Z. Phys. Chem. 227 (2013) 1543–1557 / DOI 10.1524/zpch.2013.0403
© by Oldenbourg Wissenschaftsverlag, München
Potential Energy Surfaces and Rates of Spin
Transitions
By Pavel F. Bessarab1 , 2 , ∗, Valery M. Uzdin2 , 3 , and Hannes Jónsson1 , 4
1
2
3
4
Science Institute and Faculty of Physical Sciences, VR-III, University of Iceland, Reykjavı́k, Iceland
Department of Physics, St. Petersburg State University, St. Petersburg, 198504, Russia
National Research University of Information Technologies, Mechanics and Optics, St. Petersburg,
197101, Russia
Dept. of Applied Physics, Aalto University, Espoo, FI-00076, Finland
(Received February 21, 2013; accepted in revised form May 4, 2013)
(Published online July 8, 2013)
Rate Theory / Spin Transitions / Magnetic Memory / Magnetic Nano-Islands /
Potential Energy Surface
The stability of magnetic states and the mechanism for magnetic transitions can be analyzed in
terms of the shape of the energy surface, which gives the energy as a function of the angles
determining the orientation of the magnetic moments. Minima on the energy surface correspond
to stable or metastable magnetic states and can represent parallel, antiparallel or, more generally,
non-collinear arrangements. A rate theory has been developed for systems with arbitrary number,
N, of magnetic moments, to estimate the thermal stability of magnetic states and the mechanism
for magnetic transitions based on a transition state theory approach. The minimum energy path on
the 2N-dimensional energy surface is determined to identify the transition mechanism and estimate
the activation energy barrier. A pre-exponential factor in the rate expression is obtained from the
Landau–Lifshitz–Gilbert equation for spin dynamics. The velocity is zero at saddle points so it
is particularly important in this context to realize that the transition state is a dividing surface
with 2N − 1 degrees of freedom, not just a saddle point. An application of this rate theory to
nanoscale Fe islands on W(110) has revealed how the transition mechanism and rate depend on
island shape and size. Qualitative agreement is obtained with experimental measurements both for
the activation energy and the pre-exponential factor. In particular, a distinct maximum is observed
in the pre-exponential factor for islands where two possible transition mechanisms are competing:
Uniform rotation and the formation of a temporary domain wall. The entropy of the transition state
is enhanced for those islands making the pre-exponential factor more than an order of magnitude
larger than for islands were only the uniform rotation is viable.
1. Introduction
The rate theory developed by Eyring and M. Polanyi [1] and, in a different form,
later by Wigner [2] for chemical reactions, can be applied in many contexts. It provides a method for estimating the rate of rare events by performing a statistical average
* Corresponding author. E-mail: [email protected]
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Table 1. Correspondence between atomic systems and spin systems.
Chemical reaction/Diffusion
←→
Magnetic transitions
particle rearrangements
particle coordinates
Newton’s eqn. of motion
←→
←→
←→
rotation of magnetic moments
orientation of magnetic moments
Landau–Lifshitz eqn.
over fast, oscillatory motion and focuses on obtaining the probability of significant
transitions. Thermal transitions between magnetic states in spin systems is another application of the theory. We present here a formulation of a rate theory for magnetic
transitions and describe an application to magnetization reversal in nanoscale islands
on a metal surface. The stability of magnetic states with respect to thermal fluctuations
and external perturbations is an important problem in fundamental studies of magnetism and of critical importance in the design of nanoscale recording devices. The
development of scanning tunneling microscopy (STM) and other microscopic scanning
probes has made it possible to create and control magnetic structures down to the level
of individual spins [3]. The magnetic properties of small clusters can be studied with
STM by measuring the Kondo resonance at a particular atom [4] or via measurement
of spin polarized current which depends on the relative orientation of magnetic moments in a magnetic tip and the surface atoms [5]. The lifetime of magnetic states at
a given temperature can be extracted from such measurements and the thermal rate of
transitions thereby determined. Krause et al. [6] studied the magnetization reversal process in small Fe clusters on a W(110) surface by spin polarized STM. An Arrhenius
dependence on temperature was observed and the results used to determine the activation energy as well as the pre-exponential factor over a wide range in cluster size and
shape. A theoretical estimate of the thermal stability of magnetization of nano-clusters
as a function of size and shape could help design nanoscale devices for data storage with
unprecedented capacity.
Thermally activated magnetic transitions can be rare events on the scale of the vibrational motion of the magnetic moments, analogous to rearrangements of atoms that
take place in chemical reactions or during epitaxial growth and are typically occurring on a much longer time scale than atomic vibrations. The analogy between atomic
scale systems and spin systems is summarized in Table 1. In a magnetic system, the
time of longitudinal relaxation responsible for the magnitude of magnetic moments is
much shorter than the time of transverse relaxation which determine the evolution of
the orientation of the magnetic moments. On the time scale corresponding to the reorientation transitions, the potential energy surface is, therefore, a function of just the
angles defining the orientation of the momentum vectors. This is analogous to the BornOppenheimer approximation in atomic systems where the potential energy surface is
a function of the coordinates of the atoms but the electronic degrees of freedom are
assumed to be fast enough to be integrated out.
In what follows, we will focus on temperature induced remagnetization processes
in nano-islands on a metal surface. The energy of such systems can be described quite
well with a non-collinear extension [7,8] of the Alexander and Anderson model [9].
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Potential Energy Surfaces and Rates of Spin Transitions
A brief description of this methodology is presented in Sect. 2. An example energy surface for a simple system, an Fe trimer, is discussed in Sect. 3. While the calculation of
an activation energy for a magnetic transition can, within a harmonic approximation,
be obtained from the minimum energy path on the energy surface in the same way as
for an atomic system, the evaluation of the pre-exponential factor is quite different. Assuming quasiclassical description, the equation of motion for the magnetic moments
is the Landau–Lifshitz equation for the magnetic moments, but Newton equation for
the atoms. Although the Landau–Lifshitz equation was first formulated to describe the
precession of a classical magnetic moment, it has proven also to be a good approximation for quantum magnetic systems with itinerant electrons. There, the length of
the magnetic momentum vectors is not fixed but depends on the direction and has to
be determined self-consistently with an electronic structure calculation [10,11]. Minimum energy paths are found using the nudged elastic band method as discussed in
Sect. 4. The derivation of the pre-exponential factor for magnetic systems is sketched
in Sect. 5. Finally, we apply the methodology to study thermally activated magnetization reversal processes in Fe nanoislands on W(110) surface, as described in Sect. 6.
We conclude with a discussion of refinements to this rate theory, for example by evaluating dynamical recrossings of the transition state and extension to, for example, low
temperature where tunneling becomes the dominant transition mechanism, and to larger
length scales where each magnetic moment represents a domain consisting of multiple
spins, so-called micromagnetic models [12].
2. Microscopic model for magnetic system with itinerant electrons
We make use of a non-collinear extension [7,8] of the Alexander–Anderson [9]
(NCAA) model which has been shown to describe adequately the magnetic states of
clusters of 3d transition metals on non-magnetic substrates [14]. The predictions of the
model for such clusters have been supported by subsequent density functional theory
calculations, such as the non-collinear ordering of magnetic moments and the presence of multiple metastable states [13,15]. The energy associated with the magnetic
degrees of freedom in the system is obtained by considering two electronic bands: the
3d-electrons localized on the transition metal atoms of the cluster and itinerant s- and
p-electrons of the metallic substrate.
We perform self-consistent calculations for the d-electron subsystem, while the influence of s- and p-electrons is indirectly taken into account via the renormalization of
model parameters. In particular, the s/d-hybridization leads to a non-zero width, Γ , of
the d-level. Each transition metal atom of the cluster is characterized by two dimensionless parameters. The first parameter is the position of the energy level of non-interacting
d-electrons relative to the Fermi energy of the substrate in units of the width of the dlevel, (E i0 − F )/Γ . The second parameter is the Coulomb repulsion of d-electrons on
site, Ui /Γ . These parameters can be determined from a comparison with ab initio calculations of simple systems, such as ad-atoms on the substrate or from experiments if the
number of d-electrons and magnetic moment of individual atoms can be determined.
Values of these parameters for adatom clusters have been found to be of the same order
of magnitude as those of bulk material.
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Fig. 1. Left: Energy surface for a Fe trimer. Two local minima are found, one corresponding to parallel and
the other corresponding to antiparallel ordering of the spins. The minimum energy path is indicated by the
filled circles, showing the location of images in a nudged elastic band calculation. The saddle point between these two minima is shown with a X. The uniform rotation path corresponding to the simultaneous
rotation of the second and third magnetic moments is shown with a dotted line. Right: Energy along the
minimum energy path. Insets show the direction of the spins at the minima, the saddle point and two other
intermediate spin configurations. The definition of the angles θ2 and θ3 is shown in one of the insets. The
energy corresponding to the uniform rotation path is shown with a dotted line.
The interaction between the d-electrons is described by dimensionless hopping parameters Vij /Γ , which depend on the relative position of the transition metal atoms in
the cluster. The hopping parameters include contributions from direct d-d transitions
between the atoms i and j as well as contributions from d-s(p)-d transitions via conductivity band of the substrate. Since both contributions decrease with the distance between
atoms, hopping parameters are a measure of interatomic distances and therefore define
the geometry of the cluster.
The magnetic moment of each atom in the cluster is taken to be a vector. The number of d-electrons, magnitude of the magnetic moment as well as total energy of d-states
are calculated self-consistently for given values of the angles, θi and φi , defining the
orientation of the magnetic moments. Detailed description of an efficient implementation of these self-consistent calculations can be found elsewhere [8,14,16].
3. Potential energy surface of a Fe trimer
To illustrate a potential energy surface for a non-collinear spin system, we have chosen a cluster containing three iron atoms. Several magnetic states of Fe, Cr and Mn
ad-trimers which are close in energy have previously been identified [14]. The question is how large an energy barrier separates these states and what the lifetime is of
each state at a given temperature. While a triatomic island is too small to support long
lived metastable states, we use this as a first application of the methodology because
the energy surface can be visualized
in a contour graph. The parameters in the NCAA
model are taken to be: E 0 − F /Γ = −12, U/Γ = 13, V12 /Γ = 0.95, V13 /Γ = 1.2 and
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Potential Energy Surfaces and Rates of Spin Transitions
V23 /Γ = 1.3. Note that the trimer is chosen to have asymmetric structure so the parameters V12 , V23 and V13 have different values. These values are typical for models of 3d
transition metals (see Ref. [17] and references therein).
Spin-orbit interaction is not taken into account in the NCAA model. The spin and
configuration spaces are, therefore, not connected and the direction of the magnetic moments in configurational space cannot be determined within the model. However, it is
possible to show that the magnetic moments of all the atoms tend to lie in a plane.
The energy of the system increases significantly when a magnetic moment of an atom
points out of the plane. Therefore, it is sufficient to set φi = 0 for all three atoms to
visualize the relevant part of the energy surface. It is convenient to choose the quantization axis for the system to be along the magnetic moment of one of the atoms, and
the full configuration is then specified by only two angles, θ2 and θ3 between magnetic
moments of the first and second atom and first and third atom, respectively. Figure 1
shows the coordinate system chosen as well as a contour graph of the energy surface,
E(θ2 , θ3 ). The global minimum represents a parallel state, but a metastable antiparallel state also exists. This is an example of a magnetic system with two possible states
corresponding to two different arrangements of the magnetic vectors. The question that
is addressed here is how often such a system will switch between the possible states at
a finite temperature.
4. Minimum energy paths
The minimum energy path (MEP) between two minima on a potential energy surface
represents the path of highest statistical weight for transitions between the two states.
While no classical trajectory follows the MEP, this is a convenient definition of a reaction coordinate. A maximum in the energy along an MEP is a first order saddle point,
and the highest maximum gives the activation energy within a harmonic transition state
theory approximation, as discussed below. The nudged elastic band (NEB) method can
be used to find MEPs [18]. For atomic systems, the NEB method involves taking some
initial guess of a path between the two minima, and using an iterative technique to bring
that to the nearest MEP. A path is represented by a discrete set of coordinates of all
the atoms placed along the path. Each set of coordinates is referred to as an ‘image’ of
the system. For magnetic systems, the angles θi , φi defining the direction of the magnetic vectors in the system are used to define each of the images. In order to ensure
continuity of the path, springs are introduced between adjacent images. The springs
also control the distribution of the images along the path, and thereby define the way
in which the path is discretized. These springs only serve the purpose of distributing
the images evenly along the path, preventing them from sliding down to endpoint or
possible intermediate minima. In order to ensure that the springs do not interfere with
the convergence to the MEP and to ensure that the springs only affect the distribution
of images along the path, a force projection scheme is applied, the ‘nudging’. At each
image an estimate of the local tangent to the path is made and the spring force projected
onto the tangent, while the component of the true force along the path is removed, so
as not to affect the distribution of images along the path. The MEP for the Fe trimer is
shown in Fig. 1. The location of each of the images in the converged NEB calculation is
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shown. Initially, a linear interpolation in the two angles between the two minima is generated. This corresponds to a uniform rotation. Then, a minimization algorithm is used
to iteratively move the images down to the MEP.
A more detailed description of the method is as follows. A convenient choice of the
set of variables for a system of N magnetic vectors is x ≡ {θ, φ} ≡ {θ1 , θ2 , . . ., θ N , φ1 ,
φ2 , . . ., φ N }, where θi , φi are polar and azimuthal angles, respectively, defining the direction of the i-th magnetic moment. Vectors are denoted here by bold face symbols.
The magnitude of the magnetic moments is assumed here to be a fast variable that
responds quickly to a change in the direction. The path is represented by a string of
images, i.e. a set of P values of the angles for all the magnetic vectors in the system,
X ≡ {x1 , x2 , . . . , x P }, where the end points x1 and x P are fixed, usually at or near the
minima on the energy surface. An essential part of the method is to estimate the local
tangent at each image, τ i . While a straight forward way of doing this is to use a finite
difference between the two adjacent images,
τi =
xi+1 − xi−1
|xi+1 − xi−1 |
this has been found to lead to instabilities in the path resulting in kink formation [18].
A finite difference estimate involving the current image and the neighboring image of
higher energy works better, see Ref. [19]. The force on each image, i, is calculated as
= FNCAA
− (FNCAA
· τ i ) τ i + (FSpr
Fnudged
i
i · τ i) τ i
i
i
where FNCAA
is the force evaluated from the energy surface −∇ E(xi ) and FSpr
is the
i
i
spring force which can be evaluated as ki+1 (xi+1 − xi ) − ki (xi − xi−1 ). The option here is
to use different values for the spring constant to distribute the images unevenly along
the path, but we have used the same value k = 1 eV/deg and distributed the images
evenly in the present calculations. In order to converge one of the images rigorously
on the highest energy point on the MEP, so as to obtain a good estimate of the energy
barrier, the climbing image algorithm is used [20].
Various minimization algorithms can be used to iteratively move the intermediate
images so as to zero the force Fnudged . A successful strategy has been to use a modified
velocity Verlet algorithm where only the component of the velocity parallel to the force
is retained when the inner product of velocity and force is positive, but the velocity is
zeroed when the inner product is negative (for more detail see Ref. [18]). This is a robust algorithm which can handle initial paths that are far from the MEP. But, when the
magnitude of the force has dropped to a small value, it is more efficient to switch to
some quadratically convergent method, such as conjugate gradients (see, for example,
Ref. [21]). Once convergence has been reached, the images lie on the MEP where
the force FNCAA only can have a component in the direction of the path, i.e. FNCAA−
(FNCAA · τ ) = 0.
5. Rate theory
The long life time of the magnetic states of interest means that transitions between
states are rare events on the time scale of spin oscillations and direct simulations of the
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Potential Energy Surfaces and Rates of Spin Transitions
dynamics would require prohibitively long simulations and large computational effort,
analogous to the problem of simulating atomic scale transitions. The separation of time
scales, however, makes it possible to apply statistical approaches. Monte-Carlo (MC)
simulations with Metropolis sampling do not provide such dynamical information, only
thermal averages, but transition state theory (TST) [1,2] which is based on a statistical
estimate of the probability of reaching a transition state, can give an estimate of the rate
and also reveal the transition mechanism [22,23]. It is assumed that transitions are rare
enough that a Boltzmann distribution is established and maintained in the whole reactant region up to and including the transition state. A TST approximation to the rate
is obtained by making a simple approximation for the dynamics, namely that reactive
trajectories only cross the transition state once. However, this approximation can then
be checked and corrected by calculating short time trajectories started at the transition
state to obtain the so called dynamical corrections [26]. The full TST approach is, therefore, a two step procedure where in the first step the time scale problem is overcome
and an approximate estimate of the rate is obtained, while in the second step short time
scale trajectories are calculated and the exact rate obtained. In its most elaborate form,
the transition state is determined by maximizing the free energy of the system [22,23].
But, most commonly, a harmonic approximation to the energy surface is used both at
the initial state minimum and at first order saddle points on the energy rim surrounding
the minimum. The transition state is then approximated as a mosaic of hyperplanar segments, each one containing a first order saddle point and having normal vector pointing
in the direction of the unstable mode at the saddle point [23].
The theory can be described in more detail in the following way. A general expression for the rate of escape from an initial state is given by [25]
k = δ [ f(x)] v⊥ (x)χ [η (t)]
(1)
where x represents all dynamical variables in the system, angular brackets denote the
thermal averaging with the Boltzmann distribution, f(x) = 0 defines the dividing surface separating the initial state from the rest of configuration space, and v⊥ (x) = ∇ f(x) ·
ẋ is the projection of the velocity onto the local normal of the dividing surface. χ [η(t)]
is a functional of full, dynamical trajectories described by η (t) where coupling to a heat
bath has been included. This expression says that the transition rate is obtained by integrating over the dividing surface and determining which trajectories going through
the dividing surface are reactive and what flux through the dividing surface they correspond to. χ takes the value of unity if a trajectory originating in the initial state goes
directly from the point x on the dividing surface to the final state and spends long time
there compared with the time it takes to cross the barrier. Otherwise, χ is zero. This ensures that a trajectory that recrosses the transition state and ends up again in the initial
state will not contribute to the thermal average, and a reactive trajectory that crosses the
dividing surface more than once will only contribute once to the integral.
In the TST approximation, all trajectories crossing the transition state in such a way
that the velocity is pointing away from the initial state are assumed to be reactive,
i.e. the functional χ [η (t)] is approximated by the Heaviside step function, h [v⊥ (x)].
Thereby, the effect of recrossing the dividing surface is neglected in TST. Since the
detailed trajectories are not considered at this level of approximation, the coupling to
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the heat bath does not enter the expression for the rate. The only requirement is that
the coupling is strong enough to establish the Boltzmann distribution. The TST approximation can, however, be corrected in the second step where short time dynamical
trajectories are started at the dividing surface and recrossings of the dividing surface
examined. From such trajectories, which only require short simulations because the system typically leaves the transition state rapidly, the dynamical correction factor [26], κ,
can be determined. This two step procedure, first a TST estimate and then evaluation of
dynamical corrections, in principle gives the exact rate as, k = κk TST .
A harmonic approximation to the TST rate estimate is obtained by making
a quadratic expansion of the energy at the minimum on the energy surface corresponding to the initial state and at each first order saddle point on the energy ridge
surrounding the minimum. The harmonic approximation to transition state theory
(HTST) corresponds to choosing hyperplanar segments of D − 1 dimensions for a dividing surface, where D is the number of degrees of freedom in the system. Here,
D = 2N where N is the number of magnetic moments. Each hyperplanar segment goes
through a first order saddle point and has a normal vector pointing along the unstable
mode at the saddle point, i.e. the normal mode with negative eigenvalue. Considering
now just one first order saddle point at a time, and labeling the unstable mode as q1 ,
the normal projection of the velocity is v⊥ (θ, φ) = q̇1 . The velocity can be estimated at
each point on the dividing surface using the Landau–Lifshitz–Gilbert (LLG) equations
of motion for magnetic moments (see [27])
γ ∂E
α ∂E
2
1 + α sin θi φ̇i =
,
−
Mi ∂θi sin θi ∂φi
∂E
∂E
γ
2
1 + α sin θi θ̇i = −
,
+ α sin θi
Mi ∂φi
∂θi
(2)
where γ is a gyromagnetic ratio, α is the damping constant and E is the energy of the
system. When a quadratic approximation to the energy surface is made, these equations
of motion become linear. The normal component of the velocity can then be expressed
in terms of normal mode coordinates qi , i.e. displacements along eigenvectors of the
Hessian
matrix of the energy
at the saddle point. The expansion can be written as
D
v⊥ = γ i=2 ai qi 1 + α2 , to show explicitly the dependence on α. The dependence on
Mi is implicitly included in the coefficients ai . The velocity is, according to the LLG
equation, a function of the variables θ and φ, as illustrated in Fig. 2, unlike the atomic
case where the velocity is independent of coordinates and given by the Maxwell distribution at equilibrium. While the velocity through the dividing surface is zero at the
saddle point because the gradient of the energy vanishes there (as illustrated in Fig. 2),
an integration over the hyperplanar dividing surface gives a non-zero reactive flux and
the resulting estimate of the rate constant is [24]
k
HTST
ν
=
2π
det Hm −( E s −E m )/kB T
e
det
Hs
(3)
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Fig. 2. A region around a first order saddle point on an energy surface for a magnetic system. The hyperplanar transition state used in HTST is shown as a thick line. It includes the saddle point and has a normal
vector pointing along the unstable mode. The component of the velocity normal to the transition state,
which gives the reactive flux, vanishes at the saddle point according to the LLG equation of motion. This
velocity component then increases as one goes away from the saddle point, pointing towards the product
state in one half of the hyperplane. Since the HTST rate is obtained by integrating the reactive flux over
the hyperplanar transition state, a non-zero reaction rate is obtained.
where det Hm and det Hs denote the determinants of the Hessian matrices at the minimum and at the saddle point, respectively, and ν is defined as
D 2
D
2
ai
Ms,i sin θs,i γ
ν=
2
sin θm,i i=2 i
(1 + α2 ) i=1 Mm,i
i being the eigenvalues of the Hessian at the saddle point and θ s , θ m and M s , Mm being polar angles and magnitude of magnetic moments at the saddle point and at the
minimum, respectively. The determinants in Eq. (3) are computed as a product of the
eigenvalues and the prime means that the negative one, 1 , is omitted.
Equation (3) predicts an Arrhenius type dependence of the rate on the temperature,
with an activation energy E a = E s − E m and a temperature independent pre-exponential
factor. The HTST rate given by Eq. (3) decreases as the damping constant α is increased
since this reduces the velocity. The LLG equations of motion, Eq. (2), include damping
but the effect of recrossing the transition state dividing surface is not taken into account.
This can be done in the second step where trajectories are started at the hyperplanar dividing surface and recrossings of short time trajectories counted. In the present case, we
include only the first step in this two step procedure.
For each possible final state, one or more MEP can be found. When the system is
made to follow an MEP, each degree of freedom of the system is advanced in such a way
that the energy is minimal with respect to all degrees of freedom perpendicular to the
path. A maximum along a MEP corresponds to a first order saddle point, i.e. a point
where the Hessian has only one negative eigenvalue. The NEB method [18] is used
to find MEPs between a given pair of states. In the NEB calculations presented here,
the orientation of the magnetic moment Mi of each atom is included explicitly. This
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gives a detailed description of the magnetic transitions, including complex mechanisms
involving non-uniform rotation of the magnetic moments.
The rate theory presented here for magnetic systems has been tested by comparison with direct dynamical calculations of systems where the activation energy is low
enough and the transition rate high enough to make such calculations possible. The
LLG equations are then extended to include a fluctuating force representing coupling
to a heat bath. The results of such a test has been presented in Ref. [24]. The HTST
estimated rate was found to be within a factor of 2 of the exact rate obtained from the
dynamical simulations over a wide range in the coupling strength to the heat bath. This
shows that HTST can give a good approximation to the rate of magnetic transitions.
6. Fe islands on W(110)
The rate theory presented above has been applied to study the rate of magnetization
reversal in Fe monolayer nano-islands on a W(110) surface. Recently, extensive experimental data on both activation energy and pre-exponential factor as a function of
island size and shape have been reported for this system [6]. A brief account of calculations using a Heisenberg-type Hamiltonian has been given recently [28], but here
we carry out calculations using the NCAA model which is more accurate for 3d-system
with itinerant electrons. The model is supplemented with an additional term representing anisotropy. This term represents the interaction of the magnetic system, here the Fe
island, with the substrate and is needed to make the magnetic moment vectors preferably lie along one direction within the surface plane. The origin of this term is spin-orbit
coupling. Anisotropy terms are routinely included in Heisenberg-type Hamiltonians
and can, in an analogous way, be added to the NCAA Hamiltonian. The energy of the
system is then
Kn
(M i · en )2
H = H NCAA +
n
i
where H NCAA is the NCAA Hamiltonian. The index n in the sum takes two values,
for easy-axis and ⊥ for easy-plane anisotropy representing the interaction with the
substrate. Unlike the Heisenberg model, the length of the magnetic vectors is not
fixed in the NCAA model but are calculated self-consistently for each orientation of
the magnetic moments in the system. The parameters used here to represent Fe islands on W(110) were chosen to largely reproduce the previous
with
calculations
the Heisenberg-type Hamiltonian and have the following values: E 0 − F /Γ = −12,
U/Γ = 13; hopping parameters are taken into account only between the nearest neighbors, V/Γ = 0.9. Γ is taken to be 0.7 eV. An easy-axis anisotropy K || is included along
the [11̄0] direction. The corresponding parameter value is K || μ2B = −0.2 meV. We also
include an easy-plane anisotropy K ⊥ which makes it preferable for magnetic moments
to lie in the (110) plane, with K ⊥ μ2B = 0.5 meV.
The calculations were carried out for monolayer islands of rectangular shape consisting of 5 atomic rows in the [11̄0] direction. The length of the islands along the
[001] direction lies within the range of N[001] = 5 to N[001] = 29 atom rows. The islands have two degenerate states, with spins aligned parallel or antiparallel to the
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Potential Energy Surfaces and Rates of Spin Transitions
Fig. 3. Energy barrier for magnetization reversal in a rectangular Fe island on W(110) with 5 atom rows
along the [11̄0] direction and 27 rows along [001] direction. The energy change along the MEP for a regularly shaped rectangular island is shown with a dashed line. The filled circles show the location of images
in the nudged elastic band calculation. MEPs for slightly deformed islands (see inset) are also shown with
solid lines, illustrating how the transition state energy depends on island shape.
anisotropy axis. The calculations of the MEPs for the magnetization reversal revealed
two possible transition mechanisms for these islands. Small islands, with N[001] ≤ 13,
reverse their magnetization via coherent rotation of all magnetic moments. However,
transitions in longer islands follow a more complicated path involving nucleation
and propagation of an excitation that can be described as a thin, temporary domain
wall.
Figure 3 shows MEPs of magnetic transitions in islands with N[001] = 27. The magnetization reversal starts at one of the narrower ends of the island and a domain wall
forms perpendicular to the long axis of the island. The domain wall then moves along
the [001] direction eventually leading to reversal of the magnetization of the whole
island.
The effect of changing slightly the shape of the islands is also shown in Fig. 3. By
adding 4 atoms to make the island a bit broader in the center, the activation energy increases by ca. 10%. Similarly, by removing 2 atoms from the island near the center, the
activation energy is reduced by a similar fraction. This shows how sensitive the activation energy and thereby the rate is to small changes in the shape of the island, something
that could be used to taylor the properties of such systems.
The dependence of the energy barrier on the size and shape of the islands is illustrated in Fig. 4. For small islands, the magnetization reversal occurs via coherent
rotation of all the magnetic moments. The activation energy increases steadily as the
number of rows along the [001] direction is increased. For clusters with 15 atomic rows
or more, the transition mechanism is different. Instead of a uniform rotation, a temporary domain wall forms and propagates during the transition. The domain wall is
oriented perpendicular to the long axis of the island. Since the activation energy is then
mainly determined by the length of the domain wall, the activation energy becomes
constant, independent of N[001] for long islands.
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Fig. 4. Activation energy for magnetization reversal in Fe islands on W(110) as a function of the number
of atom rows along the [100] direction. When the number of atom rows along the [100] direction is larger
than 14, a cross over occurs in the transition mechanism from a unifrom rotation mechanism to a temporary
domain wall formation.
Fig. 5. Pre-exponential factor for magnetization reversal in Fe islands on W(110) as a function of the number of atom rows along the [001] direction. The increase by an order of magnitude as the number of
atoms is increased results from increased entropy of the transition state, due to soft vibrational modes. For
N[001] = 15 two transition mechanisms are equally likely, a uniform rotation of all the magnetic moments
and the formation of a temporary domain wall along the [11̄0] direction. The presence of two active mechanisms leads to particularly large transition state entropy and a maximum in the pre-exponential factor.
This kind of variation in the activation energy is indeed observed in the experimental data [6]. However, the calculated barrier is systematically too small for the
smaller islands. This might be due to a rim effect where the anisotropy for rim atoms is
larger [6]. We note that the magnetic moment predicted by the NCAA model is about
10–15% larger for rim atoms and this means the barrier does not drop as rapidly for
the smaller islands as in a Heisenberg-type Hamiltonian, but the rim effect in the experimental data seems to be even stronger. Another possible contribution to a rim effect
is the adsorption of impurities such as hydrogen or oxygen on the island edges and/or
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that interatomic distance between atoms is different, thereby modifying the value of the
magnetic moments.
The pre-exponential factor is also strongly dependent on the size and shape of the islands. Figure 5 shows the HTST estimate evaluated using Eq. (3) as a function of N[001] .
The figure shows that the pre-exponential factor is largest for islands where the cross
over in transition mechanism occurs. A maximum in the pre-exponential factor occurs
for the island with N[001] = 15. This feature is in quite good agreement with the experimental data, which however is available only for a limited range in the island size
and for somewhat irregular islands. A contour plot in Ref. [6] shows a ridge formation
in the magnitude of the pre-exponential factor for N[001] = N[11̄0] with a maximum at
N[001] = 15. The maximum in the pre-exponential factor comes from large entropy in the
transition state resulting from soft vibrational modes of the magnetic moments that are
present when two or more mechanisms are active. The soft modes have low frequency
which enter the denominator of Eq. (3) and thereby lead to a large pre-exponential
factor.
In Ref. [6], an increase of the pre-exponential factor was ascribed to an increase in
the number of nucleation sites at the edge of an island. Each nucleation center would
then add a new pathway for a transition leading to larger pre-exponential. However,
the calculations do not show evidence of this. The calculations show temporary domain
wall formation that involves all spins along the side of an island, so the clusters studied
here are small compared to the critical nucleus. Rather, the explanation of the observed
trend lies in the entropy of the transition state.
Also, an observed decrease in the pre-exponential factor for elongated islands was
ascribed to the effect of Brownian motion and the possibility of diffusive motion which
can bring the system back to the initial state [6]. The reduction in the pre-exponential
factor evident from Fig. 5 does not result from such motion since it is obtained from
a HTST estimate where recrossings of the dividing surface are not included. Rather, the
drop is due to a reduction of the transition state entropy as the island becomes longer
and the temporary domain wall mechanism becomes the dominant mechanism while
uniform rotation becomes insignificant. The effect of recrossings of the transition state
in magnetic systems remains an important topic of future studies.
7. Discussion
While the calculated rate of the magnetization reversal using the HTST approximation
gives good qualitative agreement with the experimental data and provides an interpretation of the observed trends, two important improvements to the methodology would
be interesting to explore. First of all, a full TST calculation could be carried out where
the free energy of the transition state is evaluated by reversible work [29,30]. While
the transition state in the HTST approximation is taken to be a mosaic of hyperplanar
segments, each segment going through a saddle point in the energy ridge surrounding
the initial state minimum [23], the transition state in a full TST calculation can have an
arbitrary location and orientation. A variational principle due to Keck [26] shows that
the optimal choice for the transition state maximizes the free energy of the transition
state. One way to represent such a transition state is to variationally optimize a mosaic
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of hyperplanar segments where the location and orientation of each segment is chosen
to maximize the free energy [22,31]. A more general shape of the dividing surface can
also be used, but the optimization then becomes more involved. It remains to be seen
how well HTST can be used to estimate rates of transitions in magnetic systems and to
what extent it is necessary to apply full TST.
Another improvement in the methodology would be to carry out the second step in
the two step process, namely the evaluation of the dynamical correction. This may be
particularly important for magnetic transitions since the flatness of the energy profile
along the MEP for the longer islands makes recrossings of the transition state dividing
surface likely, thereby causing the TST expression, Eq. (3), to give an overestimate. An
alternative formulation of rate theory originally due to Kramers takes such re-crossings
into account by assuming the dynamics can be described by the Langevin model [32].
This theory was originally developed for a single variable, but was extended to multidimensional systems by Langer [33]. It was extended to magnetic systems with one magnetic vector (see review in Ref. [34]) and more recently to multiple magnetic vectors
in the context of micromagnetic modeling [35]. Using a Heisenberg-type Hamiltonian,
we have estimated the maximal recrossing effect within this level of approximation and
it turned out to be surprisingly small [28]. One advantage of the TST formulation over
that of Kramers/Langer is that it does not assume a particular form for the dynamical trajectories. An accurate estimate of the recrossing correction can be obtained by
calculating short time dynamical trajectories started from points on the transition state
dividing surface [26]. A recent example of such an approach for atomic systems is given
in Ref. [36]. A simple Langevin description of the recrossing dynamics in atomic scale
systems has, in fact, proven to be too crude [37]. It remains to be seen from direct calculations of recrossing trajectories to what extent a Langevin description of the dynamics
of moments in magnetic systems is adequate. The TST formulation of the rate theory
makes it possible to introduce various levels of accuracy of the recrossing correction.
At low enough temperature, tunneling becomes the dominant transition mechanism.
Such behavior has, in particular, been observed for adatom islands on a surface [38].
A harmonic quantum transition state theory based on an expansion at saddle points (‘instantons’) on an extended, quantum mechanical energy surface has been formulated and
used successfully for atomic scale systems [23,39]. The extension of this approach to
magnetic systems with multiple degrees of freedom remains to be formulated and is the
subject of ongoing work.
The rate theory discussed here can also be applied to models involving longer length
scales and coarse graining, such as micromagnetic models [12]. There, each magnetic
momentum vector represents a number of microscopic spins. The NEB method can be
applied in a similar way as described above and the rate expression is analogous to the
one presented here since the basic equation of motion is again the LLG equation.
Acknowledgement
We thank Prof. Björgvin Hjörvarsson and Dr. Stefan Krause for helpful discussions.
This work was supported by the Icelandic Research Fund, The University of Iceland
Scholarship Fund, RFBR Grant No. 12-02-31716, and DFG-RFBR Cooperative Grant:
DFG No. ZA 161/20-1; RFBR No. 11-02-91337.
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