Thermally assisted macroscopic quantum tunneling of a ferromagnetic particle in a magnetic field at an arbitrary angle

34 Pages·0.28 MB·English

by  Gwang-Hee Kim

Thermally Assisted Macroscopic Quantum Tunneling of a
Ferromagnetic Particle in a Magnetic Field at an Arbitrary Angle
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Gwang-Hee Kima
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Department of Physics, Sejong University, Seoul 143-747, KOREA
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h Abstract
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At finite temperature we study the quantum tunneling of magnetization for
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a a small ferromagnetic particle with the biaxial symmetry placed in a mag-
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- netic field at an arbitrary angle. We present numerical WKB exponent below
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o the crossover temperature in which the quantum tunneling is affected by the
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thermal activation, and the approximate form of the WKB exponent around

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v the crossover region. The effect of quantum fluctuations on the thermal acti-
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vation rate beyond the crossover regime are discussed.
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9 PACS numbers: 75.45.+j, 73.40.Gk, 75.30.Gw, 75.50.Gg
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Typeset using REVTEX
I. INTRODUCTION
In the last decade, the problem of quantum tunneling of magnetization(QTM) in a single
domain magnetic particle has attracted a great deal of theoretical [1] and experimental [2]
interest. There are two reasons for this. One is the rapid development of the new field of
nanomagnetism, in which a single domain magnetic particle can be prepared. The other is
that such a system meets the main criteria for observing macroscopic quantum tunneling
(MQT). There are three conditions for seeing MQT which can be stated qualitatively as
follows. First, the height and the width of the barrier should not be large. Second, the
effective moment of inertia associated with the magnetization switching should not be too
large. These two conditions are required for a tunneling time accessible to current exper-
imental situations. Third, the crossover temperature at which the transition between the
thermal activation and the quantum tunneling occurs should be of the order of milliKelvin
range or above. Based on the criteria, it has been believed that a magnetic field is a good
external parameter to make QTM observable. Also, tunneling dynamics poses many inter-
esting problems, particularly concerning the role played by the thermal fluctuations at finite
temperature. This is the main subject of this paper, which deals with thermal corrections
to quantum tunneling rate of a single domain ferromagnetic paricle in a magnetic field.
Consider a single domain ferromagnetic particle uniformly magnetized along an easy axis
determined by magnetocrystalline anisotropy. Since the anisotropy generates two or more
energetically equivalent orientations of the magnetization, at extremely low temperature
the direction of the magnetization M might change from one easy axis to the other easy
axis by the quantum tunneling process. However, according to the criteria, the rate of
change of M is too small to be observed without controlling the height and the width of
the barrier as well as the effective moment of inertia intrinsically produced by the magnetic
anisotropy energy. Applying an external magnetic field H in a proper direction, one of two
equivalent orientationsbecomes metastable andthemagnitude ofthe physical quantities can
be controlled in an appropriate way. In this situation the change of M with time is expected
2
to be able to be observed and compared with theroretical results. As temerature increases
from zero, thermal effects add to the quantum tunneling process. [3] At sufficiently high
temperature, the direction of the magnetization is changed by the pure thermal activation
whose rate is proportional to exp( U/k T) where U is an energy barrier related to the
B
−
magneticanisotropyenergyandtheexternalmagneticfield. Intheintermediatetemperature
region, it is expected that there exists a crossover temperature T . Around T both thermal
c c
fluctuations andquantum tunneling coexist, namely, either thethermally activatedquantum
tunneling below T or the quantum mechanically assisted thermal activation process above
c
T can occur. Well below the crossover temperature, i.e., at a temperature low enough to
c
neglect the thermal activation process, the rate of change of M is soley due to quantum
tunneling. [4,5]
A number of theoretical studies about QTM for the single domain ferromagnetic particle
have been performed for several magnetocrystalline anisotropy. [1] The QTM problem with
a magnetic field at an arbitrary angle was firstly studied by Zaslavskii [6] who calculated the
tunneling ratefortheuniaxialsymmetry bymappingthespinsystem ontoaone-dimensional
particle system. For the same symmetry, Miguel and Chudnovsky [4] have calculated the
tunneling rate within the imaginary time path integral method. [7] They have also discussed
the tunneling rate at finite temperature and suggested experimental procedures. Kim and
Hwang [5] have performed the calculation based on instanton approach for biaxial and
tetragonalsymmetry. Theirworkpresented thetunnelingandoscillationratebetweenangles
π/2 θ π at zero temperature, where θ is the angle between the initial easy axis and
H H
≤ ≤
the external magnetic field. This paper extends the tunneling rate for the biaxial symmetry
to a finite temperature. Within the instanton approach, we present the numerical results
for the WKB exponent below the crossover temperature and their approximate formulas
around the crossover temperature. Also, we discuss the effect of quantum fluctuations on
the thermal activation rate beyond the crossover regime.
Thispaperisstructuredinthefollowingway. InSec. II,weintroducegeneralformulation
for the tunneling rate based on the spin coherent state path integral method. We discuss the
3
angular dependence of the critical field and the critical angle by using approximate formula
of the total energy in the small ǫ(= 1 H/H ) limit, where H is defined as a critical
c c
−
field at which the barrier disappears. In Sec. III, we give a brief survey of the quantum
tunneling of magnetization in the presence of the magnetic field with a general direction at
zero temperature. In Sec. IV, we consider thermal effects on the quantum tunneling rates
below T and qunatum corrections to thermal activation rates above T and present their
c c
numerical results in the entire temperature regime. In Sec. V we summarize the analytic
results discussed and provide actual estimates of such important quantities as the tunneling
rate, crossover temperature, and so on, for several real materials.
II. GENERAL FORMULATION
Consider the spin coherent state path integral representation of the partition function
given by
Z(βh¯) = D[M(τ)]exp( S /h¯), (1)
E
−
I
where β = 1/k T, the path sum is over all periodic paths M(τ) = M(τ +βh¯), and S the
B E
Euclidean action [7] which includes the Euclidean version of the magnetic Lagrangian L as
E
S [M(τ)] = dτL [M(τ)]. (2)
E E
I
Due to strong exchange interaction in a single domain ferromagnetic particle, the magni-
tude of the magnetization is a constant M . For that reason the dynamical variable is the
0
direction of the magnetization, equivalently, θ(τ) and φ(τ) in the spherical coordinates of
M. Therefore, up to the normalization the functional measure in Eq. (1) is equivalent to
DΩ as
N 2J +1
DΩ = lim sinθ dθ dφ , (3)
k k k
ǫ 0 4π
→ kY=1(cid:16) (cid:17)
where ǫ = max(τ τ ) and J = M V/h¯γ. Here γ = gµ /h¯, with g being the g-factor,
k+1 k 0 B
−
µ the Bohr magneton, M the magnitude of the magnetization and V the volume of the
B 0
4
system. Since the tunneling rate Γ of a metastable state is proportional to the imaginary
part of the free energy of the system, ImF, and ImZ ImF, from Eq. (1) the tunneling
∝
rate in semiclassical limit, with an exponential accuracy, is
Γ exp( Smin(T)/h¯), (4)
∝ − E
where Smin(T) is obtained along the trajectory with period βh¯ that minimizes the Euclidean
E
action
β¯h M dφ(τ)
S = V i 0[1 cosθ(τ)] +E[M(τ)] dτ. (5)
E
{ γ − dτ }
Z0
The first term in Eq. (5) is the topological Wess-Zumino term [8,9], and the second term is
the energy density, which is composed of the magnetic anisotropy energy E and the energy
a
given by an external magnetic field H, given by
E[M(τ)] = E M H. (6)
a
− ·
We have selected the biaxial symmetry with magnetic anisotropy energy E = K (α2 +
a 1 1
α2)+K α2. Here α ’s are the directional cosines and K > 0, K > 0 are the parallel and
2 2 2 i 1 2
transverse anisotropy constants, respectively, whose relative magnitude can be of any value.
Applying the magnetic field in the xz plane [4,5] and expressing α , α and α in spherical
1 2 3
coordinates of M, the energy (6) is expressed as
E[θ,φ] = K sin2θ+K sin2φsin2θ
1 2
H M sinθcosφ H M cosθ+E , (7)
x 0 z 0 0
− −
where E is a constant to make E(θ,φ) zero at the initial orientation. As will be seen
0
later, while there isn’t an exact analog of kinetic energy in the action (5), there is an
effective moment of inertia in the dynamics of a single domain particle that is inversely
proportional to a linear combination of K and H . For this reason we need either the
2 x
transverse anisotropy constant K or the magnetic field H transverse to the initial easy
2 x
axis for quantum tunneling. Otherwise, QTM is not observable due to the effective moment
of inertia whose magnitude becomes infinity.
5
Introducing the dimensionless constants,
k = K /K , h = H /H , h = H /H , H = 2K /M , (8)
2 2 1 x x 0 z z 0 0 1 0
we obtain the energy (7) written as
1 k
E¯[θ,φ] = sin2θ + 2 sin2φsin2θ
2 2
h sinθcosφ h cosθ+E¯ , (9)
x z 0
− −
¯
where E(θ,φ) = E(θ,φ)/2K . As noted in Eq. (9), there are two equivalent easy directions
1
in the xy plane, which are φ = 0 and π in the azimuthal angle if h = 0. We treat the
x
problem of the easy directions given by φ = 0 in the xy plane. Even though we start the
argument from φ = π, the whole discussion is the same as the one from φ = 0 by replacing
φ by π+φ. So, the classical path becomes φ (τ) = φ (τ)+nπ where n = 0 or 1 where φ (τ)
cl 0 0
is the classical trajectory started from φ = 0. [5] However, if the external field is not along
the easy axis(z axis), i.e., h = 0, only φ = 0 in Eq. (9) is the easy direction in the xy plane
x
6
and φ = π not. Keeping this point in mind, the total energy on the easy plane φ becomes
1
E¯(θ,0) = sin2θ hcos(θ θ )+E¯ , (10)
H 0
2 − −
where weused h = hsinθ andh = hcosθ . Let usdefine θ to betheangleofmetastable
x H z H 0
state generated by the anisotropy energy and the external magnetic field, and θ the an-
c
gle at which the barrier vanishes by the external critical magnetic field H . Then, θ is
c 0
determined by [dE¯(θ,0)/dθ] = 0, and θ and h by both [dE¯(θ,0)/dθ] = 0
θ=θ0 c c θ=θc,h=hc
and [d2E¯(θ,0)/dθ2] = 0. After simple calculations, the dimensionless switching or
θ=θc,h=hc
critical field h ( H /H ) and the critical angle θ are expressed as [4,5]
c c 0 c
≡
h = (sin2/3θ + cosθ 2/3) 3/2, (11)
c H H −
| |
1 2 cotθ 1/3
H
θ = arcsin[ | | ], (12)
c 2 1+ cotθ 2/3
H
| |
where it should be noted that Eq. (11) is the well-known Stoner-Wohlfarth expression.
[10] Simple analysis for Eq. (10) reveals that the small value of ǫ is favorable for the
6
magnetization switching because the height and the width of the barrier are proportional to
the power of ǫ. The situation can be achieved by applying the external magnetic field close
to the critical field. Hence the problem will be considered in the small limit of ǫ, i.e., ǫ 1.
≪
Expanding [dE¯(θ,0)/dθ] = 0 about θ , and
θ=θ0 c
using the relations [dE¯(θ,0)/dθ] = 0 and [d2E¯(θ,0)/dθ2] = 0, we obtain
θ=θc,h=hc θ=θc,h=hc
the equation for η(= θ θ ) as [5]
c 0
−
3
sin(2θ )(ǫ η2) ηcos(2θ )(2ǫ η2) = 0. (13)
c c
− 2 − −
Simple calculations show that η is of the order of √ǫ. Thus the order of magnitude of the
second term in Eq. (13) is smaller than that of the first term by √ǫ and the value of η is
determined by the first term which leads to η 2ǫ/3. However, when θ is very close
H
≃
q
to π/2 or π, sin(2θ ) becomes close to zero, and the second term is much larger than the
c
first term. Therefore, the value of η is obtained from the second term when θ π/2 or π,
H
≃
which leads to η √2ǫ for θ π/2 and η 0 for θ π from the detailed analysis of the
H H
≃ ≃ ≃ ≃
equationswhichθ andθ satisfy. Sincethefirsttermin(13)isdominantintherangeofvalue
0 c
θ which satisfies tan(2θ ) > O(√ǫ), η 2ǫ/3 is valid for π/2+O(√ǫ) < θ < π O(√ǫ)
c c H
≃ −
q
from Eq. (12). Fig. 1 shows that this is checked by performing the numerical calculation
for the equations which θ and θ satisfy.
0 c
¯
Introducing a small variable δ(= θ θ ), we obtain an approximate form of E(θ,φ) as
0
−
k
E¯(δ,φ) = 2 sin2φsin2(θ +δ)+h sin(θ +δ)(1 cosφ)+E¯ (δ), (14)
0 x 0 1
2 −
¯ ¯ ¯
where E(θ,φ) is represented as E(δ,φ), and E (δ) is a function of only δ given by
1
1
E¯ (δ) = sin(2θ )(3δ2η δ3)
1 c
4 −
1 3 δ4
+ cos(2θ )[δ2(ǫ η2)+δ3η ]. (15)
c
2 − 2 − 4
Although cos(2θ )-term in Eq. (15) looks smaller by a factor of η which is of the order of
c
√ǫ, it can not be neglected near θ = π/2 and π because sin(2θ ) is almost zero for these
H c
regions of θ . In order to evaluate the order of magnitude of each term in action (5) and
H
simplify the calculations for ǫ 1, it is convenient to use new scaled variables
≪
7
τ˜ = ǫα/2 ω τ, δ¯= δ/√ǫ, ω = 2γK /M . (16)
0 0 1 0
Then, the Euclidean action (5) becomes
β˜ dφ
SE[δ¯(τ˜),φ(τ˜)] = h¯Jǫ−α2 dτ˜ iǫα2[1 cos(θ0 +√ǫδ¯)]
{ − dτ˜
Z0
k
+ 2 sin2φsin2(θ +√ǫδ¯)+h sin(θ +√ǫδ¯)(1 cosφ)
0 x 0
2 −
1 η 1 3η2 η δ¯4
+ sin(2θc)ǫ32(3δ¯2 δ¯3)+ cos(2θc)ǫ2[δ¯2(1 )+δ¯3 ] . (17)
4 √ǫ − 2 − 2ǫ √ǫ − 4 }
where β˜ = ǫα/2 ω βh¯ and the parameters α is fixed by the analysis of the order of magnitude
0
of each term in the action (17) subject to the situation considered.
III. QTM AT ZERO TEMPERATURE
Consider the Euclidean action (17) at T = 0 [5] by performing a simple dimensional
¯
analysis. At T = 0 (β ) the instanton starts and ends at the metastable state δ = 0,
cl
→ ∞
φ = 0 for π/2+O(√ǫ) < θ < π O(√ǫ), in which E¯(δ¯ = 0,φ = 0) = 0. Thus, from
cl H cl cl
−
the energy conservation the total energy E¯(δ ,φ ) in Eq. (14) becomes zero at any τ¯ for
cl cl
T = 0. Since sin(2θ )-term ( ǫ3/2) is larger than the last term ( ǫ2) by √ǫ in Eq. (17),
c
∼ ∼
the first, second and third term should be of the order of ǫ3/2. Since h is finite in the range
x
of angle and θ is not close to 0 and π/2, the small value of φ in the third term of Eq.
0
(17) contributes to the path integral such that φ should be of the order of ǫ3/4. This fact
is valid for any value of k because h is not zero in this range of angle and plays the role
2 x
of the effective transverse anisotropy component which is crucial for QTM. Therefore, from
E¯(δ ,φ ) = 0, the magnitude φ is approximately given by
cl cl cl
2E¯ (δ )
φ2 = 1 cl . (18)
cl −k sin2(θ +δ )+h sin(θ +δ )
2 0 cl x 0 cl
From Eq. (15) we get E¯ (δ) sin(2θ )(3δ2η δ3) because cos(2θ )-term is much smaller
1 c c
∼ −
than sin(2θ )-term. Since η 2ǫ/3, and the critical angle θ andthe angleθ of metastable
c c 0
≃
q
state have values between 0 and π/2 in this range of angle, we get φ i√M cotθ ǫ3/4,
cl c
∼ −
where the effective moment of inertia M is [11]
8
sinθ
c
M = . (19)
h +k sinθ
x 2 c
Thus, the approximate formula of the classical Euclidean action is found to be Scl
E ∼
ih¯J sinθ √ǫδ¯ φ h¯Jǫ5/4√M sinθ cosθ from the dynamical part of the first term in Eq.
c cl cl c c
∼
(17), where δ¯ O(1) and E¯(δ¯ ,φ ) = 0. By using the critical field (11) and the critical
cl cl cl
∼
angle (12), the classical action is approximately
Scl h¯Jǫ5/4g(θ ), (20)
E ≈ H
where
cotθ 1/6
H
g(θ ) = | | . (21)
H
1+ K2(1+ cotθ 2/3)
K1 | H|
q
In the case of θ = π, we have θ = θ = η = 0 and 0 δ 2√ǫ from Eq. (15). In
H c 0 cl
≤ ≤
this situation, the action (17) becomes
β˜ i dφ k 1 δ¯4
SE[δ¯(τ˜),φ(τ˜)] = h¯Jǫ−α2 dτ˜ ǫα+22δ¯2 ( )+ 2ǫδ¯2sin2φ+ ǫ2(δ¯2 ) . (22)
{2 dτ˜ 2 2 − 4 }
Z0
Since the last term in Eq. (22) is of the order of ǫ2, the order of magnitude of k sin2φ in
2
the second term should be ǫ. If k becomes much smaller than ǫ, in which the magnetic
2
particle possesses the uniaxial symmetry, [4,6] QTM is not observable. This is understood
fromthe fact as follows. Consider the action (5) as the dynamical system in the Hamiltonian
formulation which consists of the canonical coordinate φ and p = 1 cosθ. [9] The Poisson
φ
−
bracket p ,H which determines the dynamics of the spin system with the Lagrangian
φ
{ }
(5) becomes zero for θ = π because the Hamiltonian H becomes a function of only p .
H φ
Thus, θ does not depend on time. [12] If the value of k is much greater than ǫ, the small
2
value of φ, i.e., O(φ) ǫ1/2 mainly contributes to the path integral, in which α is chosen
∼
to be 1. In this case, a comparison of k -term with the last term in Eq. (22) determines
2
φ i K ǫ/K , which leads to the least action Scl ih¯Jǫδ¯ 2φ h¯Jǫ3/2 K /K . The
cl ∼ − 1 2 E ∼ cl cl ∼ 1 2
q q
approximate forms of Scl which we have discussed for π/2+O(√ǫ) < θ < π O(√ǫ) and
E H −
θ = π agree with the exact results in Ref. [5] up to the numerical factor N = 16 61/4/5
H
×
9
and 8/3, respectively. [13] From above estimates we note that the dependence of the WKB
exponent on θ , ǫ andK /K can bededuced fromthe simple dimensional analysis based on
H 2 1
the energy conservation E¯(δ¯ ,φ ) = 0 without solving the equation of motionfromδS = 0.
cl cl E
However, in order to obtain the dependence of the WKB exponent on K , K , ǫ, and θ
1 2 H
at finite temperature, we cannot make use of the dimensional estimate as above, because
E¯(δ¯ ,φ ) = 0. Thus, we need to calculate the bounce solution by numerically solving the
cl cl
6
equation of motion satisfied by the least trajectory.
IV. QTM AT FINITE TEMPERATURE
Consider the Euclidean action (17) with a periodic instanton for π/2+ O(√ǫ) < θ <
H
π O(√ǫ) and θ = π.
H
−
A. π/2+O(√ǫ) < θ < π O(√ǫ)
H
−
Since the sin(2θ )-term is much larger than cos(2θ )-term in Eq. (17), the action is
c c
simplified as
β˜ dδ¯
SE[δ¯(τ˜),φ(τ˜)] = h¯Jǫ−α2 dτ˜ iǫα+21 sin(θ0 +√ǫδ¯) φ ( )
{− dτ˜
Z0
k φ
+ 2 sin2φsin2(θ +√ǫδ¯)+2h sin(θ +√ǫδ¯)sin2( )
0 x 0
2 2
1 η
3 ¯2 ¯3
+ sin(2θc)ǫ2(3δ δ ) , (23)
4 √ǫ − }
where h = 0 and k has any value, and we performed the integration by part for the first
x 2
6
˜ ˜
term. Also, the total time derivative is neglected because φ ( β/2) = φ (β/2) = 0 by
cl cl
−
¯
noting that from the Euler-Lagrange equation φ is proportional to dδ /dτ˜ which is zero at
cl cl
˜ ¯
τ˜ = β/2 due to the periodicity of δ (τ˜). As is well known for the crystal symmetry such
cl
±
as the form K α2 K α2, the topological term dφ/dτ in Eq. (5) plays important roles in
z z − y y
the presence of a clockwise or counterclockwise winding over the barrier along the passage
from φ = 0 to φ = π in macroscopic quantum coherence(MQC). [8] Since there is no such
10