r,t) =
d
dt
+ v · ∇
n(r,t). (21)
From Fig. 1(d) we observe that during the action of the pulse
the density variation in the vicinity of the atoms appears to be
very small compared to the magnetization variation. We can
therefore safely assume that in this spatial region
˙
n(r,t) 0,
with at the same time n(r,t) = 0. From these considerations
we deduce that v(r,t) 0 is a reasonably good approximation
for the velocity field in the vicinity of the atoms (this does not
imply that the velocity field is exactly zero, but only that its
effect on the spin dynamics in this particular case is negligible).
The same argument is valid also for the state resolved density
n
j
(r,t), given that
˙
n(r,t) =
j∈O
˙
n
j
(r,t), the contribution of
the local time derivative of the Kohn-Sham state density can be
neglected. By applying the latter in Eq. (12) we finally obtain
a relation that can be considered approximately valid in this
spatial region of the simulation box,
d
dt
m(r,t) −∇ ·D +μ
B
m × B
kin
+ T
SO
, (22)
where the contribution to the spin dynamics due to the
velocity-field term has been neglected. Note that here we
have also used the condition m(r,t) × B
XC
(r,t) = 0, which
is consequential to the LSDA. In addition, the decay of B
XC
during the evolution, is not so significant as to produce a
dynamical modification of the gap between up- and down-spin
states.
In Fig. 2 we compare the behavior of the kinetic field and of
the local magnetization at two atomic sites, respectively: 1 (one
of the atoms in the base plane of the bipyramid) and 6 (an atom
at one of the apexes). It can be seen from Fig. 2(a) that these two
024412-4
ULTRAFAST DEMAGNETIZING FIELDS FROM FIRST . . . PHYSICAL REVIEW B 95, 024412 (2017)
0
10
20 30
t [fs]
4
5
S
z
I
[ h
_
/2]
I = 1
I = 6
0
10
20 30
t [fs]
-11
-10
-9
-8
-7
μ
Β
B
kin,z
I
[eV]
(a)
(b)
FIG. 2. Local spin dynamics of the Fe
6
cluster: (a) time evolution
of the magnetization S
I
z
(t) around the atomic centers and (b) time
evolution of the z component of B
I
kin
(t). All the quantities are
integrated inside a sphere of radius R = 0.9
˚
A centered on the two
atomic sites, where we have used B
I
z
(t) =
1
|S
I
R
|
S
I
R
d
3
rB
z
(r,t).
sites present different rates of demagnetization. In particular,
at site 6 the spin decay is considerably more prominent with
respect to that observed at site 1. In contrast, the fluctuations
in S
I
z
are significantly more pronounced for site 1 than for site
6. This can be understood from the fact that we have chosen
here an electric pulse with polarization vector in the basal
plane of the bipyramid. As such, the charge fluctuations for
the atoms in the basal plane are expected to be much larger
than those of the apical atoms. Finally, we note that B
I
kin,z
(t)
follows similar qualitative trends as S
I
z
(t) [see Fig. 2(b)]. In
fact, the average change following the excitation pulse is larger
for site 6 (the one experiencing the larger demagnetization),
but the fluctuations are more pronounced for site 1 [the one
experiencing the larger fluctuations in S
I
z
(t)].
The correlation between the kinetic field and the mag-
netization loss is also rather evident in Fig. 3. There the
time-averaged variations in the x component of the two fields
m × B
kin
and
˙
m(r,t) are clearly comparable in magnitude
and localized over the same regions of the simulation box.
This demonstrates that the kinetic field can be considered
as the main force driving the noncollinearity during the spin
evolution. The fact that the contrast is stronger at the apex
atoms (hot spots for demagnetization) agrees with Fig. 2(a),
while the dipole-type patterns indicate how the longitudinal
spin decays preserving global collinearity. The correlation
between the z components of m × B
kin
and
˙
m(r,t) is not as
evident as that for the transverse component x. This is due to
the fact that the x and y components of the field are much
smaller compared to the z one. Furthermore, the contribution
to the spin dynamics along z of the SO coupling, together with
the internal dissipative term due to the spin currents, cannot
be neglected.
In order to quantify the local noncollinearity we examine the
evolution of the misalignment angle θ between the z axis and
the direction of the magnetic fields (averaged over spheres). It
can be seen in Fig. 4 that at site 1 the averaged kinetic field and
FIG. 3. Contour plots of the time- and space-averaged (in the
direction perpendicular to the plane spanned by atoms 1, 3, 5 and 6,
as indicated on the plot) observables evaluated only within spheres of
radius R = 1.0
˚
A around each atom: (a) and (b) the temporal variation
of the spin density m
(x,z)
(r,t)/t for t = 0.1 fs and (c) and (d)
the x and z components of the second term on the right-hand side of
Eq. (22).
the local spin deflect very little from the quantization axis and
remain rather parallel to each other. The angle that B
1
kin
(t)
forms with the magnetization direction (the m
1
direction)
is practically negligible. Instead, at site 6, B
6
kin
(t)showsa
significant deflection from the z axis after the first 5 fs of the
-0.005
0
0.005
Δ
θ
[deg]
B
kin
1
S
1
I = 1
0
3
6
B
kin
6
S
6
I = 6
0102030
t [fs]
0
0.3
0.6
|A| [Å
-1
]
|A
xy
1
|
A
z
1
0102030
t [fs]
0
2
4
|A
xy
6
|
A
z
6
(a)
(b)
(d)
(c)
FIG. 4. Evolution of the spin noncollinearity in the Fe
6
cluster.
(a) Plot of θ = θ(t) − θ(0) at site 1 for the B
XC
[or S(t )] direction
(black curve) and the B
kin
direction (red dashed curve). (b) Same
quantities as in (a) but calculated at atomic site 6. (c) Plot of
√
¯
A
2
1
+
¯
A
2
2
,
where
¯
A =
3
i=1
A
2
i
and A
i
is introduced in Eq. (23), compared to
¯
A
3
at atomic site 1. (d) Same quantities as in (c) but calculated at atomic
site 6. The fields are measured within a sphere of radius R = 0.8
˚
A
centered on the atom center.
024412-5