spin-torque driven macrospin dynamics subject to thermal noise

Spin-Torque Driven Macrospin Dynamics subject to

Thermal Noise

by

Daniele Pinna

A dissertation submitted in partial fulfillment

of the requirements for the degree of

Doctor of Philosophy

Department of Physics

New York University

January 2015

Andrew D. Kent

Daniel L. Stein

Dedication

To the most spectacular Bird on this side of the Holocene boundary.

iii

Acknowledgements

This doctoral work would not have been possible without the support of my two

advisers: Dan Stein and Andy Kent. They have supplied me with both the directive

and incentive to explore the problems discussed in this thesis. I am indebted to

them for the freedom and the space they have given me to take my research in an

autonomous direction. Witnessing my professional development sifted and molded

by their experienced oversight could not have been more fortuitous for me.

I also thank the Physics department of New York University for accepting

and supporting me throughout these 5.5 years of studentship. The experience

accumulated through both my teaching and research assistanships wouldn’t have

gone by so smoothly without the many people handling all sorts of bureaucracy

and background logistics.

Any journey must come about with significant personal growth. In this respect,

I acknowledge the many graduate colleagues that enabled me to call this Physics

department a “home”. I particularly thank Roberto Gobbetti, Sven Kreiss, and

Colm Kelleher for allowing me to bounce ideas around and discussing the most

random topics that our minds could cook up.

More scientifically speaking, I’d also like to thank the many people that have

defined the Kent Lab throughout my stay. They have supplied me with crucial

context, without which this research work would have never made any sense to me

to begin with.

I would also like to acknowledge friends and family for the immensity of their

love and support throughout these years. It feels as if Graduate school could not

have taken place without your presence allowing me to grow into a more mature

molt of myself. Thank you for existing and being who you are.

iv

Last but not least, I am greatful to the National Science Foundation and IARPA

for providing the funding to support my Ph.D. research. My thesis research was

supported by NSF-DMR-1006575, NSF-DMR-1309202 and by IARPA contract

W911NF14-C-0089.

v

Abstract

This thesis considers the general Landau-Lifshitz-Gilbert theory underlying the

magnetization dynamics of a macrospin magnet subject to spin-torque effects and

thermal fluctuations, as a function of the spin-polarization angle. The macrospin

has biaxial magnetic anisotropy, typical of thin film magnetic elements, with an

easy axis in the film plane and a hard axis out of the plane. We will argue that

when magnetic diffusion due to spin-torque and thermal noise effects happen on a

timescale that is much larger than the conservative precessional timescale due to

material anisotropies, it becomes possible to explore steady-state dynamics per-

turbatively by analytically averaging the magnetization dynamics over constant

energy orbits. This affords the possibility to simplify the magnetization dynamics

to a 1D stochastic differential equation governing the evolution of the macropsin’s

anisotropy energy. Current induced steady-state motions are then found to appear

whenever the magnetization settles onto a stable constant energy trajectory where

a balance of spin-torque and damping effects is achieved: with the remaining gyro-

magnetic motion due to anisotropy fields driving precessions. After averaging, all

the relevant dynamical scenarios are found to depend on the ratio between hard

and easy axis anisotropies. We derive analytically the range of currents for which

in-plane and out-of-plane limit cycles exist and discuss the regimes in which the

constant energy orbit averaging technique is applicable. We find that there is a

critical angle of the spin-polarization necessary for the occurrence of such states

and predict a hysteretic response to applied current. This model can be tested

in experiments on orthogonal spin-transfer devices, which consist of both an in-

plane and out-of-plane magnetized spin-polarizers, effectively leading to an angle

between the easy and spin-polarization axes.

vi

Contents

Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiv

List of Appendices xxv

Introduction 1

1 General Formalism 7

1.1 Magnetic Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.1.1 Exchange Energy . . . . . . . . . . . . . . . . . . . . . . . . 11

1.1.2 Dipolar Energy . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.1.3 Magnetocrystalline/Shape Anisotropy Energy . . . . . . . . 14

1.1.4 Magnetostatic and Zeeman Energy . . . . . . . . . . . . . . 16

1.2 Macrospin Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.2.1 The Landau-Lifshitz (LL) equation . . . . . . . . . . . . . . 18

1.2.2 Gilbert Damping . . . . . . . . . . . . . . . . . . . . . . . . 20

vii

1.2.3 Slonczewski Spin-Torque . . . . . . . . . . . . . . . . . . . . 21

1.2.4 Thermal Effects . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.3 The Stochastic Landau-Lifshitz-Gilbert Slonczewski Equation (sLLGS) 26

2 Stochastic Calculus 32

2.1 Gaussian Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.2 Stochastic Differential Equations . . . . . . . . . . . . . . . . . . . 33

2.3 Interpretation of Stochastic Integrals . . . . . . . . . . . . . . . . . 35

2.4 The Fokker-Planck Equation . . . . . . . . . . . . . . . . . . . . . . 39

2.5 Limits of Stochastic Modeling . . . . . . . . . . . . . . . . . . . . . 41

2.5.1 Gradient Systems . . . . . . . . . . . . . . . . . . . . . . . . 42

2.5.2 Non-Gradient Systems . . . . . . . . . . . . . . . . . . . . . 43

2.6 Stochastic Macrospin Dynamics: Reprise . . . . . . . . . . . . . . . 44

3 Numerical Methods 48

3.1 Euler-Maruyama Scheme . . . . . . . . . . . . . . . . . . . . . . . . 51

3.2 Heun Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.3 Strong and Weak Convergence of a Numerical Scheme . . . . . . . . 53

3.4 GPU: The CUDA Environment . . . . . . . . . . . . . . . . . . . . 56

4 Uniaxial Macrospin Model 63

4.1 Collinear Spin-Torque Model . . . . . . . . . . . . . . . . . . . . . . 67

4.1.1 Collinear High Current Regime . . . . . . . . . . . . . . . . 67

4.2 Tilted Spin-Torque Model . . . . . . . . . . . . . . . . . . . . . . . 69

4.3 Thermally Activated Regime . . . . . . . . . . . . . . . . . . . . . . 72

4.4 Switching Time Probability Curves . . . . . . . . . . . . . . . . . . 80

viii

5 Constant Energy Orbit-Averaged Dynamics 85

5.1 Constant Energy Orbit-Averaged (CEOA) dynamics . . . . . . . . . 87

5.2 IP Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.2.1 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . 96

5.2.2 D > D0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.2.3 D < D0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.2.4 CEOA Breakdown . . . . . . . . . . . . . . . . . . . . . . . 104

5.2.5 Thermally Activated Switching . . . . . . . . . . . . . . . . 111

5.3 OOP Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.3.1 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . 116

5.3.2 CEOA Breakdown . . . . . . . . . . . . . . . . . . . . . . . 118

5.3.3 Thermal Stability, Precession Linewidth, Phase, Amplitude

and Power Fluctuations . . . . . . . . . . . . . . . . . . . . 122

5.4 Experimental Verification of OOP orbits . . . . . . . . . . . . . . . 133

5.4.1 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

5.4.2 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6 Structure of Escape Trajectories 145

6.1 Friedlin-Wentzell (FW) theory . . . . . . . . . . . . . . . . . . . . . 146

6.2 Macrospin Escape Paths . . . . . . . . . . . . . . . . . . . . . . . . 153

Conclusion 158

Appendices 163

Bibliography 172

ix

List of Figures

1 A schematic of the origin of the spin transfer torque. Layers A,B

and C are non-magnetic layers. . . . . . . . . . . . . . . . . . . . . 2

2 Illustration of a spin valve/MTJ device. . . . . . . . . . . . . . . . . 4

1.1 Constant energy trajectories for D = 10. ε < 0 trajectories are

shown in red whereas ε > 0 trajectories are shown in blue. Notice

how two distinct basins exist for positive and negative energy tra-

jectories. The separatrix (corresponding to ε = 0) separating the

different basins is shown in black. . . . . . . . . . . . . . . . . . . . 19

1.2 Relaxation trajectory of the magnetization for a free layer under the

effects of Gilbert damping and the conservative LL torque [65]. . . . 21

1.3 Relaxation trajectory for the magnetization of a free layer under

the effects of Gilbert damping, the conservative LL torque, and

stochastic thermal noise [65]. . . . . . . . . . . . . . . . . . . . . . . 25

x

1.4 A typical trilayer spin-valve consisting of a free magnetic layer sand-

wiched between an in plane magnetized reference layer and out-of-

plane magnetized polarizer layer. The net spin-torque acting on the

free layer will generally appear tilted away from the easy-axis of

the magnet. The advantage of such devices lies in the ability to in-

stantly torque the magnetic free layer without the need of a thermal

incubation time to destabilize the magnetization initially. . . . . . . 29

1.5 Easy-easy x and hard-axis z magnetic anisotropy directions are

shown along with spin-polarization direction np. The spin-polariation

is tilted by an angle ω with respect the magnetic easy axis. . . . . . 30

3.1 Floating-Point Operations per Second for the CPU and GPU [131]. 57

3.2 Memory Bandwidth for the CPU and GPU [131]. . . . . . . . . . . 57

3.3 A schematic view of a CUDA streaming multiprocessor with 8 scalar

processor cores [132]. . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.1 Histogram distribution of mx after letting the magnetic system relax

to thermal equilibrium (103 natural time units). The overlayed red

dashed line is the theoretical equilibrium Boltzmann distribution.

In the inset we show a semilog-plot of the probability vs. m2x de-

pendency. As expected, the data scales linearly with slope equal to

ξ ≡ K/kBT = 80: the ratio between total anisotropy and thermal

energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

xi

4.2 Current amplitude vs. mean switching time. Blue line shows the

fit of the ballistic limit to the numerical data (in blue crosses). Red

line shows the improvement obtained by including diffusion gradient

terms. Times are shown in units of (T · s) where T stands for Tesla:

real time is obtained upon division by µ0HK . . . . . . . . . . . . . . 70

4.3 mx: green > 0, red < 0 for applied current I = 5. The plane dissect-

ing the sphere is perpendicular to the uniaxial anistropy axis. Its

intersection with the sphere selects the regions with highest uniaxial

anisotropy energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.4 Mean switching time behavior for various angular tilts and cur-

rents larger than the critical current obtained by numerically solving

(1.29). Each set of data is rescaled by its critical current such that

all data plotted has Ic = 1. Angular tilts are shown in the legend in

units of π/36 such that the smallest angular tilt is 0 and the largest

is π/4. Times are shown in units of (T ·s) where T stands for Tesla:

real time is obtained upon division by µ0HK . . . . . . . . . . . . . . 73

xii

4.5 Mean switching time behavior in the sub-critical low current regime

obtained by numerically solving the full macrospin dynamics. Times

are shown in units of (T · s) where T stands for Tesla: real time

is obtained upon division by µ0HK . The red and green line are

fits to the data with the functional form 〈τ〉 = C exp(−ξ(1 − I)µ),

where µ is the debated exponent (either 1 or 2) and C is deduced

numerically. The red curve fits the numerical data asymptotically

better the green curve. The difference between the red line and

(4.11) is that our theoretical prediction includes a current dependent

prefactor which was not fitted numerically. The differences between

numerical data and (4.10) is due to numerical inaccuracies out to

such long time regimes. The differences between (4.10) and (4.11),

on the other hand, quantify the reach of the crossover regime. . . . 76

4.6 Mean switching time behavior in the sub-critical low current regime

obtained by numerically solving the macrospin dynamics. Various

polarizer tilts are compared by rescaling all data by the appropriate

critical current value. Times are shown in units of (T · s) where T

stands for Tesla: real time is obtained upon division by µ0HK . . . . 78

xiii

4.7 Influence of precessional orbits on transient switching as seen from

the switching time probability curve in the supercritical current

regime. The case shown is that of an angular tilt of π/3 subject

to a current intensity of 2.0 times the critical current. Data was

gathered by numerically solving (1.29). The non-monotonicity in

the probability curve shows the existence of transient switching.

Times are shown in units of (T · s) where T stands for Tesla: real

time is obtained upon division by µ0HK . . . . . . . . . . . . . . . . 83

4.8 Spin-torque induced switching time probability curves for various

angular configurations of polarizer tilt (a sample normalized current

of 10 was used) obtained by numerically solving (1.29). A log-log

y-axis is used following (4.17) to make the tails of the probability

distributions visible. Times are shown in units of (T · s) where T

stands for Tesla: real time is obtained upon division by µ0HK . . . . 84

5.1 Orbital frequencies plotted as a function of ε for differentD. To com-

pare the results, the positive portion of ε axis has been rescaled by

D. Frequency is expressed in units of (GHz/T). Physical frequency

is obtained upon multiplying by µ0HK . The sharp minimum in the

frequency is a result of the precessional period diverging at ε = 0. . 90

5.2 Critical currents versus the ratio of the hard and easy axis anisotropies

D. The blue curve is I1C and the red curve is I0

C . For D < D0, cur-

rents greater than I1C lead to deterministic switching (labeled DS).

For D > D0 currents between I0C and I1

C lead to limit cycles (LC).

Limit cycles can also occur for currents just below and approxi-

mately equal to I1C for D < D0, as shown in Figure 5.5 . . . . . . . 99

xiv

5.3 Three sample regimes of deterministic energy flow ε as a function

of energy for D > D0. Coloring is included to better distinguish

the various curves. (blue thick dashed curve) I < I1C : Subcrit-

ical regime, thermal noise must oppose a positive energy flow to

achieve switching; (green dash-dotted curve) I1C I0C : Supercritical regime,

negative flow leads to deterministic switching. . . . . . . . . . . . . 100

5.4 Three sample regimes of deterministic energy flow ε as a function

of energy for D < D0. Coloring is included to better distinguish

the various curves. (blue thick dashed curve) I < I0C : Subcritical

regime, thermal noise must oppose a positive energy flow to achieve

switching; (green dash-dotted curve) I0C I1C :

Supercritical regime, negative flow leads to deterministic switching. 102

5.5 Energy flow for a D = 4 macrospin and applied current I = 2.82 <

I0C . Circles and squares respectively represent stable and unstable

equilibria. For these parameters (D = 4 and I = 2.82), two stable

equilibria of the zero temperature dynamics coexist. . . . . . . . . . 103

xv

5.6 a) Mean switching time versus current for D = 4, α = 0.04 and

ξ = 80 at different θ angular tilts with φ = 0 kept fixed. All

currents have been rescaled by 1/ cos θ. Times are shown in units of

(s ·T ) where T stands for Tesla: real time is obtained upon division

by µ0HK . For visual guidance, the critical currents I0C , I1

C and

limit current IM have been included. In a regime where the CEOA

technique is applicable, the switching data from the various angular

configurations should all fall on top of each other. b) Double y-axis

plot of max[T (ε)|∂t|ε||] and the percent deviation of data from (a))

as a function of normalized current. In the current range where the

deterministic flow achieves its minimum, the deviation of the data

does also. As the critical current I0C is approached, deviation spikes

are observed analogously to what can be inferred by the theory. . . 108



currents have been rescaled by 1/ cos θ. Times are shown in units

of (s · T ) where T stands for Tesla: real time is obtained upon

division by µ0HK . For visual guidance, the critical currents I0C , I1

C

and limit current IM have been included. b) Double y-axis plot

of max[T (ε)|∂t|ε||] and the percent deviation of data from (a)) as

a function of normalized current. In the current range where the




xvi



currents have been rescaled by nz = 1/ cos θ. Times are shown in

units of (s ·T ) where T stands for Tesla: real time is obtained upon

division by µ0HK . For visual guidance, the critical currents I0C , I1

C

and limit current IM have been included. b) Double y-axis plot

of max[T (ε)|∂t|ε||] and the percent deviation of data from (a)) as

a function of normalized current. In the current range where the




xvii

5.9 a) Scaling dependence of mean switching time as a function of ap-

plied current I for models with varying D < D0. ξ is the energy

barrier height and I1C the critical current threshold for deterministic

switching. b) Fit of (5.45) to the form (1 − I/I1C)β. Dashed lines

represent continuation of analytical results outside the technique’s

regime of validity. Fitting exponent β is plotted as a function of

applied current for models with varying D. In the limit of small

D the exponent approaches the constant value β = 2 consistent

with previous uniaxial macrospin results [136,155,159]. For D > 0,

the exponent β depends nonlinearly on the applied current inten-

sity. Only for values D ∼ D0 do we notice that in the limit of

small applied currents, β → 1 as suggested by similar energy diffu-

sion studies from the literature [139, 154]. For intermediate values

D0 > D > 0 the low current limit of β can be obtained analytically

(5.47). On the other hand, in the limit I → I1C the exponent β can

be shown to diverge for all non-zero values of D. . . . . . . . . . . . 113

xviii

5.10 Three regimes of deterministic energy flow ε as a function of energy

for D = 10. (blue-dashed) I < IOOP: Subcritical regime. Energy

flows from positive to negative energy basins due to dynamics being

globally dissipative (overdamped). (red-dashdotted) I > Imax: Su-

percritical regime. Energy flows towards limiting stable value ε = D

due to dynamics being overdriven by applied current. (green-dotted)

IOOP < I < Imax: Oscillator regime. Energy flow will stabilize at

a fixed point corresponding to a precessing oscillator state. In this

regime, the fixed point represents a constant energy trajectory where

spin-torque and damping effects balance. . . . . . . . . . . . . . . . 119

5.11 Steady-state ensemble energy as a function of dimensionless ap-

plied current I (rescaled by IOOP = (2/π)√D + 1) for a model with

D = 10, ξ = 80 and α = 0.04. Red line shows an analytic fit to

numerical data within the current limits defined by the theory (for

reference Imax/IOOP ≈ 4.97). Insets show density plots in spheri-

cal coordinates of 5120 numerical trajectories for a sample with a

2.56ωC tilt between easy and spin-polarization axes, driven by a cur-

rent of I/IOOP = 4 (top), and I/IOOP = 15 (bottom). The dotted

line denotes the conservative trajectory. . . . . . . . . . . . . . . . . 121

5.12 Standard deviation of the energy distribution plotted as a function of

dimensionless applied current I (rescaled by IOOP = (2/π)√D + 1)

for D = 10, ξ = 80 and α = 0.04. The solid blue line shows

the theoretical prediction (5.64) calculated within the current limits

defined by the theory (for reference Imax/IOOP ≈ 4.97). . . . . . . . 124

xix

5.13 Inverse quality factor (5.66) vs. applied current for D = 10 set at

room temperature (ξ = 80). Red dashed line denotes the upper

bound of the validly of the CEOA formalism: Imax/IOOP ≈ 4.97 for

the parameters chosen. . . . . . . . . . . . . . . . . . . . . . . . . . 126

5.14 Switching probability vs. spin-current pulse length for a macrospin

model with D = 10, ω = 2.12ωC) driven by a spin-current intensity

of I = 2.75 IOOP in the absence of thermal noise. Times are shown

in units of (s · T ) where T stands for Tesla: real time is obtained

upon division by µ0HK . Before the current pulse is switched on,

the magnetic ensemble is taken to be antiparallel to the easy-axis of

the magnetic film. Switching probability is defined as the ensemble

fraction that relaxes into a parallel configuration upon switching

the current pulse off. The right-hand vertical axis plots the evo-

lution of the average 〈mz〉 component. In the absence of thermal

noise the oscillator remains coherent at all times and its periodic

motion is clearly seen. Due to the deterministic nature of the zero-

temperature dynamics, the macrospin will deterministically switch

either into the parallel or antiparallel state at all times. . . . . . . . 130

xx

5.15 Switching probability vs. spin-current pulse length for a macrospin

model with D = 30, ω = 3ωC driven by a spin-current intensity

of I = 5 IOOP in the presence of thermal noise corresponding to

ξ = 80 (left) and ξ = 1200 (right). Times are shown in units

of (s · T ) where T stands for Tesla: real time is obtained upon

division by µ0HK . Before the current pulse is switched on, the

magnetic ensemble is taken to be antiparallel to the easy-axis of the

magnetic film. Switching probability is defined as the fraction of the

ensemble that relaxes into a parallel configuration upon switching

the current pulse off. For long pulse times the switching probability

converges to a value indicating that the phase of the OOP precession

has decohered. The red dashed lines are a qualitative graphical

representation of the decoherence time. . . . . . . . . . . . . . . . . 131

5.16 Log-log plot of ensemble decoherence time vs. energy barrier height

to thermal energy ratio ξ for a macrospin model with D = 30,

ω = 3ωC driven by a spin-current intensity of I = 1.5 Iswitch. Times

are shown in units of (s · T ) where T stands for Tesla: real time is

obtained upon division by µ0HK . Linear regression (solid lines) of

data points demonstrates a transition between a phase noise domi-

nated regime τdec ∝ 1/T below a certain critical inverse temperature

ξ < ξC . Above ξ > ξC (T < TC), both amplitude and phase noise

contribute to ensemble decoherence. . . . . . . . . . . . . . . . . . . 132

xxi

5.17 Resistance versus in-plane applied field hysteresis loops. The major

loop (black curve) shows the switching of both the free and reference

layers. The minor loop (blue curve) shows the response of just the

free layer. The loop is centered at 41 mT due to dipolar interactions

between the reference and free layer. Inset: Schematic of the spin-

valve’s layer stack showing the out-of-plane magnetized polarizing

layer (OP), in-plane magnetized free layer (FL) and reference layer

(RL). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5.18 Differential resistance versus current at various easy axis applied

fields starting from the P state. The magnitude of the current |Idc|

is increased (black curves) and then decreased (red curves). (a) The

black curve shows switching from P to AP at 1.5 mA and also -1.2

mA, i.e. the switching occurs for both polarities of the current. At

larger positive and negative current the resistance change is inter-

mediate of that of the P to AP transitions. On reducing the current

there is a transition from the intermediate resistance (IR) state into

an AP state. (b) At 40 mT switching from P to AP only occurs

for positive polarity current and on reducing the current there is an

IR to AP state transition for |Idc| ≤ 1 mA. (c) At 42 mT switching

from P to AP again only occurs for positive polarity current. How-

ever, on reducing the applied field the transition is from IR to P for

|Idc| ≤ 1 mA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

xxii

5.19 Current swept state diagram of an OST spin valve device showing

the threshold currents for switching as a function of applied easy

axis field. IP−APC and IAP−IRC are labeled by solid and open blue

symbols. IAP−PC and IP−IRC are labeled by solid and open red sym-

bols. The green curves indicate the IIR−APC (crosses) and IIR−PC

(dashes), showing the bistability range of the IR states. . . . . . . . 138

5.20 (a) Representative FL minor hysteresis loops measured with a slowly

swept field at several fixed currents. The scale bar shows ∆RAP−P =

0.1 Ω, the resistance difference between the AP and P states. (b)

State diagram constructed from dV/dI|H hysteresis loops. The

color represents ∆R, the resistance difference between field increas-

ing and field decreasing measurements. The central zone (orange

color) corresponds to the AP/P bistable zone. Black dashed curves

trace the boundaries between P, AP and IR states. . . . . . . . . . 139

5.21 Simulations of an ensemble of 5000 macrospins represented as a

state-diagram with current increasing (a) and current decreasing

(b). The three relevant states AP, P and IR, are color coded as

red, green and blue respectively. Each data point is represented by

a RGB color that is determined by the proportion of the ensemble

populating each corresponding state. Currents are shown in units

of switching positive current at zero field and room temperature.

Applied fields are shown in units of anisotropy field. The parameters

used in the simulation are described in the main text. . . . . . . . . 144

6.1 Diagram of momentum ellipse parametrized by γ. γ0 and π + γ0

correspond to instanton and anti-instanton solutions respectively. . 148

xxiii

6.2 Contour plot of the norm of the drift field taken from the Maier-

Stein model [177]. On the left, for α = 3, two global minima are

present at the unstable (0, 0) and stable (1, 0) equilibria respectively

along with a saddle along the x-axis. On the right, for α = 5, the

previous saddle has become a local maxima due to the appearance

of two new local minima off the x-axis. . . . . . . . . . . . . . . . . 152

6.3 Escape trajectory for a uniaxial macrospin model with j = 0.3 and

ω = 0. Blue line shows the least-action result of numerical integra-

tion of FW dynamics. Dashed red line is the analytical result shown

in equation (6.28). . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

6.4 Escape trajectory for a uniaxial macrospin model with j = 0.3 and

ω = 0.2π. Blue line shows the least-action result of numerical in-

tegration of FW dynamics. Dashed red line is the analytical result

shown in equation (6.28). . . . . . . . . . . . . . . . . . . . . . . . . 157

xxiv

List of Appendices

Appendix A Constant Energy Orbit Averaging of sLLGS Equation . . . . . 163

Appendix B Orbit averaging of a Stratonovich Equation . . . . . . . . . . . 169

xxv

Introduction

Over the past three decades, the study of spintronics has set the stage for emer-

gent technologies capable of exploiting the intrinsic “spin” of the electron, and its

associated magnetic properties, to operate, probe and manipulate novel solid-state

devices. The first major acknowledgement of this success came with the award-

ing of the 2007 Nobel prize in Physics to Albert Fert [1] and Peter Grunberg [2]

for their discovery of the giant magnetoresistance effect (GMR). In the GMR ef-

fect, the resistance of a magnetic multilayer is found to depend strongly on the

relative magnetization orientations of the various layers composing it. When the

magnetizations are parallel the measured resistance is small, and when the magne-

tizations are antiparallel the resistance is higher. This influence of magnetization

on current flow later led Slonczewski [3] and Berger [4] to suggest that there may

also be an influence of the current on the magnetization. The practical implica-

tions of this line of thoughthas led to great research interest in what are known

as spin-transfer phenomena, where spin-polarized currents are used to manipulate

the magnetic moments of ferromagnetic structures without the need for external

magnetic fields. Particularly, sweeping advances capable of employing the electron

“spin” to store data in switcheable magnetic states have proceeded to define the

backbone of today’s non-volatile data storage industry.

Spin-transfer torque (STT), occurs when a current of spin-polarized electrons

travels through a ferromagnetic material and deposits angular momentum via an

exchange interaction with the ferromagnet’s macroscopic magnetization (see Fig-

ure 1). Due to conservation of the total angular momentum in this interaction, the

net change in the angular momentum of the current flowing through the ferromag-

netic layer must be equal and opposite to the net change in angular momentum of

1

Figure 1: A schematic of the origin of the spin transfer torque. Layers A,B and Care non-magnetic layers.

the ferromagnetic layer itself along with any potential excitations of the underlying

lattice. Moreover, since angular momentum is proportional to magnetization, with

a conversion factor given by the gyromagnetic ratio γ, the absorption of angular

momentum results in a change in the magnetization direction of the ferromagnet.

Since its experimental confirmation, the study of STT has grown into a thriving

field of both experimental [5–10] and theoretical research [11–17]. Over the last

decade STT has shown tremendous practical potential as a data storage technology

via its ability to quickly read and write information into high density, non-volatile

memory structures by reversing the orientation of magnetic bits. It has also shown

promise in its ability to drive magnetic spin-torque oscillators (STO) at GHz fre-

quencies with narrow linewidths, which has a variety of uses [18–25], as well as

its ability to move magnetic domain walls, which may have uses in magnetic data

storage and logic devices [26–32].

From a practical standpoint, the typical material structures consist of a sandwich-

2

like layering (called a stack) of different nanometer thick magnetic metals separated

by a non-magnetic layer made of either a metallic conductor (spin valve) or a thin

insulator through which electrons can tunnel quantum mechanically (magnetic

tunnel junction). The various layers can generally differ in the coercive fields they

exhibit, with one layer being fabricated to possess a particularly weak coercive field

leading it to be “softer” in terms of its preferred magnetic orientation. This layer

is known as the free layer. The ability to tune, switch and control the magnetic

orientation of the free layer lies at the basis of all spin-transfer technologies. The

other layers in the stack, are built to retain a fixed magnetic orientation. If on one

hand they serve to spin-polarize the charge current traversing through the device,

on the other, they allow for the experimental determination of the free layer’s rela-

tive orientation via the GMR effect. As a result, fixed layers are also often referred

to as polarizer or reference layers.

Figure 2 presents an illustration of a spin valve/MTJ device. When electrons

are made to flow through the device from the bottom towards the top, the electrons

become spin-polarized by the reference layer. When this transmitted spin-current

encounters the free layer it induces a spin-torque on the free layer which tries

to align the free and fixed layer magnetizations. Alternatively, by applying the

current in the opposite direction (from top towards the bottom), the direction of

the spin-torque can be reversed. In the latter case, electrons which have a net

spin-polarization opposite to the direction of the fixed layer’s magnetization, will

be reflected off the fixed layer’s interface inducing a spin-torque on the free layer.

The resulting effect tries to anti-align the free and fixed layer magnetizations [16].

The spin-torque also allows for a number of dynamical regimes.

For data storage applications, the free layer serves as the fundamental unit

3

Figure 2: Illustration of a spin valve/MTJ device.

of memory, the magnetic bit, and operation involves using spin-torque to switch

the orientation of the free layer from one metastable state to another. This use of

STT was confirmed shortly after its theoretical prediction [33,34] with much of the

early research focusing on understanding the effects of magnetic anisotropy [35,36],

temperature [37, 38], and spin-current strength [39, 40] on the switching dynam-

ics. Another important application involves using spin-torque effects to generate

large-angle steady-state magnetic excitations in the free layer [41, 42]. Persistent

magnetization oscillations have been predicted to be used as wide-band tunable

RF oscillators [43] operating in the GHz to THz frequency range. To these ends, it

is of importance to understand the physics of current induced magnetic excitations

in the presence of noise with attention to the parameters that determine the tun-

ability of quality factors in these systems. Overall, whereas much of the research

into these phenomena has focussed on experimental and micromagnetic simula-

tions [44], a simple theoretical model for describing the richness of the observed

free layer dynamics has been lacking.

4

This thesis is organized as follows. In Chapter 1, we will outline the macrospin

model used to describe the free layer dynamics throughout this thesis. The model

approximates the free layer as an evolving, single domain, magnetic spin subject to

anisotropic, spin-torque and thermal effects. As this results in the simplest possible

reduction of the true microscopic degrees of freedom existing in reality, it forms

the basis for spin-torque driven magnetic phenomena. A proper understanding of

its dynamical regimes will represent a platform upon which the true complexity

of experimental phenomena can be quantified and tested. Due to the stochastic

nature of the model’s dynamics and the non-conservative quality of its dynamical

flow, Chapter 2 will then summarize a few basic results regarding the stochastic

modeling of physical systems. One of the main successes of the work underlying this

thesis has been the development of an efficient and massively parallelized code for

the numerical simulation of the dynamics derived in Chapter 1. It’s peculiarity lies

in the use of computer graphics cards (rather than CPU and computational cluters)

to perform all numerical integrations. This allowed study of the ensemble evolution

of the stochastic macrospin dynamics by focusing on the underlying Langevin

behavior rather than solving a partial differential equation for the evolution of the

probability density of magnetization states over time. Chapter 3 will outline both

the numerical methods used and the hardware employed for their execution.

Numerical methods in hand, the macrospin dynamics will be analyzed in depth

throughout Chapters 4 and 5. Chapter 4 will focus on the simplest anisotropic

scenario: the uniaxial macropin model. In this chapter we will derive the statis-

tical properties underlying thermally activated switching phenomena. The main

result of this chapter will be the resolution of a long standing debate regarding the

scaling of mean switching time with applied current. In particular we show that

5

the orientation of the polarized current relative to that of the macrospin’s prefer-

ential magnetization axis does not significantly influence the thermally activated

dynamics. In Chapter 5, we will develop an averaging theory based on time-scale

separation to significantly reduce the complexity of the macrospin’s dynamics. We

will derive a one-dimensional stochastic differential equation describing the evo-

lution and diffusion of the macrospin’s energy over its energy landscape. After

initially verifying that the results presented in Chapter 4 can be accurately repro-

duced with this technique, we will proceed to study the various dynamical regimes

accessible by a macrospin presenting a more general biaxial anisotropy energy

landscape. The main result of this chapter will consist in a surprisingly simple

set of physical conditions relating the spin valve’s spin-torque efficiency and the

magnetic anisotropy strengths to predict whether a macrospin should be expected

to switch or behave like a spin-torque oscillator. We argue that these details are

of importance to both researchers and industry professionals alike in establish-

ing guidelines for the development of practical devices. Finally, in Chapter 6, we

will briefly touch upon the properties of thermally driven dynamical systems sub-

ject to non-gradient flow. We do so to explore the structure and topology of the

macrospin’s random transitions between equilibrium orientations. We will argue

that the spin-torque’s non-gradient nature cannot be expected to alter the most

probable escape paths significantly.

6

Chapter 1

General Formalism

The precise derivation of the term magnet, which has now become the most

common one, is difficult to ascertain. Lucretius (99-55 BC) says it was called

“magnet” from the place from which it was obtained in the native hills of the

Magnesians. However, Pliny the Elder (23-79 AD) relates a more colorful legend,

as copied from the poet Nicander (second century B.C.), that the shepherd Magnes,

while guarding his flock on the slopes of Mount Ida, suddenly found the iron crook

of his staff clinging to a stone, which has become known after him as the ‘Magnes

stone’, or magnet. [45]

What is certain, and well known, is that the first quantitative investigations of

electric and magnetic phenomena were performed by Charles-Augustin de Coulomb

(1736-1806) [46]. Their unified theoretical description was then obtained by James

Clerk Maxwell (1831-1879) [47] whereby the dynamical behavior of electric and

magnetic fields were characterized on a macroscopic length scale. However, on an

atomic length scale, quantum theory must be employed for the proper microscopic

description of the physical properties of matter to emerge.

7

The investigation of magnetic processes in the fine ferromagnetic particles we

will be interested in is, however, on a somewhat intermediate level. The size of the

particles is on the order of nanometers or micrometers leading to the necessary con-

sideration of magnetic domain formation, for which Maxwell’s equations will not

be a sufficient description. For one, effects which originate from the atomic struc-

ture of solids have to also be take into account. Magnetocrystalline anisotropy, for

example, is caused by the crystal lattice: the periodical positions of the atoms com-

posing the solid. Furthermore, the exchange interaction between spin momentum

of electrons is a typical quantum mechanical effect.

However, employing full quantum mechanical models to study the properties of

fine ferromagnetic particles is impossible with today’s available computing powers.

One is forced to ‘neglect’ quantum mechanics, ignoring the atomic nature of mat-

ter, and consider a semi-classical approximation in a continuous medium. Such a

theory, was first introduced by Landau and Lifshitz [48] in an attempt to study the

structure of the domain wall between two anti parallel magnetic domains. William

Fuller Brown contributed several works to the discipline and is responsible for

naming the theory micromagnetics [49]. He wanted to emphasize the fact that this

theory should be capable of describing the details of the walls which separate mag-

netic domains as opposed to domain theory which instead considers the domains,

neglecting the walls in between.

In outlining the elements of the micromagnetic theory, this chapter will ulti-

mately attempt to channel the reader’s interest towards the behavior of the finer

magnetic particles, whose low nanometer size scale can be approximated further

as a single magnetic domain known as a Stoner-Wohlfarth model. The derived

physical principles governing the behavior of such a system will allow us to avoid

8

the full spatially extended detail formalized by the micromagnetic theory. This

‘reduced’ theory will, however, retain very rich dynamical properties which have

found fertile applications in describing recent experimental developments in the

field of spin-transfer phenomena.

1.1 Magnetic Energies

To attempt a thermodynamics theory of spontaneous magnetism, we must as-

sume that the magnetic moment M of a material together with any applied external

field H and temperature T , provide a complete characterization of the states of our

magnetic system. It is then known that the thermodynamic potential controlling

spontaneous transformations under fixed H and T is the Gibbs free energy G. If

H and µoM (with µ0 the permeability of free space) are conjugate work variables,

denoting by F the Helmholtz free energy, one has that

G(M,H, T ) = F − µ0M ·H, (1.1)

and thermodynamics equilibrium is reached when G attains its globally minimum

value [50]. In our magnetic system, the internal degrees of freedom, which give

rise to spontaneous transformations, can be represented by the magnetic moment

M itself. It will be assumed to vary during internal microscopic processes on a

characteristic relaxation time which is much shorter that the time scale over which

M varies significantly due to thermodynamic forces driving the system towards

global equilibrium. This implies that the system relaxes by passing through a

sequence of non equilibrium states, each characterized by a well-defined value of

M.

9

The equation of state of the system for the conjugate work variables H and M

will then be given by

H =1

µ0

(∂F

∂M

)T

, µ0M = −(∂G

∂H

)T

. (1.2)

So far we have not considered the dependence of the magnetization on spatial

coordinates. Magnetic materials, can exhibit very complex magnetization patterns.

We, thus, subdivide our ferromagnetic body into elementary volumes large enough

to contain a statistically significant number of atoms and yet small enough with

respect to the typical length scale over which the magnetization varies significantly.

The assumption which led to the definition of the Gibbs free energy implies, then,

that the relaxation time over which individual volumes reach thermal equilibrium

with respect to the given value of the magnetization M(r) is much shorter than

the time over which the system as a whole approaches equilibrium through time

changes of M(r).

There are five significant contributions to the Gibbs free energy of a ferromag-

netic body: the exchange energy, the dipolar energy, the magnetocrystalline/shape

anisotropy energy, the magnetostatic energy, and the Zeeman energy due to the

interaction with the external field H. [51] We omit the magnetoelastic energy,

which arises from magnetostriction for two reasons. When a ferromagnet is mag-

netized, it shrinks (or expands) along the direction of magnetization resulting in a

volume change and, with it, the saturation magnetization defined as the magnetic

moment per unit volume. In micromagnetics, however, it is a basic assumption

that the saturation magnetization remains constant. Secondly, a large part of the

10

magnetostrictive effects can be expressed in the same mathematical form as the

magnetocrystalline anisotropy. If the anisotropy constants are extracted from ex-

periment, as they often are, all magnetostrictive effects are virtually included for

free, thus allowing us to ignore its effects explicitly in our treatment.

1.1.1 Exchange Energy

The Coulomb interaction between electrons with overlapping orbits of neigh-

boring atoms can be very strong on short length scales. In ferromagnets (antifer-

romagnets), the minimum Coulomb energy will be achieved when their orbits are

anti-symmetric (symmetric) causing their spins to align (anti-align) due to Pauli’s

exclusion principle. This interaction can have a field strength as strong as 100

Tesla for neighboring spins. It’s effects are typically captured by introducing the

Heisenberg Hamiltonian of the exchange energy, which is usually written in the

form

Hex = −M∑i,j=1

JijSi · Sj, (1.3)

where Jij is the exchange integral (taken to be positive for ferromagnets and

negative for antiferromagnets), which can be calculated using quantum mechan-

ics [52,53]. It decreases rapidly with increasing distance between atoms with spin

operator S, allowing the sum to be taken only over nearest neighbors. Writing J for

Jij and replacing the spin operators by classical vectors, we rewrite the exchange

energy as

Eex = −JS2∑i,j|i 6=j

cosφi,j. (1.4)

11

Further developing the cosine into its Taylor series expansion for small φi,j and

noting that, in this limit |φi,j| ' |mi −mj| ' |(r · ∇)m| with ri the displacement

vector between neighboring lattice points (where we define from now on m =

M/MS), the exchange energy can be written as

Eex = JS2∑NN

φ2i,j = JS2

∑i

∑ri

[(r · ∇)m]2 . (1.5)

Changing the summation over i to an integral over the entire ferromagnetic

body, we finally get:

Eex =

∫V

A[(∇mx)

2 + (∇my)2 + (∇mz)

2] d3r. (1.6)

The exchange constant appearing is A = JS2c/a, where a is the distance between

nearest neighbors and c = 1, 2, 4 for a simple cubic, body centered cubic and

face centered cubic crystal structure. The typical length scale over which the

exchange energy acts, known as the exchange length, is intimately related to the

exchange constant: lex =√

2A/µ0MS. Within the range of the exchange length,

the exchange interaction dominates over magnetic body and spin directions can

be expected to not change significantly. For common magnetic materials such as

Co, Fe, Ni and Permalloy, the exchange lengths are all several nanometers in size.

Assuming that the size of the magnetic element we wish to model is on the order

of the exchange length scale, we can safely assume that the element will behave as

a single coherent domain with Eex = 0.

12

1.1.2 Dipolar Energy

As the exchange interaction weakens with the size of the magnetic sample, it

begins to compete with an opposing effect: the dipolar interaction. At large dis-

tances, atomic spins see themselves as magnetic dipoles. Contrary to the exchange,

the dipolar interaction will attempt to make spins align or anti-align with each

other depending on their relative position. The minimum energy configuration of

the magnetic texture will be such that a balance of short distance alignment and

long distance anti-alignment takes place. The magnetic sample can be expected

to form domains, where within each domain the spins are all oriented in the same

direction while the spin directions in different domains may be different.

Without burdening ourselves with a long derivation, we limit ourselves to stat-

ing the continuum representation of the dipolar energy, namely [54]:

Edipole =

∫V

1

2Hd(r) ·m(r)d3r (1.7)

where Hd is known as the demagnetization field. It is the sum of the dipolar

interaction between the magnetic moment at position r and all other positions r′.

It will necessarily depend on the shape and surface of the magnetic material:

Hd(r) = ∇(∫

V

∇ ·m(r)

|r− r′|d3r′ −

∮S

n(r′) ·m(r′)

|r− r′|d2r′

). (1.8)

The same consideration reserved for the exchange interaction applies for the

dipolar interaction also. If we wish to study nanometer size magnetic elements with

strong exchange interactions, the system is expected to behave monodomain-like

and dipolar effects can be disregarded.

13

1.1.3 Magnetocrystalline/Shape Anisotropy Energy

The Heisenberg “exchange” Hamiltonian is completely isotropic and its energy

levels do not depend on the direction in space in which the crystal element is magne-

tized. In the absence of other interaction terms, its magnetization direction would

not have a preferred orientation. Real magnets, however, are not isotropic, and

exhibit preferential magnetic orientations. The most common type of anisotropy

is the magnetocrystalline anisotropy, which is caused by the spin-orbit interaction

of the electrons. The structure of electron orbitals is, in fact, closely linked to the

crystallographic structure making the net magnetic moment prefer to align along

well-defined crystallographic axes. As such, a magnetic material will be magnetized

more easily along certain spatial directions than others. The magnetocrystalline

anisotropy is usually small compared to the exchange energy. The preferred di-

rection of magnetization, however, will be entirely determined by this anisotropy

along with any corrections due to the element’s shape: shape anisotropy.

Even though the spin-orbit interactions leading to the anisotropy can be eval-

uated from first principles [55], it is typically easier to consider it as arising phe-

nomenologically from the crystal symmetries (via power series expansions) and

extracting the relevant coefficients from experiment. In hexagonal crystals, the

anisotropy energy is a function of only one parameter, the angle between the mag-

netization and the preferential axis: known as the c-axis or easy-axis. Experiments

show that it is typically symmetric with respect to the basal plane of crystal (lead-

ing to vanishing odd power contributions of cos θ in the power series expansion

which are not allowed by time reversal invariance). To second order, we can then

write the anisotropy energy as

14

Eani =

∫V

[−K1 cos2 θ(r′) +K2 cos4 θ(r′)

]d3r′

=

∫V

[−K1m

2x(r′) +K2m

4x(r′)]

d3r′, (1.9)

where the easy-axis is chosen to be parallel to the x-axis. It is known from experi-

ment, that terms of higher order, and in most cases even K2, are small. If K1 < 0,

then the easy axis is in the basal plane of the crystal whereas if K1 > 0 the easy-

axis will be out of the plane of the crystal. The actual shape of the sample will

further influence the anisotropy energy due to dipolar interaction between spins

preferring that the magnetization reside in the crystal plane. This is known as

shape anisotropy. In some cases, the net effects of shape and magnetocrystalline

anisotropies can be written in a mathematical form as

Eanitot =

∫V

[N1m

2x(r) +N2m

2y(r) +N3m

2z(r)

]d3r. (1.10)

If we further suppose that the sample behave like a monodomain m(r) ≡ m

for r in the material, and that its magnetization is normalized |m| = 1, the total

anisotropy can be rewritten as:

Eanitot = K(Dm2z −m2

x), (1.11)

where K > 0 is the easy-axis (uniaxial) total anisotropy energy, and we further take

the convention of referring to the z-axis (the hard-axis) as that least energetically

favorable for the magnetization orientation. D is then the dimensionless ratio of

hard- and easy-axis anisotropies.

15

1.1.4 Magnetostatic and Zeeman Energy

Lastly, a magnetic potential will be induced whenever a magnetic body is in the

presence of an externally applied field H. This will attempt to align the sample’s

magnetization along it. Its contribution is known as the Zeeman energy, and can

be written as

EZ = −µ0MS

∫V

m ·H d3r. (1.12)

For a strongly magnetized body, even in the absence of external fields, a magneto-

static energy is generated by the presence, within the magnetic body, of an internal

field HD (known as the demagnetization field) which acts in the opposite direction

to the overall magnetization of the body itself. A strong magnetic body will typi-

cally develop magnetic domain structures in order to minimize this magnetostatic

energy. It’s form is derived from Maxwell’s equations and results in:

ED = −µ0MS

2

∫V

m ·H d3r, (1.13)

where for a monodomain magnet, this contribution is included in the shape anisotropy.

1.2 Macrospin Dynamics

Due to the linearity of Maxwell’s equations, the superposition principle allows

for the addition of all the terms contributing to the Gibbs free energy G. A

full micromagnetic treatment will generally employ all those discussed thus far.

However, as hinted in the first part of this chapter, this work will be interested in

modeling a Stoner-Wohlfarth monodomain magnetic body with magnetization M

16

of constant magnitude MS. The body is assumed to have a size lm along the y

direction, and size a in both the x and z directions. The total volume of the object

is then V = a2lm. The energy landscape experienced by M is generally described

by only two of the terms described: the Zeeman energy due to any externally

applied field Hext, and the magnetocrystalline/shape anisotropy energy with easy-

and hard-axes chosen along the x and z directions respectively. We take advantage

of the sample’s constant magnetization to define a normalized unit magnetization

vector m = M/MS.

The total energy landscape of the macrospin can then be written as:

E(m) = KPm2z −Km2

x −MSVm ·Hext, (1.14)

where KP = µ0M2SV and K = (1/2)µ0MSV HK are the hard- and easy-axis

anisotropy energies respectively with µ0HK the Stoner-Wohlfarth switching field

(in units of Tesla) fixing the coercivity of the magnetic model. In the spirit of the

previous sections, we rescale the magnetic energy by K, allowing us to simplify

the energy landscape of the model into

ε = E(m)/K =[Dm2

z −m2x − 2h ·m

], (1.15)

where we define h = Hext/HK and D ≡ KP/K = MS/HK as the ratio of the

anisotropies. Such an energy landscape, in the absence of applied fields, generally

selects stable magnetic configurations parallel and anti-parallel to x.

17

1.2.1 The Landau-Lifshitz (LL) equation

The torque ~τ acting on a physical system will relate directly to the rate of

change in angular momentum L of the system by Newton’s second law: ∂tL = ~τ .

The torque acting on a magnetic moment M subject to a magnetic field H is in

turn ~τ = M ×H. Since the magnetic moment of a point particle is linked to its

angular momentum by the gyromagnetic ratio1 M = −γL, we must have

ΓLL =dm

dt= −γm×H (1.16)

as the equation of motion for the element’s magnetic moment (known as the

Landau-Lifshitz (LL) equation).

If we are interested in the dynamical properties and time evolution of the

macrospin’s magnetization, we must consider its precession in an effective magnetic

field generated by (1.15). The effective interaction field Heff is then given by

Heff = − 1

µ0MSV∇mE(m) = −HK [Dmzz−mxx− h] . (1.17)

The dynamics describe an undamped precession of the magnetization vector m

about the effective field’s direction with Larmor frequency ωL = γµ0|Heff |. As

Heff is the gradient of the energy, the motion of the magnetization described by

(1.16) preserves the energy of the macrospin. As a result, under the effect of the

LL torque, the magnetization m travels along closed contours of constant energy

(see Figure 1.1).

1γ = gµB

~ ' 1.76× 1011 rads·T with g ' 2 is the Lande factor and µB the Bohr magneton.

18

Figure 1.1: Constant energy trajectories for D = 10. ε < 0 trajectories are shownin red whereas ε > 0 trajectories are shown in blue. Notice how two distinct basinsexist for positive and negative energy trajectories. The separatrix (correspondingto ε = 0) separating the different basins is shown in black.

19

1.2.2 Gilbert Damping

From experiments, however, it is known that changes in the magnetization

decay in finite time. For instance, in hysteresis curve measurements, applying a

sufficiently strong field to a magnetic material causes the magnetization to saturate

along the direction of the field. This is in stark contrast to the simple precessional

motion described by the LL torque (1.16) and indicates that dissipative effects con-

tribute to the magnetization dynamics. The basic mechanism driving the damping

is due to electron scattering [56, 57], although other mechanisms have also been

considered in the literature. These range from magnon-induced currents [58], to

magnon-magnon scattering [59], spin-orbit coupling [60], and spin-shot noise [61].

The simplest phenomenological way to incorporate these effects into the magneti-

zation dynamics was suggested by Gilbert [62]:

ΓD =α

MS

m× m (1.18)

with dimensionless Gilbert damping parameter α. This Gilbert damping (GD)

torque term in the equation of motion leads to the decay of the LL precessions and

aligns the magnetization along the effective magnetic field. The strength of the

Gilbert dissipation is proportional to the phenomenological dimensionless damping

constant α. In modern nanomagnetic devices its value is can be as small as α =

0.01 [33, 63, 64], allowing for hundreds of precessional cycles of the magnetization

about the effective field prior to equilibration (see Figure 1.2).

The resulting dynamics m = ΓLL + ΓD, arising from both damping and pre-

cession about an effective field, is known as the Landau-Lifshitz-Gilbert (LLG)

equation. They are sufficient to describe the evolution of a magnetization subject

20

Figure 1.2: Relaxation trajectory of the magnetization for a free layer under theeffects of Gilbert damping and the conservative LL torque [65].

to applied fields in real world scenarios. It is however not yet suited to capturing

effects due to thermal noise and spin-transfer torques. We now proceed to outline

both.

1.2.3 Slonczewski Spin-Torque

In 1996 both Slonczewski [3] and Berger [4] proposed a novel way of manipu-

lating the magnetization of nanoscale magnetic objects by passing a spin-polarized

current through the magnetic layer. The resulting effect has become known as

spin-torque (ST), or spin-transfer torque (STT), due to the spin-angular momen-

tum that is transferred from the spin-polarized current to the magnetic moment.

A simple explanation for this effect is as follows. Spin-polarized electrons entering

the free layer find themselves either aligned (with the amplitude ∝ cos(θ/2)), or

anti aligned (with the amplitude ∝ sin(θ/2)) with the free layer magnetization di-

rection, where θ is the angle between the current’s polarization and the macrospin’s

magnetization axes. At the interface, two main quantum processes contribute to

21

transferring ~ angular momentum from the itinerant electron to the macroscopic

magnetization of the free layer [66]. The first is the result of spin-dependent reflec-

tion/transmission due to differing energy bands of the ”spin-up” and ”spin-down”

conducting electrons across the interface. The other is generated by classical de-

phasing of spins in the ferromagnet due to their differing spin-dependent preces-

sional frequency around the local magnetization texture. When summed over all

Fermi surface electrons, these processes reduce the transverse component of the

transmitted and reflected spin currents to nearly zero for most systems of interest

within a short span of the ferromagnet/spacer layer interface. It follows that, to a

good approximation, the torque on the magnetization is proportional to the trans-

verse component of the incoming spin current and that maximal effects should be

most prominent in thin magnetic layers.

The corresponding non-conservative term in the macroscopic equation of mo-

tion takes the form

ΓS = −γj [m× (m× np)] , (1.19)

where spin-torque effects are assumed to be brought about by a flow of current

polarized in the direction np proportional to the spin-angular momentum deposited

per unit time j = (~/2e)ηJ/µ0MSHKd, with η = (J↑ − J↓)/(J↑ + J↓) the spin-

polarization factor of incident current J and thickness d of the magnetic free layer.

Here the spin-current is represented by an effective magnetic moment j = jnp

whose orientation and magnitude can depend on the actual state of the macrospin’s

magnetization. For the purpose of this work, though, we will assume it to be a

static quantity that acts on the magnetization dynamics. In passing, we note that

the self-induced magnetic field of the current is ignored here as the dimension a,

22

characterizing the thickness of the magnetic system, is considered to be smaller

than 1000A and spin-current effects are expected to dominate over any current

induced magnetic fields at this scale.

The form of the spin-torque effects thus state that the amount of angular mo-

mentum absorbed by the macrospin is proportional to the orthogonal component

of the spin-polarization axis np. As we will soon show, this exchange of angular

momentum can act to pump (subtract) energy into (from) the system depending

on the polarization of the spin-current in relation to the easy-axis x of the free

layer.

An additional “field like” torque may result from non-equilibrium spin-accumulation

in the free layer

ΓFST = γσj (m× np) , (1.20)

which causes the free layer magnetization to precess about the direction of the spin-

current polarization np, and σ represents the relative strength of the “field like”

torque compared to the STT. This spin-accumulation arises from the fact that the

transverse components of the spin-current persist with a characteristic relaxation

length λS, which may vary from only a few angstroms to several nanometers [67].

While some experimental [68] and theoretical [69] literature has suggested the

strength of this additional “field like” torque can be comparable in strength to the

STT given by (1.19), most have found σ 1 [70–74]; thus it will not be considered

in the remainder of this thesis.

23

1.2.4 Thermal Effects

The deterministic description of the previous sections is incomplete, especially

for small enough magnetic domains, due to the absence of thermal noise. Noise

must accompany the presence of Gilbert damping for the fluctuation-dissipation

theorem (FDT) to be satisfied. This was first realized by W.F. Brown [75]. Ther-

mal effects are included by considering uncorrelated fluctuations in the effective

interaction field: Heff → Heff + Hth. We model the stochastic contribution Hth by

specifying its correlation properties, namely:

〈Hth〉 = 0

〈Hth,i(t)Hth,k(t′)〉 = 2Cδi,kδ(t− t′). (1.21)

Thermal noise manifests itself as random fluctuation of the magnetization away

from the deterministic trajectory (see Figure 1.3). At low energies it acts to push

the magnetization away from the easy-axis, giving the free layer positive average

initial energy and an average initial deviation from the easy-axis. This effect will

be critical in some of the switching dynamics discussed.

We also note that current flow is a source of shot noise, which at low frequencies

acts like a white-noise source in much the same way as thermal noise [76]: j →

j+δjS, where δjS(t) is an isotropic Gaussian random component of the spin-current

with

〈δjS,i(t)δjS,k(t′)〉 = 2C(θ)δi,kδ(t− t′), (1.22)

24

Figure 1.3: Relaxation trajectory for the magnetization of a free layer under theeffects of Gilbert damping, the conservative LL torque, and stochastic thermalnoise [65].

where the noise correlator is dependent on the angle between the spin-polarization

axis np and the magnetization m. The noise correlator C(θ) is calculated in [14]

as

C(θ) =~jSF(θ)

2MSVcoth

(eΦ

2kBT

)(1.23)

where Φ is the voltage bias across the magnetic layer and jSF(θ) is the spin-flip

current given by

jSF(θ) =~Φ

4eVMS

[GP sin2

(θ

2

)+GAP cos2

(θ

2

)], (1.24)

where GP and GAP are the conductances for the electrons with spin parallel and

anti parallel to the free layer respectively. Since the parallel conductance is al-

ways larger than the anti-parallel conductance (GP > GAP ), the spin-shot noise is

strongest when the fixed and free layers are anti-aligned. It is therefore interesting

to understand when this additional source of noise plays a role.

25

For a magnetic layer coupled to unpolarized leads, the current induced noise

on the magnetization dynamics was found to be ΓL/ΓR(1+ΓL/ΓR)2

Φ [77], where Φ is the

voltage drop across the magnetic layer, while ΓL/ΓR is a dimensionless ratio char-

acterizing the coupling strength of the magnetic layer to the left (L) and right (R)

leads. Thus the noise is maximal (Φ/4) for perfectly symmetrical couplings, and

is smaller in the limit of highly asymmetric contacts. This basic behavior, and the

order of magnitude of the effect, is not likely to be modified by polarized leads. We

argue that the temperatures at which experiments have been performed, current

noise effects are not important. For an all metallic device, such as a spin-valve

nanopillar, the couplings are nearly symmetrical and, at the critical current, a

typical voltage drop across the magnetic layer is less than 10 mV or, equivalently,

1 K. For a magnetic tunnel junction device Φ can be ∼ 1 V. However, in this case

the coupling is asymmetric. One lead (L) forms a magnetic tunnel junction with

the nanomagnet, while the other (R) a metallic contact. This gives ΓR/ΓL > 104

and a relevant energy ∼ 1 K, again far lower than room temperature. Therefore,

since experiments are typically performed at room temperature where T = 300K,

where thermal noise dominates, only thermal noise is considered hereafter.

1.3 The Stochastic Landau-Lifshitz-Gilbert Slon-

czewski Equation (sLLGS)

The contribution to the magnetization dynamics brought about by thermal

noise and spin-torque will significantly alter the magnetization dynamics. The

original LLG equation m = ΓLL + ΓD used to describe the dynamics will be

modified into:

26

m =− γm× (Heff + Hth) + α (m× m)

− γjm× (m× np) , (1.25)

where the first term proportional to the gyromagnetic ratio γ is conservative and

precessional in nature, the second is dissipative and attributed to damping with

strength α, while the third represents a non-conservative forcing due to spin-torque

effects brought about by a flow of current polarized in the direction np. Having

assumed that the system magnetization is constant in magnitude will then imply

that m will evolve over the surface of a unit sphere and satisfy the condition that

m ·m = 0. This notion can be used to solve for m and rewrite the Landau-Lifshitz

dynamics in the more convenient form:

m =− γ′m× (Heff + Hth)− αγ′m× [m× (Heff + Hth)]

− γ′jm× (m× np) + γ′αjm× np, (1.26)

where γ′ = γ/(1 +α2) is known as the Gilbert ratio, Heff is given by (1.17) and we

have essentially rewritten the dynamics by solving explicitly for m. In passing, we

note that the manipulation just performed has given rise to the “field like” torque

γ′αjm× np, which acts like an applied external field pointed along the direction of

the spin-current polarization. Due to the magnitude of the leading factor α, this

torque is typically small and can be ignored for most purposes.

The macrospin’s geometry is fully determined by two angles: ω the angle be-

tween the spin-polarization np and x easy-axis and the azimuthal angle ψ char-

acterizing the extent to which np, x and z are coplanar (see Fig. 1). A tilted

27

spin-polarization axis allows modeling a spin-torque that results from more than

one “polarizing” layer in a spin-valve (or MTJ) stack or, more generally, a free

layer that has an easy-axis tilted relative to the spin-polarization axis. This is par-

ticularly relevant to experiments employing a perpendicular polarizer layer with an

in-plane magnetized spin-valve, consisting of a free and reference layer [63, 78–83]

(see Fig.1.4). The two fixed layer contributions lead to a net spin-torque which

can be formally thought to arise from a tilted spin polarizer [84–87]. In this case,

the effective spin-polarization will be tilted with respect to the easy-axis of the

free layer. Without loss of generality, taking for simplicity the ‘reference’ and ‘po-

larizer’ magnetization to be aligned with the x- and z- axes respectively (collinear

with the anisotropy axes), the net spin polarization axis can be written as:

j = jnp =j√

η2ref + η2

pol

(ηref x + ηpolz) , (1.27)

where ηref and ηpol are the spin polarization factors of the two magnetic layers

taken separately. The effective tilt angle ω can then be written in terms of the

ratio of the spin-torque efficiencies ω = atan(ηpol/ηref). The net polarization factor

appearing in (1.19) will then be η = 1.

For the main body of this work, externally applied magnetic fields will not be

considered. Considerations regarding their effects will be given in the conclusion.

With this in mind, we write the deterministic drift terms appearing in (1.26):

28

Figure 1.4: A typical trilayer spin-valve consisting of a free magnetic layer sand-wiched between an in plane magnetized reference layer and out-of-plane magnetizedpolarizer layer. The net spin-torque acting on the free layer will generally appeartilted away from the easy-axis of the magnet. The advantage of such devices liesin the ability to instantly torque the magnetic free layer without the need of athermal incubation time to destabilize the magnetization initially.

29

Figure 1.5: Easy-easy x and hard-axis z magnetic anisotropy directions are shownalong with spin-polarization direction np. The spin-polariation is tilted by an angleω with respect the magnetic easy axis.

mx = Dmzmy

+ α[(Inx +mx)(1−m2

x) + Inzmx(√

1− n2mz − nmy) +Dmxm2z

]my = −(D + 1)mxmz

+ α[mymz(Inz

√1− n2 +Dmz)−mxmy(Inx +mx) + Inzn(1−m2

y)]

mz = mxmy

− α[(Inz√

1− n2 +Dmz)(1−m2z) +mxmz(Inx +mx) + Inznmzmy

].

(1.28)

where nx = cosω, nz = sinω and n = sinψ. One of these equations is neces-

sarily redundant due to the |m| = 1 restraint.

In general, however, taking into account all stochastic contributions due to

thermal noise, the full magnetization dynamics will read:

30

mi = Ai(m) +Bik(m) Hth,k, (1.29)

where the expressions for the drift vector A(m) and diffusion matrix B(m) terms,

written in vectorial form, are

A(m) = m× heff − αm× (m× heff)

− αIm× (m× np)− α2Im× np, (1.30)

Bik(m) =√C[−εijkmj − α(mimk − δik)]. (1.31)

Where we define I = j/(αµ0MSHK), heff = Heff/HK and introduced the natu-

ral timescale τ = γ′µ0HKt. We will refer to this set of three, coupled, stochas-

tic differential equations describing the magnetization dynamics as the stochastic

Landau-Lifshitz-Gilbert-Slonczewski (sLLGS) equation. The notation chosen for

expressing the stochastic contribution ‘Hth,k’ implies that noise is to be inter-

preted in the sense of Stratonovich calculus [88] whose meaning will be presented

and justified in the following chapter.

31

Chapter 2

Stochastic Calculus

As we have seen in the previous chapter, the effect of thermal fluctuations can

be introduced in the macrospin dynamics by adding a random fluctuation field

to the effective magnetic field. A typical magnetization trajectory can then be

obtained by integrating the equations of motion. In this chapter we discuss the

mathematical framework under which one can model numerically a differential

equation with stochastic contributions.

2.1 Gaussian Noise

When one considers the effect of temperature on a dynamical system, it is as-

sumed that the resulting perturbations are of very high frequency. “Very high”

means that the timescale of the fluctuations is much smaller than the typical

timescale of the deterministic portion of the dynamics. In our case that the fre-

quency is expected to be well above the typical precession frequency of the mag-

netization vector. Thus, the fluctuating field used to simulate thermal activation

is chosen to be represented by a stochastic process. Furthermore, it is assumed

32

to be Gaussian white noise because the fluctuations emerge from the interaction

of the magnetization with a large number of independent microscopic degrees of

freedom, themselves subject to equivalent stochastic processes: phonons, conduct-

ing electrons, nuclear spins, etc. [89] As a result of the central limit theorem, the

fluctuation field is Gaussian distributed.

A stochastic process η(t) is called Gaussian white noise [90], if its two time

covariance is delta function correlated:

〈η(t)〉 = 0 (2.1)

〈η(t)η(t′)〉 = 2Cδ(t− t′) (2.2)

The Fourier transform of the covariance function can be shown to be indepen-

dent of frequency

F (ω) =

∫ds〈η(t)η(t+ s)〉 exp(iωs)

= 2C

∫dsδ(s) exp(iωs)

= 2C (2.3)

due to the absence of a correlation time.

2.2 Stochastic Differential Equations

Let us consider, for simplicity, a 1D stochastic differential equation (also called

Langevin equation or SDE) with additive noise:

33

x(t) = A(x(t), t) + η(t). (2.4)

The first term appearing in the equation is the deterministic drift term and its

effect on the dynamics of x(t) are perturbed by the noisy diffusion term η(t) which

is a Gaussian random variable.

The increase dx of the dynamical variable over an infinitesimal time step will

then be (to first order)

dx(t) = A(x(t), t

)dt+ dW (t) (2.5)

where we define

dW (t) =

∫ t+dt

t

dt′η(t′). (2.6)

Interpreting the above integral as the limit of a sum, dW will also be a Gaussian

random variable due to it being a sum of Gaussian random variables. Thus we must

have 〈dW (t)〉 = 0 and, employing the covariance properties of η(t), the variance

of dW will be

〈dW (t)2〉 =

∫ t+dt

t

dt1

∫ t+dt

t

dt2〈η(t1)η(t2)〉

=

∫ t+dt

t

dt1

∫ t+dt

t

dt22Cδ(t1 − t2)

= 2Cdt. (2.7)

If intervals [t, t+dt] and [t′, t′+dt] were to not overlap, we must have 〈dW (t)dW (t′)〉 =

0.

A first basic result of stochastic calculus should now be apparent to the reader.

34

The second moment of dW (t) is linear in dt. On the other hand, dW (t) is only of

order√

dt, which can be stated clearly by writing

dW (t) =√

2Cη(t)√

dt. (2.8)

In mathematics, dW (t) is known as a Wiener process and, by slight abuse of nota-

tion, stochastic differential equations such as (2.5) are often conveniently written

in the following form:

x(t) = A(x(t), t

)+ W (t). (2.9)

2.3 Interpretation of Stochastic Integrals

Let us now complicate the matter further by considering a scenario with mul-

tiplicative noise as opposed to simple additive noise:

x(t) = A(x(t), t

)+B

(x(t), t

)η(t), (2.10)

where from now on, for notational clarity, we assume that the variance of η(t) has

been absorbed inside B(x(t), t

)such that 〈η(t)η(t′)〉 = δ(t − t′). The increment

dx over a short time interval dt will then be:

dx(t) =

∫ t+dt

t

dt′A(x(t′), t′

)+

∫ t+dt

t

dt′B(x(t′), t′

)η(t′). (2.11)

If B(x(t), t) and η(t) were both continuous, the second integral term could be

shown to equal B(x(t), t)η(t)dt in the limit of small dt according to the first inte-

gral mean-value theorem. The noise η(t), however, has been chosen by definition

35

to represent a stochastic process and is not continuous. It is therefore not clear at

what time ti ∈ [t, t+ dt] the dynamical variable x(ti), appearing in the multiplica-

tive factor B(x(ti), t), is to be computed. Choosing a convention for computing∫ t+dt

tdt′B(x(t′), t′)η(t′) amounts to selecting a specific type of calculus that one

wishes to work in, resulting in a stochastic integral which can generally be written

as

∫ t+dt

t

dt′B(x(t′), t′)η(t′) = B(x(t) + αdx(t), t

) ∫ t+dt

t

dt′η(t′), (2.12)

with α ∈ [0, 1]. The integral of η(t) can now again be interpreted as defining a

Wiener process dW (t) allowing us to write (2.11) implicitly as:

dx = A(x+ αdx, t

)dt+B

(x+ αdx, t

)dW (t), (2.13)

where from now one we drop the explicit temporal dependence of x(t) for notational

convenience. Expanding the rhs to first order in dx(t) will then result in

dx = A(x, t)dt+ α∂xA(x, t)dtdx

+B(x, t)dW (t) + α∂xB(x, t)dxdW (t)

+O(dx2) (2.14)

which, upon substituting (2.13) back in to obtain an expansion in powers of dt one

finds

36

dx = A(x, t)dt+B(x, t)η(t)√

dt

+ αB(x, t)∂xB(x, t)η2(t)dt

+O(dt3/2)

=[A(x, t) + αB(x, t)∂xB(x, t)η2(t)

]dt+B(x, t)η(t)

√dt, (2.15)

where we have once again used (2.8) to express the Wiener process in terms of the

time interval explicitely. The above equation shows how the particular choice of α

will generally result in an additional drift term, which contains both α and η2(t).

The latter can be replaced by 1 for terms up to the order of dt. The increment dx

must then satisfy the two following noise averaged statistics:

〈dx〉 = [A(x, t) + αB(x, t)∂xB(x, t)] dt (2.16)

〈dx2〉 = B2(x, t)dt (2.17)

In particular, setting α = 0, we get

dx = A(x, t)dt+B(x, t)η(t)√

dt, (2.18)

known as the Ito interpretation of the stochastic differential equation. This is often

indicated by writing (2.10) in the form

x(t) = A(x, t) +B(x, t) · W. (2.19)

Setting α = 1/2, we get

37

dx =

[A(x, t) +

1

2B(x, t)∂xB(x, t)

]dt+B(x, t)η(t)

√dt, (2.20)

known as the Stratonovich interpretation of the stochastic differential equation.

This is often indicated by writing (2.10) in the form

x(t) = A(x, t) +B(x, t) W. (2.21)

We observe at once that Ito and Stratonovich interpretations differ from each

other only through the additional noise induced drift term:

1

2B(x, t)∂xB(x, t). (2.22)

This implies that one is free to switch from one calculus convention to the

other simply by taking into account the corresponding diffusion drift term. As

an example, to convert a Stratonovich SDE into its corresponding Ito form, one

simply writes:

x = A(x, t) +B(x, t) W

= A(x, t) +1

2B(x, t)∂xB(x, t) +B(x, t) · W. (2.23)

Due to the different drift terms, the two interpretations can be expected to yield

different dynamical properties. In the mathematics community, the Ito calculus

is most commonly used due to its conceptual simplicity arising from the property

that noise increments∫ t+dt

tdsη(s) and x(t) are statistically independent as implied

by (2.12): 〈B(x, t)η(t)〉 = 0 [91]. On the other hand, in the physics community,

38

the Stratonovich interpretation is preferred due to its deeper physical origin. Since

the noise term in (2.10) models, in a coarse-grained sense, the effect of microscopic

degrees of freedom that have finite (albeit short) correlation times τ such that

〈η(t)η(t′)〉 ∝ 2Cτ exp[−|t− t′|/τ ], (2.24)

this term should be physcally interpreted as the limit in which these correlation

times go to zero. The Wong-Zakai theorem [92] then says that in the zero cor-

relation time limit, the colored noise becomes white noise and we obtain the

Stratonovich interpretation of a SDE driven by white noise.

2.4 The Fokker-Planck Equation

The net effect of a dynamic driven by (2.10) will consist in a deterministic flow

due to A(x)1 accompanied by a random diffusion in configuration space due to

stochastic contributions. Since no two trajectories can be expected to be identical,

it is often of interest to know the probability density of finding the dynamical sys-

tem in a particular dynamical state at a given moment in time: ρ(x, t). Assuming

that the system is initially identified with a precise initial condition x(t0) = x0,

one must then have ρ(x0, t0) = δ(x − x0). We now proceed to derive an equation

governing the evolution of ρ(x, t) as a result of an underlying stochastic dynamic

driving the evolution of x(t).

We start by considering the continuity equation for ρ(x, t) expressed in integral

form

1From now on we will omit the explicit time dependence unless necessary.

39

ρ(x, t+ dt) =

∫dx′ρ(x, t+ dt|x′t)ρ(x′, t), (2.25)

where ρ(x, t+ dt|x′t) is the conditional probability distribution of x at time t+ dt

given that it was x′ at time t. It is defined by:

ρ(x, t+ dt|x′t) = 〈δ [x(t+ dt)− x]〉x′t, (2.26)

where the average is over the random noise η(t) and x(t + dt) − x(t) = dx(t) is

determined, to first order, by (2.16). Taylor expanding the conditional probability

around x0 yields:

ρ(x, t+ dt|x′t) = δ(x− x0)− 〈dx(t)〉∂xδ(x− x0)

+1

2〈dx2(t)〉∂2

xδ(x− x0) + ... (2.27)

Using then equation (2.25), we obtain

∂tρ(x, t) = ρ(x, t+ dt)− ρ(x, t)

= ∂x [−A(x)− αB(x)∂xB(x)] ρ(x, t) +1

2∂2x

[B2(x)ρ(x, t)

]= −∂x

[A(x) + (α− 1)B(x)∂xB(x)− 1

2B2(x)∂x

]ρ(x, t), (2.28)

where we show the effects of the calculus α explicitly. This is generally known as

the Fokker-Planck (FP) equation although many mathematicians refer to it also

as the Kolmogorov forward equation [93].

The above procedure can be repeated identically (albeit with a bit more alge-

40

bra) for a multi dimensional SDE:

x = A(x) + B(x) · W, (2.29)

where the dynamical state and deterministic drift are now represented by vectors,

and the stochastic effects are due to a stochastic vector W and state dependent

variance matrix B which need not be square. In writing down the above equation

only we have used “·” to signify simply matrix-vector multiplication instead of a

particular choice of stochastic calculus. The corresponding FP equation for the

evolution of the probability density can then be written as:

∂tρ(x, t) = −∇x ·

G ·[G−1 ·

(A + (α− 1)

(B · ∇x

)· BT

)−∇x

]ρ(x, t)

,

(2.30)

where we have defined G ≡ B · BTas the product between B and its transpose. It

is relevant to note that the FP equation can be written as a standard continuity

equation ∂tρ(x, t) = ∇x · J(x, t) where J is the probability current:

J(x, t) = G ·[G−1 ·

(A + (α− 1)

(B · ∇x

)· BT

)−∇x

]ρ(x, t). (2.31)

2.5 Limits of Stochastic Modeling

The work discussed up to this point is premised on the notion that true physical

systems are expected to be approximated by a SDE similar to (2.10) or (2.29). It

is not immediately apparent though, whether the deterministic drift A(x) should

41

be identical to the physical force field driving the dynamics in the absence of

noise. This is particularly important for a model employing the Stratonovich

interpretation as we have shown that it leads to the appearance of an extra drift-

diffusion term to the deterministic dynamics. Often, for example, one may have

physical intuition as to how the noise may be affecting the system without having

a clear idea as to what the deterministic forces are [94,95]. In this section we will

consider two separate scenarios that may arise when modeling physical systems

with SDEs.

2.5.1 Gradient Systems

Gradient systems correspond to physical phenomena for which a well defined

energy landscape U(x) is known. Equilibrium statistical mechanics then imposes

that, at long times, the physical system’s statistical behavior should be described

by the equilibrium Boltzmann distribution ρ(x, t → ∞) = ρB(x) ∝ exp[−βU(x)]

with β = 1/kBT the inverse temperature. Another property of gradient systems

at equilibrium is that they must obey detailed balance [96]:

ρ(xf tf |x0t0)ρB(x0) = ρ(x0tf |xf t0)ρB(xf ), (2.32)

implying that all thermally induced transitions are just as likely as their reverse

process. The result of this is that, at equilibrium, the probability current J must

vanish identically. Plugging then the equilibrium Boltzmann distribution into

(2.31), drift and stochastic contributions must be such to satisfy:

− ∇xU(x)

kbT= G

−1 ·[A + (α− 1)

(B · ∇x

)· BT

]. (2.33)

42

The scenario is very simple to work out for systems with additive noise G ≡

B = 1, resulting in the drift term satisfying usual gradient dynamics

A = −∇xU/kBT. (2.34)

If, however, the same system is to be modeled with multiplicative noise, equilibrium

conditions are satisfied by the following conditions:

A = A− (α− 1)(B · ∇x

)· BT

(2.35)

G−1 · A = −∇xU/kBT, (2.36)

Implying that the specific form of the stochastic contributions must be known, or

at least suggested, in advance for the system of equations to be solved.

2.5.2 Non-Gradient Systems

Contrary to gradient systems, non-gradient systems do not admit an equi-

librium Boltzmann distribution ρB(x) ∝ exp[−βU(x)] in terms of a precise en-

ergy landscape. An equilibrium distribution can, however, still be mathematically

shown to exist in terms of an effective action or quasi-potential S(x) such that

ρeq(x) ∝ exp[−βS(x)], where the equilibrium stastical properties will not satisfy

detailed balance anymore. Violations of detailed balance will, in turn result in a

non-vanishing equilibrium probability current J which may be observed experimen-

tally in the form of Brownian vortices [97,98]. In this case there is no unambiguous

way to associate the drift A, effective action S(x) or stochastic variance B of the

43

Langevin equation to the experimentally observed probability current. There is

simply no physical intuition to guide the choice of an effective potential such that

limt→∞ρ ∝ exp[−S(x)/kT ] unless the knowledge of both the explicit form of noise

driving the system as well as the drift field are given.

The quasi-potential S(x) may be non-differentiable in certain regions of con-

figuration space leading to highly non-trivial, thermal dynamical behavior. A

typical outcome of this scenario is the appearance of caustics and soliton-like es-

cape trajectories [99–103]. This is of particular importance since knowledge of the

system’s quasi-potential is necessary to estimate escape rates within the framework

of Kramer’s transition-state theory [90,104].

2.6 Stochastic Macrospin Dynamics: Reprise

In developing the theoretical model for the macrospin’s dynamics, we derived

expressions for the drift vector A(m) and diffusion matrix B(m) terms appearing

in

mi = Ai(m) +Bik(m) Hth,k. (2.37)

Written in vectorial form they read:

A(m) = m× heff − αm× (m× heff)

− αIm× (m× np)− α2Im× np, (2.38)

Bik(m) =√C[−εijkmj − α(mimk − δik)], (2.39)

44

where now it should be clear what we mean by Stratonovich multiplicative noise

‘Hth,k’.

The precise noise correlation C is set by the fluctuation-dissipation theorem. In

fact, in the absence of applied currents, the macrospin model is a gradient system

expected to attain thermal equilibrium satisfying a Boltzmann distribution set by

the magnetization energy:

ρB(m) ∝ exp

(−E(m)

kBT

)(2.40)

An associated Fokker-Planck equation (first constructed by Brown [75]) de-

scribing the evolution of the probability density ρ(m, t) on the m-sphere can be

written, as already explained, in the form of a continuity equation for ρ(m, t)

∂ρ(m, t)

∂t= −∇ · J(m, t) (2.41)

with

J(m, t) ≡ Aρ(m, t)−∇m

[2CB · BT

ρ(m, t)]

(2.42)

the probability current (not to be confused with the spin-angular momentum de-

posited by the spin-polarized current discussed in (1.16))2. The divergence and

2The additional drift-diffusion term arising from the Stratonovich convention can be shownto vanish identically. One has (using Einstein summation):

(B · ∇x

)· B

T= Bik

∂Bjk∂mj

= −αmk [εijkmj + α(δik −mimk)]

= −α2(mim

2k − δikmk

)= 0, (2.43)

since m2k = |m| = 1

45

gradient are defined with respect to the magnetization components on the unit m-

sphere. Writing out the FP equation explicitly (in the absence of applied current)

leads to: 3

∂ρ

∂t= −∇m ·

[−m× heff − αm× (m× heff )

+ 2C(1 + α2)m× (m×∇m)]ρ. (2.45)

Plugging (2.40) into the FP equation and noting that:

∇m · (m× heff ) = 0 (2.46)

∇mρ = KkBT

heff ρ, (2.47)

Boltzmann equilibrium is achieved at long times (∂tρ→ 0) with diffusion constant

given by:

C =αkBT

2K(1 + α2)=

α

2(1 + α2)ξ, (2.48)

with ξ ≡ K/kBT the energy barrier height divided by the thermal energy. This

general relationship between variance of thermal effect and damping is known as

the Fluctuation-Dissipation theorem, originally formulated by Nyquist [105], and

later proven by Callen and Welton [106].

3 Given a Langevin equation in the form dX = A(t,X)dt+ B(t,X) · dW(t) the FP equationfor the probability distribution of state variable X, written in component form, is:

∂P

∂t= − ∂

∂Xi

[(Ai −DBjk

∂Bik∂Xj

−DBikBjk∂

∂Xj

)P

](2.44)

46

The validity of the stochastic modeling of the macrospin dynamics is thus

based on our ability to reproduce the Boltzmann equilibrium properties in the

absence of applied currents. The incorporation of non-vanishing current effects

alters an initially gradient-like physical system into a non-gradient one. As already

mentioned, one of the main results of this thesis is the development of a theory

capable of capturing steady-state limit cycle dynamics which imply that a non-

vanishing probability current exists in the macrospin dynamics at long times.

47

Chapter 3

Numerical Methods

The translation of a numerical integration scheme valid for deterministic dif-

ferential equations does not necessarily yield a proper algorithm in the stochastic

case. Depending on the selected deterministic scheme, its unconditional transla-

tion might converge to an Ito solution, to a Stratonovich solution, or to neither of

them. Even when a scheme is shown to converge, usually the order of convergence

is lower than that of its corresponding deterministic counterpart. This has to be

considered when choosing the size of the discretization time step.

We note that the macrospin dynamics introduced, rely on the saturation mag-

netization |m| = 1 remaining constant. This is a very important assumption when

modeling such a system. It in fact allows for the phase space of the dynamics

to be 2D due to the magnetization being constrained on the surface of a sphere

along with being an inertia-less physical system (the differential equations are first

order in time). By the Poincare-Bendixson theorem [107, 108], the deterministic

portion of the evolution will not exhibit any chaotic behavior. This fact is of great

importance when choosing to construct a numerical integration scheme capable of

48

capturing the long time behavior of the sLLGS equation.

The numerical solution of the Fokker-Planck equation (2.30) is a challenging

problem. Being a partial differential equation, the Fokker–Planck equation can be

solved analytically only in special cases. Various approaches have been explored

in the literature for obtaining numerical solutions. Suzuki’s scaling theory [109]

and normal mode analyses [110] have both proved useful for obtaining approx-

imate solutions. However, scaling theory is accurate only to a few percent for

intermediate times (i.e. those between the initial and equilibrium states) in the

case of a bistable system [111] and normal mode analyses may suffer from slow

convergence for general problems. A cumulant moment method has been used

succesfully by Desai and Zwanzig [112] for a nonlinear self-consistent dynamic

mean-field theory model [113]. The slow convergence of the cumulant hierarchy

was later observed in a study of a transient bimodality carried out by Brey, Casado

and Morillo [114]. Path-integral methods have been utilized by a number of au-

thors [115–117]. Wehner and Wolfer [118] have presented a practical formalism

that numerically evaluates the path integrals involving Onsager-Machlup func-

tionals and reduces errors to a few percent. Monte Carlo techniques [119] are

useful for providing information about certain properties of the system in terms

of the moments of the stochastic processes without the need for direct reference

to the probability density distribution. In the case where the entire distribution

function is required, direct approaches, such as those based on an eigenfunction

expansion [120, 121] or finite-difference methods [122–124], are frequently used.

The eigenfunction expansion method is applicable to a general class of linear prob-

lems. Through this approach, various spectral methods can be used to provide

extremely accurate solutions of the Fokker-Planck equation, albeit at the cost of

49

extensive computational time and resources. Furthermore, the finite-difference

method is known to lead often to stiff systems of ordinary differential equations.

All the methods listed typically require advanced parallel computation techniques

running on large computational clusters.

In this work, we will follow a different approach focusing on the direct numer-

ical integration of the underlying Langevin SDE (1.29). Stochastic simulation has

gained acceptance due to its straightforward implementation and robustness with

respect to different sorts of problems. The continuous increase of the efficiency of

available computer hardware has been acting in favor of stochastic simulation, mak-

ing it increasingly more popular. The recent evolution of computer architectures

towards multiprocessor and multicore platforms also resulted in improved perfor-

mance of stochastic simulation. Let us note that in the case of a low-dimensional

system, stochastic simulation often uses ensemble averaging to obtain the values

of observables, which in turn is an example of a so-called “embarrassingly parallel

problem” and it can directly benefit from a parallel architecture. In other cases,

mostly where a large number of interacting subsystems are investigated, the imple-

mentation of the problem on a parallel architecture is less trivial, but still possible.

The recent emergence of general-purpose computing techniques on graphics pro-

cessing units (GPUs) has been a breakthrough in computational science. Current

state of the art GPUs are now capable of performing computations at a rate of

about 5 TFLOPS (Trillion FLOating Point operations per Second) per single sili-

con chip. It must be stressed that 1 TFLOPS, as a performance level, was achieved

in 1996 only throught he use of huge and expensive supercomputers such as the

ASCI Red Supercomputer (which had a peak performance of 1.8 TFLOPS [125]).

The numerical simulations of SDEs can easily benefit from the parallel GPU ar-

50

chitecture. This however requires careful redesign of the algorithms along with a

particular consideration of their convergence properties.

In this chapter we present a practical introduction to solving SDEs numerically

with particular focus to applications involving NVIDIA GPUs using Compute Uni-

fied Device Architecture (CUDA) [126]. In the first two Sections we outline two

practical stochastic integration schemes and discuss their convergence properties

in Section 3. We will then conclude with a section on GPUs and outline code de-

ployment using the CUDA architecture. Unless otherwise stated, we shall consider

a 1D Ito process1 x = x(t), t0 ≤ t ≤ T satisfying a scalar stochastic differential

equation with multiplicative noise

dx(t) = A(x(t), t

)dt+B

(x(t), t

)dW (t) (3.1)

with the initial value x(t0) = x0.

3.1 Euler-Maruyama Scheme

For a given discretization t0 = τ0 < τ1 < · · · < τn < · · · < τN = T of the

time interval [t0, T ], a Euler-Maruyama [127] approximation is a continuous time

stochastic process y = y(t), t0 ≤ t ≤ T satisfying the iterative scheme

yn+1 = yn + A(yn, τn

)∆n +B

(yn, τn

)∆Wn, (3.2)

for n = 0, 1, 2, ..., N−1 with initial value y0 = x0, where yn = y(τn), ∆n = τn+1−τn

denotes the time discretization interval and ∆Wn = Wτn+1−τn is the increment of

the stochastic process. If the diffusion coefficient B ≡ 0, the stochastic iteration

1All schemes presented can be straightforwardly extended to the multidimensional case.

51

scheme (3.2) reduces to the deterministic Euler scheme for the ordinary differential

equation x = A(x, t). The random increments ∆Wn are independent Gaussian

random variables with zero mean 〈∆Wn〉 = 0 and variance 〈(∆Wn)2〉 = 2C(τn+1−

τn).

For the integration of the Langevin equation (1.29) with constant step size ∆t

the Euler-Maruyama scheme results in

mi = mi(t) + Ai(m, t

)∆t+Bik

(m, t

)∆Wk (3.3)

with 〈∆Wk〉 = 0 and 〈∆Wk∆Wl〉 = 2Cδkl∆t. C is the variance of the thermal

field as derived in (2.48) through the Fluctuation-Dissipation theorem.

In the context of Stratonovich stochatic calculus, the deterministic drift has to

be augmented by a noise induced drift-diffusion term (2.22) which gives

mi(t+ ∆t) = mi(t) +

[Ai(m, t

)+ 2C

1

2Bjk

∂Bik

∂mj

]∆t+Bik

(m, t

)∆Wk. (3.4)

3.2 Heun Scheme

The improved Euler-Maruyama or Heun scheme [89] is an example of a predictor-

corrector method. The predictor is given by a simple Euler-Maruyama type inte-

gration. If we consider the Langevin equation (1.29), the predictor is

mi = mi(t) + A(m, t

)∆t+Bik

(m, t

)∆Wk, (3.5)

52

where ∆t is again the discretization time step and ∆Wk are Gaussian random

numbers, whose first two moments are 〈∆Wk〉 = 0 and 〈∆Wk∆Wl〉 = 2Cδkl∆t.

The Heun scheme is then given by

mi(t+ ∆t) = mi(t) +1

2

[Ai(m, t+ ∆t

)+ Ai

(m, t

)]∆t

+1

2

[Bik

(m, t+ ∆t

)+Bik

(m, t

)]∆Wk. (3.6)

3.3 Strong and Weak Convergence of a Numeri-

cal Scheme

The definition of convergence is similar to the concept for ordinary differential

equation solvers, aside from the differences caused by the fact that a solution to a

SDE is a stochastic process, and each computed trajectory is only one realization

of that process. Each computed solution path y(t), using the Euler-Maruyama

scheme for example, gives a random value at T , so that y(T ) is a random variable

as well. The difference between the values of the Ito process and its numerical

approximation at time T , e(T ) = x(T ) − y(T ), is therefore a random variable.

We will say that a time discrete approximation y∆t with maximum step size ∆t

converges strongly to x at time T if

lim∆t→0|〈x(T )− y∆t(T )〉| = 0. (3.7)

We say that a solver converges strongly with order γ > 0 if the error in the moments

is γth order in the step size, or

53

|〈x(T )− y∆t(T )〉| = O (∆tγ) (3.8)

for sufficiently small step size ∆t. Although the Euler method for ordinary differ-

ential equations has order 1, the strong order for the stochastic Euler-Maruyama

scheme for stochastic differential equations is only 1/2 [128].

Strong convergence allows accurate approximations to be computed on an in-

dividual basis. For some applications, such detailed pathwise information may be

required. In other cases, however, the goal is to ascertain the probability distribu-

tion of the solution x(T ), and single realizations are not of primary interest.

Weak solvers seek to fill this need. They can be simpler than corresponding

strong methods, since their goal is to replicate the probability distribution only.

In this respect, we will say that a discrete time approximation y∆t with maximum

step size ∆t converges weakly to x at time T if

lim∆t→0|〈f(x(T )

)− f

(y∆t(T )

)〉| = 0 (3.9)

for all polynomials f(x). According to this definition, all moments converge as

∆t → 0. If the stochastic contributions to the Langevin equation are zero and

the initial value is deterministic, the definition agrees with the strong convergence

definition, and the usual ordinary differential equation definition.

Weakly convergent methods can also be assigned an order of convergence. We

say that a solver converges weakly with order γ > 0 if the error in the moments is

γth order in the step size, or

|〈f (x(T ))− f(y∆t(T )

)〉| = O (∆tγ) (3.10)

54

for sufficiently small step size ∆t.

In general, the rates of weak and strong convergence do not agree. Unlike

the case of ordinary differential equations, where the Euler method has order 1,

the Euler-Maruyama method for SDEs has strong order 1/2. However, Euler-

Maruyama is guaranteed to converge weakly with order 1. If the drift and diffusion

coefficients are almost constant, the Euler-Maruyama scheme gives good numerical

results. In practice, however, this is rarely the case and then the results can become

very poor. Therefore, the use of higher integration schemes is recommended.

The stochastic Heun scheme, on the other hand, converges with strong order

1 and weak order 2 to the solution of the general system of Langevin equations.

There are two main reasons for the choice of the Heun scheme for the numerical

integration of the stochastic sLLGS equation. The first, is that the Heun scheme

can be shown to converge towards Stratonovich solutions of the stochastic differ-

ential equations without alterations to the deterministic drift term. Secondly, the

deterministic part of the differential equations is integrated with a weak second

order accuracy in ∆t, which renders the Heun scheme numerically more stable

than Euler type schemes. In the remainder of this work, we will be interested

solely in weak order results (averages, variances, mean first passage times, etc.)

and, as such, the Heun scheme will prove sufficient for these tasks. Specifically,

we will numerically solve the dynamical macrospin equation (1.29) employing the

fluctuation-dissipation theorem result (2.48) to compute the strength of the ran-

dom kicks perturbing the system. Unless stated otherwise, statistics are gathered

from an ensemble of 5120 events with a natural integration stepsize of ∆τ = 0.01.2

For concreteness, we set the Landau damping constant α = 0.04. Different barrier

2Corresponding to a physical step size ∆t = γ′HK∆τ ' 10−12 − 10−11s depending on thevalue of HK .

55

heights were explored although the main results are shown for a barrier height of

ξ = 80 (unless stated otherwise) which, for experimentally relevant results, corre-

sponds to room temperature [81, 129,130]. To explore the simulations out to long

time regimes, events were simulated in parallel on NVidia Tesla C2050, M2070 and

Quadro FX 580 graphics cards.

3.4 GPU: The CUDA Environment

Driven by the insatiable market demand for realtime, high-definition 3D graph-

ics, the programmable Graphic Processor Unit or GPU has evolved into a highly

parallel, multithreaded, manycore processor with tremendous computational horse-

power and very high memory bandwidth, as illustrated by Figures 3.1 and 3.2.

The reason behind the discrepancy in floating-point capability between the CPU

and the GPU is that the GPU is specialized for compute-intensive, highly parallel

computation and therefore designed such that more transistors are devoted to data

processing rather than data caching and flow control.

CUDA (Compute Unified Device Architecture) is the name of a general purpose

parallel computing architecture of modern NVIDIA GPUs. The name CUDA is

commonly used in a wider context as a means of referring not only to the hardware

architecture of the GPU, but also to the software components used to program that

hardware. In this sense, the CUDA environment also includes the NVIDIA CUDA

compiler and the system drivers and libraries for the graphics adapter.

From the hardware standpoint, CUDA is implemented by organizing the GPU

around the concept of a streaming multiprocessor (SM). A modern NVIDIA GPU

contains tens of multiprocessors. A multiprocessor consists of 8 scalar processors

56

Figure 3.1: Floating-Point Operations per Second for the CPU and GPU [131].

Figure 3.2: Memory Bandwidth for the CPU and GPU [131].

57

Figure 3.3: A schematic view of a CUDA streaming multiprocessor with 8 scalarprocessor cores [132].

58

(SPs), each capable of executing an independent thread (see Figure 3.3). The

multiprocessors have four types of on-chip memory:

• a set of 32-bit registers (local, one set per scalar processor),

• a limited amount of shared memory (48 kB for devices having Compute

Capability3 2.0, shared between all SPs in a MP),

• a constant cache (shared between SPs, read-only),

• a texture cache (shared between SPs, read-only).

The amount of on-chip memory is very limited in comparison to the total global

memory available on a graphics device (a few kilobytes vs hundreds of megabytes).

Its advantage lies in the access time, which is two orders of magnitude lower than

the global memory access time.

The CUDA programming model is based upon the concept of a kernel. A

kernel is a function that is executed multiple times in parallel, each instance run-

ning in a separate thread. The threads are organized into one-, two- or three-

dimensional blocks, which in turn are organized into one- or two-dimensional grids.

The blocks are completely independent of each other and can be executed in any

order. Threads within a block however are guaranteed to be run on a single mul-

tiprocessor. This makes it possible for them to synchronize and share information

efficiently using the on-chip memory of the SM.

In a device having Compute Capability 2.0 or higher, each multiprocessor is

capable of concurrently executing at least 1536 active threads [131]. In practice,

3The compute capability of a device is represented by a version number, also sometimes calledits “SM version”. This version number identifies the features supported by the GPU hardwareand is used by applications at runtime to determine which hardware features and/or instructionsare available on the hardware being accessed.

59

the number of concurrent threads per SM is also limited by the amount of shared

memory and it thus often does not reach the maximum allowed value. The CUDA

environment also includes a software stack. For all CUDA versions greater than

v2.1 (the latest version is v5.2), it consists of a hardware driver, system libraries

implementing the CUDA API, a CUDA C compiler and two higher level mathe-

matical libraries (CUBLAS and CUFFT). CUDA C is a simple extension of the C

programming language, which includes several new keywords and expressions that

make it possible to distinguish between host (i.e. CPU) and GPU functions and

data.

For the macrospin model described by (1.29), we use a single CUDA kernel,

which is responsible for advancing the system by a predefined number of timesteps

of size ∆t. We exploit the parallelization capabilities if the GPU by allowing each

path to be calculated in a separate thread. For CUDA devices, it makes sense to

keep the number of threads as large as possible. This enables the CUDA scheduler

to better utilize the available computational power by executing threads when

other ones are waiting for global memory transfers to be completed [131]. It also

ensures that the code will execute efficiently on new GPUs, which, by the Moore’s

law, are expected to be capable of simultaneously executing exponentially larger

numbers of threads.

Depending on the precise graphics card being used, initial testing was per-

formed to establish the optimal amount of macrospins that the kernel can accept

without losing performance speed. For the case of the Tesla C2050 (448 cores) it

was found that requesting more than 5120 independent realizations via one single

kernel call typically overflowed the GPU memory banks.

The majority of the investigations were typically interested in long time be-

60

havior of the macrospin dynamics. As such, to increase the efficiency of the code,

multiple timesteps were calculated in a single kernel invocation. The results of the

intermediate integration steps did not need to be copied to the host (CPU) mem-

ory. This makes it possible to limit the number of global memory accesses in the

CUDA threads thus, noticeably improving code performance. In fact, for typical

simulations exploring macrospin behavior out to the microsecond timescale (∼ 105

integration time steps), the time required for accessing the global memory (gener-

ally at kernel call and, successively, at kernel exit) was found to be comparable to

the actual time taken by the GPU to evolve the system numerically.

When the CUDA kernel is called, initial magnetizations configurations are

loaded from the global system memory and cached into local variables. All cal-

culations are then performed using these variables and at the end of the kernel

execution, their values are written back to the global memory where they can be

read and written to file. Often an initial kernal call was made with random initial

magnetizations and then evolved for 105 time steps to allow the ensemble elements

to thermalize.

Each path is associated with its own state of the random number generator

(RNG), guaranteeing independence of the noise terms between different threads.

The initial RNG seeds for each thread are chosen randomly using a standard

integer random number generator available on the host system. Since CUDA does

not provide any random number generation routines by default, to generate the

large number of necessary random numbers, we chose a proven combination [133] of

the three-component combined Tausworthe “taus88” [134] and the 32-bit “Quick

and Dirty” LCG [135]. The hybrid generator provides an overall period of around

2121 which is over 60 orders of magnitude larger than the total number of random

61

numbers needed for the production of this thesis. In our kernel, two pairs of

uniform variates are generated per time step and then transformed into Gaussian

variates using the Box–Muller transform. As stated in the previous section, the

integration is performed using a weak order 2 Heun scheme, which uses three of

the four Gaussian variates for a single time step (discarding the fourth).

62

Chapter 4

Uniaxial Macrospin Model

The role of spin-torque can be considered from an energy landscape point of

view. This chapter, we will be dedicated to studyng a uniaxial model in which an

axially symmetric shape anisotropy leads to the presence of a hard-plane. As we

will see, this allows for a detailed exact treatment of the stochastic magnetization

dynamics which will serve as both the starting point for our theoretical develop-

ment as well as a benchmark for the stochastic numerical schemes employed. The

choice of a uniaxial sample corresponds to setting D = 0 or vanishing hard-axis

anisotropy energy (1.11). Such a system is typically realized by using a circu-

lar magnetic domain which suppresses the shape anisotropy of the magnet. The

magnetic energy of the monodomain is:

E(m) = −Km2x. (4.1)

In general, the change in the macrospin’s energy over time can be obtained

after some straightforward algebra starting from (1.26) and is found to be1:

1In deriving we neglect contributions due to the thermal field Hth

63

1

µ0MsV HK

E = − [αm× heff − I(αnp −m× np)] · (m× heff) . (4.2)

This expression shows how current pumps energy into the system. In the absence

of current, the damping dissipates energy and, as one would expect, the dynamical

flow is toward the minimum energy configuration (mx = 1 or mx = −1). The

sign preceding the current term allows the expression to become positive in certain

regions of magnetic configuration space. Furthermore, by averaging over constant

energy trajectories, one can construct an equivalent dynamical flow equation in

energy space. We will develop this approach in the next chapter and use it to

prove the appearance of stable limit cycles by considering which constant energy

trajectories lead to a canceling of the flow E = 0 in (4.2).

Starting from an initially stable magnetic state, spin-torque effects will tend

to drive the magnetization toward the current’s polarization axis (np). Once the

current is turned off, the projection of the magnetization vector along the uni-

axial anisotropy axis (mx) will determine which stable energy state (parallel or

anti-parallel to the x-axis) the magnetic system will relax to as long as the energy

rati ξ ≡ K/kBT is large enough to prevent random fluctuations from acciden-

tally reversing the magnetization on experimentally relevant timescales. As such,

switching dynamics are best studied by studying the x component of equation

(1.29), namely:

mx = α[(I cos(ω) +mx)(1−m2

x) + I sin(ω) cos(ψ)mx(mz − tan(ψ)my)]

+ α2I sin(ω)my +

√α

ξ(1−m2

x) W, (4.3)

where ω and ψ were defined in reference to Figure 1.4, and the multiple stochastic

64

contributions in (1.29) have been compounded by employing general rules for the

addition of random variables (see Appendix B for details). W is, as before, a stan-

dard mean zero, variance 1, Wiener process, and its prefactor explicitly expresses

the strength of the compounded stochastic effects. Whereas, in general, (4.3) is

not useful as it explicitly depends on the dynamics of both the mz and my compo-

nents of the magnetization, it is a convenient dynamical tool in the limit of small

ω where the dynamical equations for mx decouple completely from the other mag-

netization components thus reducing the complexity of (1.29) to a straightforward

1-D problem.

A crucial test of our numerical scheme will be to succesfully reproduce the

thermal equilibrium properties of the uniaxial model. We solve (1.29) directly by

initially assigning to the elements in our ensemble a random magnetization with

negative mx component and allowing the system to relax in the absence of applied

currents. We assume ξ large enough2 such that, before current affects the system,

thermalization will have only been achieved within the antiparallel energy well with

no states having had time to thermally switch to the parallel orientation on their

own. A typical histogram of thermalized magnetic orientations resulting from this

exercise is shown in Figure 4.1 and confirms proper functioning of the numerical

stochastic model by accurately reproducing the expected Boltzmann equilibrium

distribution. Once the system is properly thermalized, we can proceed to turn on

a current and allow the system to evolve for a fixed amount of time. Once this

time has passed, we can let the system relax again at zero current and study what

fraction of the ensemble has switched due to the thermally assisted spin-torque

effects.

2The assumption is valid as long as the mean first passage time for thermally diffusing acrossenergy wells is much larger than the experimental timescale.

65

Figure 4.1: Histogram distribution of mx after letting the magnetic system relax tothermal equilibrium (103 natural time units). The overlayed red dashed line is thetheoretical equilibrium Boltzmann distribution. In the inset we show a semilog-plot of the probability vs. m2

x dependency. As expected, the data scales linearlywith slope equal to ξ ≡ K/kBT = 80: the ratio between total anisotropy andthermal energy.

66

4.1 Collinear Spin-Torque Model

Having derived the necessary expressions for our uniaxial macrospin model’s

dynamics, it is useful to consider the following simplification. Let us take the

uniaxial anisotropy and spin-current axes to be collinear, namely, np ≡ x (or

ω = 0). In such a scenario, the stochastic LLG equation simplifies significantly. In

particular, (4.3) reduces to the simplified form:

mx = α(I +mx)(1−m2x) +

√α

ξ(1−m2

x) W. (4.4)

In this symmetric scenario, as anticipated, magnetization dynamics have been suc-

cesfully reduced to a straightforward 1-D problem. For a general value of I < 1,

the evolution of mx has two local minima and a saddle. The two stable configura-

tions are at mx = −1 and mx = 1, while the saddle is located at mx = −I. For

currents I > Ic = 1 there is only one stable minimum. Above the critical current,

spin torque pushes all magnetic configurations toward the mx = 1 state. This

regime is particularly important not just for its simplicity but also for its similar-

ity to the pure field switching model. The collinear spin-torque model is, in fact,

mathematically identical to a field switching model with applied field of intensity

I applied parallel to the uniaxial anisotropy axis of the magnetic system [109].

4.1.1 Collinear High Current Regime

In the high current regime I Ic we expect the deterministic dynamics to

dominate over thermal effects. We refer to this interchangeably also as ballistic

evolution. The determistic (drift) contribution of (4.4) can then be solved ana-

lytically given an initial configuration mx = −m0. The switching time τswitch will

67

simply be the time taken to get from some mx = −m0 < 0 to mx = 0 and reads:

τswitch(m0) =1

α

∫ 0

−m0

dm

(I +m)(1−m2)

=1

2α(I2 − 1)

I log

[1 +m0

1−m0

]− log

[1−m2

0

]− 2 log

[I

I −m0

]. (4.5)

Since the magnetic states are considered to be in thermal equilibrium before the

current is turned on, one should average the above result over the equilibrium

Boltzmann distribution in the starting well to obtain the average switching time

〈τswitch〉B. Such an initial distribution will be:

ρB(mx) =

√ξ exp[−ξ]F [√ξ]

exp[ξm2x], (4.6)

where F [x] = exp(−x2)∫ x

0exp(y2)dy is Dawson’s integral [158]. This expression

can be used to compute the average switching time numerically.

As the intensity of spin-currents becomes closer to Ic, thermal effects increas-

ingly contribute. Moreover, diffusion gradients add to the deterministic drift, which

can be shown explicitly by writing (4.4) in its equivalent Ito form. Doing so leads

us to a first correction of the ballistic dynamics due to thermal influences. The

x-component behavior then reads:

mx = α(I +mx)(1−m2x)−

α

2ξmx +

√α

ξ(1−m2

x)W (4.7)

The first term on the right hand side is still the ballistic flow that we have just

discussed. The second term is the diffusion-drift term discussed in Chapter 2. The

68

contribution of such a term generates a net motion away from the stable minima of

the ballistic equations as one expects to see under the influence of thermal effects.

Again, we can solve the drift dominated flow analytically to compute the switching

time. Including the effects due to the diffusion-drift term, we find:

τswitch(m0) =1

α

∫ 0

−m0

dm

(I +m)(1−m2)− (m/2ξ)

=1

α

∑j

log [1 + (m0/wj)]

3w2j + 2Iwj − (1− 1

2ξ)

(4.8)

Where the wj are the three zeros of the cubic equation w3+Iw2−(1− 12ξ

)w−I = 0.

As before, the average switching 〈τswitch〉B time will simply be given by averaging

numerically over the Boltzmann distribution ρB. In Figure 4.2, fits are shown

comparing theory to simulation data. As expected both expressions coincide in

the limit of very large currents (I Ic).

4.2 Tilted Spin-Torque Model

In the high current regime (I Ic), where np = x (i.e. the polarizer tilt is

aligned with the x-axis), the ballistic equation for mx was shown to decouple from

the other components, and the dynamics became one dimensional and determin-

istic. For the more general case where the uniaxial anisotropy axis may have any

tilt with respect to the x-axis, such a critical current is not as intuitively defined.

Unlike the collinear limit, a critical current, above which all magnetic states per-

ceive a net flow towards an increasing global mx > 0, does not exist. One can in

fact plot mx over the unit sphere to see what regions allow for an increasing and

69

Figure 4.2: Current amplitude vs. mean switching time. Blue line shows the fitof the ballistic limit to the numerical data (in blue crosses). Red line shows theimprovement obtained by including diffusion gradient terms. Times are shown inunits of (T · s) where T stands for Tesla: real time is obtained upon division byµ0HK .

70

Figure 4.3: mx: green > 0, red < 0 for applied current I = 5. The plane dissectingthe sphere is perpendicular to the uniaxial anistropy axis. Its intersection with thesphere selects the regions with highest uniaxial anisotropy energy.

decreasing projection as the current is changed. An example of this is shown in

Figure 4.3.

Unfortunately, regions characterizing mx < 0 flow can be shown to persist at

all currents. The approach is refined by requiring that on average, over constant-

energy precessional trajectories, the flow is toward the positive uniaxial anisotropy

axis: 〈mx〉 > 0 [35]. Such trajectories are found by solving the flow equations

with I = α = 0. Solutions correspond to circular librations about the uniaxial

anisotropy axis. The critical current is then redefined to be the minimum current

71

at which 〈mx〉 > 0 at all possible precessional energies. This is easily done and

results in:

I ≥ maxε

[−ε

cosω

]=

1

cos(ω)= Ic, (4.9)

thus allowing for a direct comparison of dynamical switching results between dif-

ferent angular configurations with polarizer tilt ω. In our discussion of (4.3) we

mentioned how in the general ω 6= 0 tilted polarizer case there is no way to reduce

the dimensionality of the full dynamical equations. In such a state, precessional

trajectories might allow for a magnetization state to temporarily transit through a

mx > 0, “switched” configuration, even though it might spend the majority of its

orbit in a mx < 0, “unswitched” configuration. This allows for a much richer mean

switching time behavior, especially for currents greater than the critical current,

as shown in Figure 4.4, and discussed more in depth later.

4.3 Thermally Activated Regime

For currents I < Ic, switching relies on thermal effects to stochastically push

the magnetization from one energy minimum to the other. It is of interest to un-

derstand how switching probabilities and switching times depend on temperature

and applied current. This is easily done in stochastic systems with gradient flow.

In such cases an energy landscape exists and a steady state probability distribution

can be constructed via Kramer’s theory for noise-induced escape from a potential

well [90] to derive approximate low-noise switching probabilities.

Unfortunately, though, spin-torque effects introduce a non-gradient term, and

the resulting LLG equation does not admit an energy landscape in the presence of

72

Figure 4.4: Mean switching time behavior for various angular tilts and currentslarger than the critical current obtained by numerically solving (1.29). Each set ofdata is rescaled by its critical current such that all data plotted has Ic = 1. Angulartilts are shown in the legend in units of π/36 such that the smallest angular tilt is0 and the largest is π/4. Times are shown in units of (T · s) where T stands forTesla: real time is obtained upon division by µ0HK .

73

applied currents. The collinear simplification, however, is an exception. As already

described, in the absence of a tilted polarizer the dynamics become effectively

one dimensional since the mx component decouples from the other magnetization

components. Consider then (4.4): because it is decoupled from the other degrees of

freedom, we can construct a corresponding one-component Focker-Planck equation.

The evolution in time of the distribution of mx is then:

∂tρ(mx, t) = L[ρ](mx, t),

where

L[f ] = −α∂mx[(mx + I)(1−m2

x)−1

2ξ(1−m2

x)∂mx

]f.

For high energy barriers and low currents, the switching events from one basin

to the other are expected to be rare. The probability of a double reversal should

be even smaller. We therefore model the magnetization reversal as a mean first

passage time (MFPT) problem with absorbing boundaries at the mx = −I saddle

point. The MFPT will then be given by the solution of the adjoint equation

(L†〈τ〉(mx) = −1) [90]:

α

2ξexp(−ξ(mx + I)2)∂mx

[(1−m2

x) exp(ξ(mx + I)2)]∂mxτ(mx) = −1

subject to the boundary condition 〈τ〉(0) = 0. This can be solved to give:

〈τ〉(mx) =2ξ

α

∫ 0

mx

duexp(−ξ(u+ I)2)

1− u2

∫ u

−1

ds exp(ξ(s+ I)2). (4.10)

The rightmost integral can be computed explicitly. Retaining only dominant terms,

74

the final integral can be computed by a saddlepoint approximation to give:

〈τ〉 ' 2

√π

α

exp(ξ(1− I)2)F (√ξ(1− I))

1− I2. (4.11)

Such a square exponential dependence has been previously derived by Taniguchi

and Imamura [136, 137] as well as Butler et al. [138], although a τ ∝ exp(ξ(1 −

I)) dependence, proposed elsewhere in the literature [27, 72, 139], has also been

successfully used to fit experimental data [129]3.

To decide between these experimental dependences, we fit the scaling behaviors

in Figure 4.5. The square exponential dependence fits the data better, confirming

analytical results. Furthermore, comparison of the asymptotic expression (4.11) to

the full theoretical prediction obtained by solving (4.10) numerically demonstrates

that even for mean switching times of the order 10−6 T · s, asymptoticity still is

not fully achieved.

All that remains is to consider the effects of angular tilt on the switching

properties in the thermally activated regime. Insight into this problem can be

obtained by invoking (4.3) again. For small values of α, the term in square brackets

is of leading order over the second ballistic term depending on my. This allows us,

in the small α regime, to neglect the second ballistic term altogether.

We now concentrate on the behavior of the term in square brackets. For sub-

critial currents, switching will depend on thermal activation for the most part.

We expect an initially anti-parallel configuration to not diffuse very far away from

its local energy minima. It will remain that way until a strong enough thermal

kick manages to drive it over the energy barrier. Because of this, the second term

3It is important to note that experiments at fixed temperature are by necessity performedover a limited range of I/Ic and thus cannot truly distinguish an exponent of 1 or 2

75

Figure 4.5: Mean switching time behavior in the sub-critical low current regimeobtained by numerically solving the full macrospin dynamics. Times are shownin units of (T · s) where T stands for Tesla: real time is obtained upon divisionby µ0HK . The red and green line are fits to the data with the functional form〈τ〉 = C exp(−ξ(1 − I)µ), where µ is the debated exponent (either 1 or 2) andC is deduced numerically. The red curve fits the numerical data asymptoticallybetter the green curve. The difference between the red line and (4.11) is thatour theoretical prediction includes a current dependent prefactor which was notfitted numerically. The differences between numerical data and (4.10) is due tonumerical inaccuracies out to such long time regimes. The differences between(4.10) and (4.11), on the other hand, quantify the reach of the crossover regime.

76

appearing in the square brackets will generally be close to zero as the macrospin

awaits thermal switching. To make the statement more precise, one can imagine

the magnetic state precessing many times before receiving a thermal kick strong

enough to take it over the saddle. The second term can then be averaged over a

constant energy trajectory where the average 〈mz〉 will vanish identically. Hence,

in the subcritical regime, (4.3) can be rewritten in the following approximate form:

mx ' α(I cos(ω) +mx)(1−m2x) +

√α

ξ(1−m2

x) W. (4.12)

This, is reminiscent of the 1D dynamics discussed in relation to the collinear limit,

and shown explicitly in (4.4). The only difference between the two is the substitu-

tion I → I cos(ω). In other words, the thermally activated dynamics are identical

for all angular tilts up to a rescaling by the critical current. We then expect that

the mean switching time dependences remain functionally identical to the collinear

case for all polarizer tilts. We have confirmed this by comparison with data from

our simulations, and the results are shown in Figure 4.6. As predicted, all mean

switching time data from different polarizer tilts collapse on top of each other after

a rescaling by each tilt’s critical current.

As already hinted in the introduction to the collinear model, a strong mathe-

matical analogy exists with the field switching model studied in the literature [140–

144]. By considering a macrospin model with external magnetic field applied

collinear to the uniaxial anisotropy axis one obtains a dynamical equation for

the magnetization vector analogous to (4.4) with the field strength in place of

the applied current. In fact, one can think of writing an effective energy land-

scape U(mx) = −K(m2x − 2Imx), in terms of which, the equilibrium Boltzmann

distribution has precisely the same form as that given by Brown [75]. The ther-

77

Figure 4.6: Mean switching time behavior in the sub-critical low current regimeobtained by numerically solving the macrospin dynamics. Various polarizer tiltsare compared by rescaling all data by the appropriate critical current value. Timesare shown in units of (T · s) where T stands for Tesla: real time is obtained upondivision by µ0HK .

78

mally activated behavior discussed in the literature also reproduces an exponent

2 scaling dependence such as what we have shown here. One must understand,

though, that spin torque effects are generally non conservative and it is only in

the collinear scenario that they may be interpreted in this way. Upon introduc-

ing an angular tilt between uniaxial and spin-polarization axes, the analogy with

the field switching model will break down (see (4.3)). It is interesting, though,

to quantify the crossover between the spin torque and field switched macrospin

model. Coffey [140, 141] has already discussed the effect of angular tilt between

anisotropy and applied switching field axes. We introduced noise in the macrospin

model by considering a random applied magnetic field. In (1.29), we showed the

full form of the dynamical equations. To write the dynamical equations for the

field switched model, we need to suppress current effects and simply introduce a

term identical to the stochastic contribution with the exception that now the ap-

plied field will not be random but fixed at a specific angular separation from the

uniaxial anisotropy axis. Writing the dynamical equations for the field switched

model is straightforward:

mx = α[(h cos(ω) +mx)(1−m2

x) + h sin(ω)(my − αmxmz)]

+

√α

ξ(1−m2

x) W,

where instead of an applied current I, the dynamics depend on an applied field

with strength h. Comparison with (4.3) shows how, in general, the two evolution

equations are very different from each other. In the thermally activated regime,

however, where one is able to average over constant energy trajectories due to their

timescales being much smaller than those required for actual diffusion or magnetic

79

torque (h hcrit), the second term in square brackets will be averaged away and

one is left with a stochastic evolution equation identical to (22). All thermally

activated switching will then again be functionally identical for all angular tilts up

to a trivial rescaling of the applied field.

In comparing our scaling relationships between current and mean switching

time with the previous literature, a subtle issue must be addressed. Results ob-

tained by Apalkov and Visscher [139] rely upon an initial averaging of the dy-

namics in energy space over constant energy trajectories (limit for small damping)

and only subsequently applying weak noise methods to extrapolate switching time

dependences. The small damping and weak noise limits are singular and the or-

der in which they are taken is important. Both limits radically alter the form of

the equations: whereas both limits suppress thermal effects, the first also severely

restrains the deterministic evolution of the magnetic system. Our approach con-

siders the weak noise limit and, only in discussing the effects of an angular tilt

between polarization and easy axes do we employ the small damping averaging

technique to obtain functional forms for the mean switching time. The switching

time data shown seems to justify, in this particular case, an interchangeability be-

tween these two limits. More generally, however, one should not expect the two

limits to commute.

4.4 Switching Time Probability Curves

Up until now, we have analyzed the main properties of spin-torque induced

switching dynamics by concentrating solely on the mean switching times. In exper-

iments, though, one generally constructs full probability curves. The probability

80

that a given magnetic particle has a switching time τswitch ≤ τ can be explicitly

written as:

P [τswitch ≤ τ ] =

∫ m(τ)

0

dxρB(x)

= exp[−ξ(1−m(τ)2)]F [√ξm(τ)]

F [√ξ]

, (4.13)

where m(τ) is the initial magnitization that is switched deterministically in time τ .

Once one has evaluated m(τ), the probability curve follows. Ideally, in the ballistic

regime, one would like to invert the ballistic equations. Unfortunately, though, the

solutions of such ballistic equations are generally transcendental and cannot be

inverted analytically. Even in the simpler collinear case, as can be seen from

equations (4.5) and (4.8), no analytical inversion is possible. One must instead

compute the inversion numerically4. Nonetheless, one can construct appropriate

analytical approximations by inverting the dominant terms in the expressions. In

the case of (4.5), for example, one has that for currents much larger than the

critical current:

τ(m) ' I

2α(I2 − 1)log[

1 +m

1−m] (4.14)

which can be inverted to give:

m(τ) = tanh[ατI2 − 1

I]. (4.15)

Plugging into expression (4.13) for the τ probability curve, one has:

4Easily achieved thanks to the monotonicity of their form.

81

P [τswitch ≤ τ ] = exp

[− ξ

cosh2[ατ I2−1I

]

]F [√ξ tanh[ατ I

2−1I

]]

F [√ξ]

, (4.16)

This expression can be truncated to a simpler form by noting that the leading

exponential term dominates over the ratio of Dawson functions. Furthermore, if

one considers the limit of large values for τ (or, analogously, I 1), the ‘cosh’ can

be also approximated by its leading exponential term. We are finally left with:

P [τswitch ≤ τ ] ' exp

[− ξ

cosh2[ατ I2−1I

]

]

∼ exp

[−4ξ exp[−2ατ

I2 − 1

I]

], (4.17)

which is very similar in form to what has already been reported and used for fitting

experimental data in the literature [35, 129].

In the low current regime, one constructs probability curves by considering the

mean switching time and modeling a purely thermal reversal as a decay process

with rate given by equation (4.11). The fraction of switched states then vary in

time as:

P (mx > 0) = 1− exp(−t/〈τ〉). (4.18)

Upon introducing a tilt polarizer, precessional effects can be seen directly on the

switching probability curves in the super-critical regime. One expects that in the

initial phases of switching, the fraction of switched states is sensitive to the time

at which the current is turned off. One may accidentally turn off the current

82

Figure 4.7: Influence of precessional orbits on transient switching as seen fromthe switching time probability curve in the supercritical current regime. The caseshown is that of an angular tilt of π/3 subject to a current intensity of 2.0 timesthe critical current. Data was gathered by numerically solving (1.29). The non-monotonicity in the probability curve shows the existence of transient switching.Times are shown in units of (T · s) where T stands for Tesla: real time is obtainedupon division by µ0HK .

during a moment of transient passage through the switched region along the pre-

cessional orbit. This was checked and verified from our numerical simulations (see

Figure 4.7). More generally, effects similar to the “waviness” seen in the mean

switching time curves can be seen in the probability curves as well, as the angle of

the polarizer tilt is allowed to vary (see Figure 4.8). Only a numerical solution of

the LLG equation can bring such subtleties to light.

83

Figure 4.8: Spin-torque induced switching time probability curves for various an-gular configurations of polarizer tilt (a sample normalized current of 10 was used)obtained by numerically solving (1.29). A log-log y-axis is used following (4.17) tomake the tails of the probability distributions visible. Times are shown in units of(T · s) where T stands for Tesla: real time is obtained upon division by µ0HK .

84

Chapter 5

Constant Energy Orbit-Averaged

Dynamics

Within the macrospin picture, current-induced steady-state motions appear

when the magnetization settles into a stable oscillatory trajectory that balances

the spin-torque and damping [145]. This oscillatory behavior will be characterized

by magnetization precession at a frequency associated with the element’s magnetic

anisotropy energy, which can tuned, for example, by modifying the element’s shape

(i.e. magnetic shape anisotropy) or magnetocrystalline anisotropies. Thermal noise

can, however, alter the frequency and amplitude of the motion as well as change the

conditions under which steady-state precession occurs. As a result, it is important

to know both how an applied current will influence the amplitude and frequency of

a stable magnetic oscillation and how thermal noise will perturb this configuration

by inducing amplitude and phase noise.

If amplitude and phase diffusion due to spin-torque and thermal noise effects

occur on a timescale much larger than that of magnetization precession, it becomes

85

possible to analyze the steady-state dynamics perturbatively [12]. In this case, the

magnetization dynamics will consist of a fast gyromagnetic precession whose am-

plitude slowly changes over time due to spin-torque and thermal effects. This

has successfully been used to study the dynamical and thermal stability of nano-

magnets subject to spin-polarized currents [146]. This separation of dynamical

timescales falls under the framework of multiscale analysis, which can be applied

in various ways.

Three different approaches have been proposed in the literature in the context

of spin-transfer. Apalkov and Visscher [139] employed an effective Fokker-Planck

(FP) equation, which described the diffusion of a macrospin’s energy under the

influence of both spin-transfer torque and thermal noise. This has been used to in-

terpret results on studies of thermally activated magnetic switching [129,147]. Kim,

Slavin and Tiberkevich [12, 148] studied the Landau-Lifshitz-Gilbert-Slonczewski

(LLGS) equation by noting its analogy to the van der Pol oscillator equation [149].

This resulted in an elegant treatment of the leading nonlinear effects governing the

oscillatory equilibrium steady-state dynamics of the spin-wave eigenmodes. The

approach [9] has had success in explaining the experimentally observed dependence

of the oscillator’s output power on bias current for spin-valves and magnetic tun-

nel junctions [8, 73, 150–152], as well as providing a framework for the extension

of multiscaling methods to spatially extended magnetic systems in which multiple

coupled spin-wave modes may be excited [153].

Finally, macrospin dynamics subject to thermal noise have been modeled using

a stochastic Langevin equation for the time evolution of the macrospin energy by

Newhall and Vanden-Eijnden [154] and in previous work by the Author [155]. This

allows for a reduction in complexity of the sLLGS equations to a 1D stochastic

86

differential equation. In this Chapter, stochastic energy space dynamics will be

developed and used to describe the full nonlinear dependence of mean switching

time on applied currents [156,157] for biaxial macrospin models (log τ ∝ (1−I)β(I))

as an analytic continuation of the uniaxial macrospin model. Recently, Dunn

and Kamenev have extended this approach to propose AC current-driven resonant

switching [65].

Furthermore, recent research on spin-torque oscillators has focused on the ex-

citation of stable in-plane (IP) and out-of-plane (OOP) precessions about the easy

and hard magnetic anisotropy axes of thin film nanomagnets with biaxial mag-

netic anisotropy. In deriving our reduced constant energy orbit-averaged (CEAO)

theory, we present a treatment of these precessional dynamics valid over a wide

range of parameters. We will focus particularly on the OOP dynamics and show

the conditions under which precessional motion about the hard axis occurs. The

oscillator behavior we find is reminiscent of that observed in experiments on a

spin-valve where spin-torque effects are due to the influence of both a perpendic-

ularly magnetized polarizer and in-plane magnetized reference layer [42] like the

one discussed in Chapter 2. The precessional dynamics are found to be stable at

room temperature and, as a result, have great potential for the development of

spin-torque nano-oscillators.

5.1 Constant Energy Orbit-Averaged (CEOA) dy-

namics

In the absence of damping and thermal noise, the dynamics (1.29) preserve the

macrospin’s energy which, expressed in dimensionless form, reads:

87

ε =E(m)

K= Dm2

z −m2x, (5.1)

The conservative trajectories come in two different types. For −1 < ε < 0 the

magnetization gyrates around the easy axis x and is said to be precessing “in-

plane” (IP). For 0 < ε < D, the magnetization precesses about the hard axis z

and is said to be precessing “out-of-plane” (OOP) (refer to Figure (1.1)). The

evolution of such trajectories can be described analytically by solving the sLLGS

equation in the absence of noise, damping and spin-transfer torque [162]:

m0x = −Dm0

zm0y

m0y = (D + 1)m0

zm0x

m0z = −m0

ym0x (5.2)

For IP trajectories [156] one has

m0x(t) = ±

√D − εD + 1

dn[√

D − εt, k2IP

](5.3)

m0y(t) =

√1 + ε sn

[√D − εt, k2

IP

](5.4)

m0z(t) =

√1 + ε

D + 1cn[√

D − εt, k2IP

], (5.5)

where k2IP ≡ D 1+ε

D−ε and sn[·], dn[·], cn[·] are Jacobi elliptic functions [158]. The

period of these trajectories as a function of energy can be expressed as a complete

88

elliptic integral of the first kind:

T (ε) =4√D − ε

∫ 1

0

dx√(1− x2)(1− k2

IPx2)

=4√D − ε

K(k2IP). (5.6)

The amplitudes of an orbit’s precession, projected onto the z-y plane, are1

Az(ε) =

√1 + ε

D + 1(5.7)

Ay(ε) =√

1 + ε (5.8)

Analogously, for OOP trajectories

m0x(t) =

√D − εD + 1

cn[√

D(1 + ε)t, k2OOP

](5.9)

m0y(t) =

√D − εD

sn[√

D(1 + ε)t, k2OOP

](5.10)

m0z(t) = ±

√1 + ε

D + 1dn[√

D(1 + ε)t, k2OOP

], (5.11)

with k2OOP ≡ D−ε

D(1+ε). The projected precession amplitudes in the x-y plane are:

Ay(ε) =

√D − εD

(5.12)

Ax(ε) =

√D − εD + 1

, (5.13)

and the period of the librations is given by:

1precession is around the x-axis

89

Figure 5.1: Orbital frequencies plotted as a function of ε for different D. To com-pare the results, the positive portion of ε axis has been rescaled by D. Frequencyis expressed in units of (GHz/T). Physical frequency is obtained upon multiplyingby µ0HK . The sharp minimum in the frequency is a result of the precessionalperiod diverging at ε = 0.

T (ε) =4√

D(1 + ε)

∫ 1

0

dx√(1− x2)(1− k2

OOPx2)

=4√

D(1 + ε)K(k2

OOP) (5.14)

A sample of these trajectories for positive and negative energies is shown in

Figure 1.1, and orbital frequency as a function of energy is plotted in Figure 5.1.

The unit magnetic sphere can be separated into four distinct basins, two corre-

sponding to ε < 0 dynamics and the others two to ε > 0. For large values of D

the ε > 0 OPP basin can lead to larger oscillatory resistance signals than ε < 0 IP

basin due to the larger precessional amplitudes (5.13).

Upon introducing the contributions of spin-torque, damping and thermal noise,

a macrospin’s dynamical evolution will deviate from a constant energy trajec-

90

tory. Applied currents can reorient the magnetization by pumping energy into the

magnetic system. We may then ask how the constant energy trajectories will be

perturbed. This can be expressed mathematically by computing how the magneti-

zation energy changes as a result of sLLGS evolution. Taking the time derivative

of (5.1), we write 2:

ε = 2 [Dmzmz −mxmx] (5.15)

as the dynamical evolution equation for the macrospin’s energy. Expressing the

time derivatives of the magnetization components in terms of the full stochastic

sLLGS dynamics by using (1.29), one obtains a stochastic evolution equation of

the form

ε = f(m) + g(m) W. (5.16)

We now consider qualitatively how the macrospin dynamics change if the

timescale for energy pumping/sinking, due to the collective effects of damping,

spin-torque and thermal noise, is much larger than the precessional period of the

conservative dynamics. In such a scenario, the full stochastic sLLGS dynamics

might be expected to follow constant energy trajectories fairly closely, with the

macrospin drifting slowly from one constant energy trajectory to the other. Av-

eraging the right hand side (RHS) of (5.16) over constant energy trajectories will

then lead to a single stochastic differential constant-energy orbit-averaged (CEOA)

equation for the evolution of the macrospin’s energy

2The chain rule for stochastic variables is unchanged if the multiplicative noise follows theStratonovich convention.

91

〈∂tε〉 = f(ε) + g(ε) Wε. (5.17)

This approach is justified when the energy drift over the period of a single conser-

vative orbit T (ε)ε is sufficiently small. As will be shown in detail in the next two

sections, due to the different structure of the constant energy orbits in the ε < 0

and ε > 0 basins, the averaging procedure must be performed separately in the

two basins using solutions (5.3) and (5.9) respectively.

Under such assumptions, thermal noise will influence the dynamics in two dis-

tinct ways. The first is by nudging the magnetization onto different energy orbits

(effectively diffusing over the macrospin’s energy landscape). The second, is by

perturbing the precessional phase of the magnetization along a given constant en-

ergy orbit. As such, (5.17) must be supplemented by an equation describing the

stochastic evolution of the dynamical phase. This can be written down by not-

ing that noise must influence energy and phase diffusion identically because it is

isotropic:

〈∂tχ〉 =2π

T (ε)+ g(ε) Wχ, (5.18)

where T (ε) is the period of the orbit at energy ε. We will distinguish between the

two independent noise terms Wε and Wχ by the fact that they act in orthogonal

directions: respectively away and along the constant energy orbit. Whereas (5.17)

will be shown to not depend explicitly on the phase χ, (5.18) will however depend

explicitly on the energy ε.

92

5.2 IP Dynamics

We proceed first by consider damping, applied current and thermal noise ef-

fects on an IP ε < 0 orbit. We must first average (5.16) over conservative negative

energy trajectories (5.3) (see Appendix A for more information). The qualita-

tive properties of such trajectories are better understood by parametrizing them

geometrically as follows. The energy landscape (5.1) determines the geometrical

conditions these trajectories must satisfy. For ε < 0 we have

Dεs2 − 1

εq2 = −1 (5.19)

s2 + q2 + p2 = 1. (5.20)

These are satisfied by the parametrization:

m0x(w) = ±

√|ε| cosh(s)

m0y(w) = ±

√D + |ε|D

√1− γ2

− cosh2(s)

m0z(w) =

√|ε|D

sinh(s),

γ2− =

|ε|(D + 1)

D + |ε|, (5.21)

with limits −acosh(1/γ−) < s < acosh(1/γ−).

We are now ready to average (5.15) over constant energy orbits. The averaging

procedure is simplified by noting that we are interested in switching behavior

starting from the ε < 0 basin and that in that basin the constant energy trajectories

are symmetric with respect to the x axis since they necessarily precess about the

macrospin’s easy axis. As such, the majority of terms obtained by inserting (1.28)

93

into (5.15) will average to zero. The remaining nonzero terms lead to the averaged

energy equation:

〈∂tε〉 = −2α[I cos(ω)(1 + ε)〈mx〉+ (D + 1)〈m2

x〉+ ε(D − ε)]

+ 2

√α(D + 1)

ξ

√〈m2

x〉+D − εD + 1

ε W (5.22)

where 〈·〉 implies averaging over constant energy orbits as described. Construc-

tion of the stochastic term requires averaging the variance of (5.15) following the

rules of additivity for Gaussian random variables (see Appendix B). It is seen

from (5.22) that the applied current factors into this equation only in the form

I cos(ω), where cos(ω) is the cosine of the angular tilt between easy and spin po-

larization axes. This leads to a more general form of the tilt scaling result obtained

in the previous chapter [159], namely that scalings between switching times and

applied current will be functionally identical independent of the angle of the in-

coming spin-polarized current. The only caveat here is that every current must

be rescaled by multiplying it by cos(ω). Analogously, the azimuthal tilt parame-

ter ψ does not appear at all in the energy-averaged equation, implying that only

coplanar setups between polarizer, easy and hard axes need to be considered in

what follows. With this understanding, the geometrical tilts will be absorbed into

I ≡ I cosω.

The averages appearing in (5.22) can be computed analytically (see Appendix

A) giving:

〈mx〉 = ± 2π

T (ε)√D + 1

〈m2x〉(ε) =

4

T (ε)√D + 1

√D

D + 1− γ2−η1 (5.23)

94

with

T (ε) = 4

√D + 1− γ2

−

D(D + 1)K(1− γ2

−) = 4

√D + 1− γ2

−

D(D + 1)η0(γ−) (5.24)

η0(ε) =

∫ 1

0

dx√(γ2− + (1− γ2

−)x2)(1− x2)=

K(1− 1γ2−

)

γ−= K(1− γ2

−) (5.25)

η1(ε) =

∫ 1

0

dx

√γ2− + (1− γ2

−)x2

1− x2= γ−E(1− 1

γ2−

) = E(1− γ2−) (5.26)

γ−(ε) =|ε|(D + 1)

D + |ε|, (5.27)

where K(x) and E(x) are elliptic functions of the first and second kind respec-

tively (we have simplified the results by employing a non-trivial identity proved

in Appendix A.3). The ± appearing in (5.23) represents whether the dynamics

are taking place in the parallel or antiparallel IP basin respectively. The energy

equation representing the dynamics starting in the antiparallel negative energy

well (0 < |ε| < 1,mx < 0) then reads (expressed in terms of γ−3):

∂t|ε|(γ−) =8α

T (γ−)

(D(D + 1)1/3

D + 1− γ2−

)3/2

×[η1(γ−)− γ2

−η0(γ−) +1− γ2

−

D

(η1(γ−)− πI

2

√D + 1− γ2

−

D

)]

+ 4

√α

ξT (γ−)

(D(D + 1)

D + 1− γ2−

)1/4√η1(γ−)− Dγ2

−

D + 1− γ2−η0(γ−) W.

(5.28)

3We can allow ourselves the freedom to switch between expressions involving γ− and ε. γ2− =|ε|(D+1)D+|ε| is a monotonically increasing function of |ε| with the convenient property that |ε| = 0→γ− = 0 and |ε| = 1→ γ− = 1. As such limits written in terms of γ− and ε are equivalent.

95

In outlining the steps above, we have succeeded in reducing the multidimen-

sional complexity of the full magnetization dynamics to a one-dimensional stochas-

tic differential equation whose properties we now proceed to study. The relation-

ship to the previous chapter [156,159] on uniaxial macrospins can be rederived by

considering equations (5.2) and (5.28) in the limit D → 0. Doing so leads to the

much simplified energy diffusion equation [155]

∂t|ε| = 2α√|ε|(1− |ε|)(

√|ε| − I) + 2

√α

ξ|ε|(1− |ε|) W, (5.29)

with a stable energy minimum at ε = −1 and saddle point at ε = −I2. This equa-

tion also shows that, for currents I > Ic = max|ε|∈[0,1]

√|ε| = 1, the flow becomes

negative for all values of |ε|, so that in this regime all states will deterministically

switch. More specifically, substituting |ε| = m2x and ∂t|ε| = 2mxmx one rederives

(4.12) exactly.

5.2.1 Stability Analysis

In the absence of applied currents, ∂t|ε| > 0 as determined by (5.28) and ε

flows toward its minimum value of −1 (the stable fixed point of its dynamics).

To understand switching one must therefore investigate under what circumstances

ε may become greater than zero, thus implying a transition from the red to the

green trajectories discussed in Figure 1.1. To this aim, it suffices to understand the

behavior of the energy flow at the stable fixed point of the zero current dynamics

(|ε| = γ = 1) and at the energy threshold for switching (|ε| = γ = 0). These

96

limiting flows are found by computing the integrals η0(γ) and η1(γ). One finds:

limγ−→1

η0(γ−) ∼ π2

+ π8

DD+1−γ2−

(1− γ2−) + o((1− γ2

−)2) (5.30)

limγ−→1

η1(γ−) ∼ π2− π

8D

D+1−γ2−(1− γ2

−) + o((1− γ2−)2), (5.31)

and

limγ−→0+

η0(γ−) ∼ log(γ−) + o(log(γ−)γ2−) (5.32)

limγ−→0+

η1(γ−) ∼ 1 + o(γ−). (5.33)

Using these results in (5.28), one finds that in the absence of thermal noise (i.e.,

the zero temperature limit), the limiting flows are:

limγ−→0+

∂t|ε| =8α

T

√D

D + 1

[D + 1− πI

2

√D + 1

D

](5.34)

limγ−→1

∂t|ε| =4πα

T

√D + 1

D(1− γ2

−)[D + 2− 2I

]. (5.35)

As expected, both eqs. (5.34) and (5.35) show how the qualitative behavior of

the deterministic energy flow can be tuned by the value of applied current. The

stability of the zero current stable fixed point (|ε| = γ− = 1) can in fact be be

rendered unstable once (5.35) changes sign at the applied current value:

I |ε|=1c ≡ I1

C =D + 2

2. (5.36)

Analogously, at |ε| = γ− = 0 the flow is either positive or negative depending on

the applied current. Using (5.34), we find that the sign of the flow switches from

97

positive to negative at

I |ε|=0c ≡ I0

C =2

π

√D(D + 1). (5.37)

Identical expressions for the critical currents have recently been derived by Taniguchi [147]

via different means. These two critical stability thresholds are identical at a critical

value of D:

D0 = 2π2 − 4

16− π2

[1 + 2

√4 + 2π2

π2 − 4

]' 5.09. (5.38)

The value of D relative to D0, as well as the applied current, will select between

two qualitatively different dynamical regimes. We now proceed to explore both.

5.2.2 D > D0

When the ratio between hard and easy axis anisotropies is greater than the

threshold value D0, I0C > I1

C (see Figure 5.2). Unless the applied current is greater

than I0C , deterministic switching of the monodomain cannot be achieved. This

differs from conclusions drawn by previous work [35] where the critical current

for deterministic switching was assumed to be that which renders the stable fixed

point (|ε| = γ− = 1) unstable.

For small currents such that I < I1C , on the other hand, the deterministic flow

pushes the energy toward the zero current stable fixed point. In such a scenario,

switching can only occur via thermal activation over the |ε| = γ− = 0 effective

barrier.

Finally, for currents I0C > I > I1

C , the zero current stable fixed point has

been rendered unstable while the flow at the switching energy threshold remains

98

Figure 5.2: Critical currents versus the ratio of the hard and easy axis anisotropiesD. The blue curve is I1

C and the red curve is I0C . For D < D0, currents greater

than I1C lead to deterministic switching (labeled DS). For D > D0 currents between

I0C and I1

C lead to limit cycles (LC). Limit cycles can also occur for currents justbelow and approximately equal to I1

C for D < D0, as shown in Figure 5.5

positive. The monodomain will move (deterministically) from the |ε| = γ− = 1

fixed point, but switching over the |ε| = γ− = 0 must still take place via thermal

activation. This implies that a new stable energy equilibrium exists for some value

0 < γS < 1. The precessional macrospin dynamics will then manifest itself in the

form of stable limit cycles. This has been observed both experimentally [41] and

numerically [154]. These three dynamical scenarios are displayed in Figure 5.3

where the deterministic energy flow is plotted as a function of energy.

99

Figure 5.3: Three sample regimes of deterministic energy flow ε as a function ofenergy for D > D0. Coloring is included to better distinguish the various curves.(blue thick dashed curve) I < I1

C : Subcritical regime, thermal noise must oppose apositive energy flow to achieve switching; (green dash-dotted curve) I1

C I0C : Supercritical regime, negative

flow leads to deterministic switching.

100

5.2.3 D < D0

When the ratio between hard and easy axis anisotropies is smaller than the

threshold value D0, we have I1C > I0

C (see Figure 5.2). The threshold critical

current that switches all magnetic states deterministically is now I1C [35]. Above

this value of the applied current, switching will occur independently of thermal

noise effects.

For applied currents such that I0C < I < I1

C , a saddle point emerges for some

value 0 < γU < 1. For switching to occur, thermal activation must move the

monodomain energy past this saddle point. As the current is lowered, γU becomes

progressively smaller until the limiting value γU = 0. At this point, switching

requires thermal activation throughout the whole energy range. These three dy-

namical scenarios are displayed in Figure 5.4 where the deterministic energy flow

is plotted as a function of energy.

It is interesting to note that the condition D < D0 can also result in the

appearance of limit cycles. Limit cycle regimes in fact can be seen for very small

applied ranges of current less than I0C (Figure 5.5).

The uniaxial macrospin model is a particular case of a D < D0 model (D = 0).

By taking the limit D → 0, we find I0C → 0 and I1

C → 1, as discussed in the

previous section. [159] For all values of the applied current strictly between 0

and 1, uniaxial macrospin switching must take place via thermal activation over

an effective energy barrier.

101

Figure 5.4: Three sample regimes of deterministic energy flow ε as a function ofenergy for D < D0. Coloring is included to better distinguish the various curves.(blue thick dashed curve) I < I0

C : Subcritical regime, thermal noise must oppose apositive energy flow to achieve switching; (green dash-dotted curve) I0

C I1

C : Supercritical regime,negative flow leads to deterministic switching.

102

Figure 5.5: Energy flow for a D = 4 macrospin and applied current I = 2.82 < I0C .

Circles and squares respectively represent stable and unstable equilibria. For theseparameters (D = 4 and I = 2.82), two stable equilibria of the zero temperaturedynamics coexist.

103

5.2.4 CEOA Breakdown

In this section we will determine under what conditions the CEOA approach is

valid. The fundamental assumption is that the deterministic precession timescale

of the constant energy orbits be small compared to the timescale over which signif-

icant energy diffusion due to noise and spin-torque transfer takes place. We now

quantify this condition and obtain precise limits in terms of the applied current

intensity I and the anisotropy ratio D. For our approximations to be valid, the

averaged energy flow (T (ε)∂t|ε|) over any given orbit must be small compared to

the maximum allowable energy diffusion range (0 < |ε| < 1):

εmax T (ε)|∂t|ε|| = γ−

max T (γ−)|∂t|ε|| 1, (5.39)

where we continue to use the variable γ−(ε) = |ε|(D + 1)/(D + |ε|). To proceed,

we consider the deterministic drift and random components of the energy flow

separately. Noise averaging (5.28) will give the contribution of the drift to the

energy flow. The intensity of the orbit averaged drift then reads:

T (γ−)|〈∂t|ε|〉| =8α

(D(D + 1)1/3

D + 1− γ2−

)3/2

×∣∣∣∣∣η1(γ−)− γ2−η0(γ−) +

1− γ2−

D

(η1(γ−)− πI

2

√D + 1− γ2

−

D

)∣∣∣∣∣ ,(5.40)

where angular brackets denote noise averaging. This expression can be shown to

be finite for all values of γ−, I and D. Furthermore, for applied currents greater

than I0C , (5.40) has a maximum at γ− = 0; its value in this limit is given by (5.34).

104

Enforcing the conditions for validity discussed above results in an upper bound for

the current:

I (1 +1

8α√D

)I0C ≡ IM . (5.41)

In the limitD → 0, this expression converges to the correct uniaxial limit inequality

I (4πα)−1, discussed previously in [155].

When switching occurs deterministically (and I > I0C), this bound is the sole

limit to the validity of the CEOA approximation. This is because random con-

tributions to the energy flow dynamics have zero mean. However, in scenarios

where switching occurs due to thermal activation, one must consider the standard

deviation of random contributions to the energy flow. These determine the rate

of the rare events that drive the system over its confining energy barrier. The

energy flow’s standard deviation is√〈[∂t|ε| − 〈∂t|ε|〉]2〉. The orbit-averaged rate

of random events driving noise-induced energy diffusion can then be written as:

T (γ)√〈[∂t|ε| − 〈∂t|ε|〉]2〉 = 8

√α

ξη0(γ)

(η1(γ)− Dγ2

D + 1− γ2η0(γ)

), (5.42)

which can be shown to be both a monotonically decreasing function of γ− and

logarithmically divergent for γ− → 0. The latter is due to the limiting behavior

of η0(γ−) (see (5.32)) for all D > 0. The physical origin of this behavior lies in

the divergence of the constant energy precessional period (5.24) as the zero energy

switching threshold is reached (a fact also pointed out by Taniguchi et al. [147]).

The relevant maxima of (5.42) will be located at the energy flow’s saddle point

γU . Unfortunately, solving for γU while imposing that the RHS of (5.42) be much

less than 1 results in a set of transcendental equations which can only be solved

numerically. However, we can set a precise bound by requiring that the saddle

105

point never be at γU = 0 (where (5.42) diverges for all D > 0). The applied

current at which such a saddle point occurs has already been derived in (5.37),

leading us to the lower bound inequality

I I0C . (5.43)

as the validity condition for CEOA in thermally activated scenarios. The reason

for limiting ourselves to applied currents greater than I0C in deriving the upper

bound condition (5.41) is now justified. Condition (5.43) is to be considered valid

at all finite temperatures.

As a result of (5.43), there are only two scenarios in which thermally activated

switching can be analyzed and understood using CEOA dynamics. The first is in

the limit D → 0 (the uniaxial macrospin limit) where one has γ(ε) → 1 indpen-

dently of ε. Divergences in the intensity of random contributions are then avoided

for all ε and (5.42) takes the simple form 4π√

(α/ξ)(1− ε). At ε = 0 it converges

to the finite value 4π√α/ξ (which may still be large, however, depending on the

ratio α/ξ). The second scenario is for models with D < D0 and applied current

values I > I0C . In these, the saddle point to be reached via thermal activation

shifts to nonzero values γU 6= 0 where (5.42) does not diverge and the validity

condition may be satisfied.

The above discussion demonstrates that thermally activated switching between

dynamical limit cycle equilibria cannot be modeled with our technique. Thermal

switching from a limit cycle, in fact, must proceed via noise diffusion up to a

saddle point of the type γU = 0 where noise contributions diverge. Nonetheless, it

should be noted that this does not necessarily invalidate the conditions for their

106

existence discussed in the previous section. In fact, relaxation to a limit cycle is

a drift-driven process for which noise related contributions average to zero. Our

analysis merely shows that, once the macrospin has relaxed to its limit cycle state,

further dynamics must proceed via thermal activation for which CEOA does not

provide a suitable description.

In Figure 5.6, the dynamical behavior for D = 4 is displayed. Here D < D0

and I0C < I1

C . Fig 5.6a shows the dependence between rescaled current and mean

switching time for different polar angular tilts θ and fixed azimuthal angle φ. For

reference, the critical currents I0C , I1

C and IM are shown. For large applied currents,

deviations between the different sets of data are clearly visible. This discrepancy

decreases as the current is lowered.

However, for I < I0C , the procedure is expected to fail due to the previously

discussed divergence of the orbital period. To show this, we calculate the maximum

deviation of the mean switching time between all the angular data points at each

value of the rescaled current (see Figure 5.6b). As a further test, we have also

numerically computed the LHS of (5.39) and compared it to the angular deviations.

The critical currents I0C , I1

C and the limit current IM are again shown for reference.

The theory is in satisfactory agreement with numerical simulations as can be seen

from both the good alignment of the deviation minimum with the minimum of the

maximum flow and the rapid spike in deviation as I0C is approached.

This discussion implies that the CEOA technique applies best to small D

macrospin models, and becomes increasingly inaccurate for larger D. Figures 5.7

and 5.8 show the same analysis of a macrospin with D = 50 and one with D = 7.

Fig. 5.7a indicates that the technique is not applicable for sufficiently high values

of D. Nonetheless, theory and numerical results approach each other near the

107

Figure 5.6: a) Mean switching time versus current for D = 4, α = 0.04 and ξ = 80at different θ angular tilts with φ = 0 kept fixed. All currents have been rescaledby 1/ cos θ. Times are shown in units of (s · T ) where T stands for Tesla: realtime is obtained upon division by µ0HK . For visual guidance, the critical currentsI0C , I1

C and limit current IM have been included. In a regime where the CEOAtechnique is applicable, the switching data from the various angular configurationsshould all fall on top of each other. b) Double y-axis plot of max[T (ε)|∂t|ε||] andthe percent deviation of data from (a)) as a function of normalized current. In thecurrent range where the deterministic flow achieves its minimum, the deviation ofthe data does also. As the critical current I0

C is approached, deviation spikes areobserved analogously to what can be inferred by the theory.

108

Figure 5.7: a) Mean switching time versus current for D = 50, α = 0.04 andξ = 80 at different θ angular tilts with φ = 0 kept fixed. All currents have beenrescaled by 1/ cos θ. Times are shown in units of (s · T ) where T stands for Tesla:real time is obtained upon division by µ0HK . For visual guidance, the criticalcurrents I0

C , I1C and limit current IM have been included. b) Double y-axis plot

of max[T (ε)|∂t|ε||] and the percent deviation of data from (a)) as a function ofnormalized current. In the current range where the deterministic flow achievesits minimum, the deviation of the data does also. As the critical current I0

C isapproached, deviation spikes are observed analogously to what can be inferred bythe theory.

minimum of the deterministic flow, as seen in Fig. 5.7b. As in Fig. 5.6, as currents

drop below I0C , deviations in the data rapidly spike.

The analysis and its comparison to numerical data indicate that the CEOA

technique is best suited for studying macrospin dynamics in the crossover regime

separating ballistic from fully activated thermal switching. Furthermore, the sta-

bility analysis of the energy dynamics (5.28) manages to capture the emergence of

limit cycles even though it fails to describe thermally activated processes proceed-

ing from them.

109

Figure 5.8: a) Mean switching time versus current for D = 7, α = 0.04 and ξ = 80at different θ angular tilts with φ = 0 kept fixed. All currents have been rescaledby nz = 1/ cos θ. Times are shown in units of (s · T ) where T stands for Tesla:real time is obtained upon division by µ0HK . For visual guidance, the criticalcurrents I0

C , I1C and limit current IM have been included. b) Double y-axis plot

of max[T (ε)|∂t|ε||] and the percent deviation of data from (a)) as a function ofnormalized current. In the current range where the deterministic flow achievesits minimum, the deviation of the data does also. As the critical current I0

C isapproached, deviation spikes are observed analogously to what can be inferred bythe theory.

110

5.2.5 Thermally Activated Switching

Scenarios in which thermally activated switching can be studied using the ap-

proach described above are generally rather limited. Nonetheless, in the previous

section, the technique was shown to be applicable in models with D < D0 for

currents I0C < I < I1

C . Starting from (5.28) one could in principle construct a

Fokker-Planck equation describing energy diffusion and then attempt to solve an

appropriate mean first passage time problem numerically. Instead, we will simplify

the thermal activation problem by deriving the exponential scaling dependence be-

tween mean switching time and current, using an analytical tool described in [155].

In that paper, it was shown how, starting from energy diffusion dynamics analo-

gous to (5.28), the exponential scaling dependence can be written in terms of an

integral from the initial stable state to the saddle using a Friedlin-Wentzell [172]

type formulation:

log(〈τ〉) ∝ 2

∫ 1

|εU |

f(ε′)

g2(ε′)dε′ (5.44)

where f(ε) and g(ε) are, respectively, the drift and noise terms of (5.28), now

expressed in terms of the energy ε. This integral can be computed numerically

once the saddle point of the energy flow has been identified (e.g. via a Newton

algorithm). The integrand can be further expanded for D >> 1 and approximated

around the stable point |ε| = 1. This gives, to first order in 1− |ε|,

log(〈τ〉) ∝ ξ

∫ 1

|εU |

(1− (I/I0

C)

√D

D + ε′

(1− ε′

η1(ε′)− ε′η0(ε′)

))dε′

' ξ

∫ 1

|εU |

(1− (I/I0

C)1− ε′

1− ε′ζ(D)

)dε′ (5.45)

111

where ζ(D) is given by

ζ(D) =I1C

I0C

=π

4

(D + 2√D(D + 1)

). (5.46)

The approximated integral can now be written in closed form in terms of hyper-

geometric functions. The resulting expression is accurate to within 2% for values

D > 0.1.

In Fig. 10 we plot the dependence of the mean switching time as a function of

applied current, and compare it to log(τ) ∝ ξ(1 − I/IC)β. In the limit D → 0,

the uniaxial switching exponent β = 2 is recovered. For larger values of D, the

exponent β depends non-linearly on the applied current I. In the limit I → 0 the

limiting value of β can be derived analytically:

limI→0

β(I) = limI→0

log(1− q(D)

I0CI)

log(1− I/I1C)

= q(D)I1C

I0C

, (5.47)

with

q(D) ≡∫ 1

0

√D

D + x

(1− x

η1(x)− xη0(x)

)dx. (5.48)

The same calculation can be performed in the limit I → I1C and used to show

that the exponent β diverges for all non-zero values of D. The mean switching

time is nonetheless well behaved, as can be seen in Fig. 5.9a. This differs from the

results obtained by Taniguchi [147], in which the limiting value of the exponent β is

roughly 2.2, as the applied current approaches the critical threshold for determinisic

switching. Unfortunately, the numerical verification of β in regimes where CEOA

is applicable is a complicated matter. To do so, one would need to know the form

of the exponential prefactor in the scaling which, in turn, is also expected to be

112

Figure 5.9: a) Scaling dependence of mean switching time as a function of appliedcurrent I for models with varying D < D0. ξ is the energy barrier height and I1

C

the critical current threshold for deterministic switching. b) Fit of (5.45) to theform (1− I/I1

C)β. Dashed lines represent continuation of analytical results outsidethe technique’s regime of validity. Fitting exponent β is plotted as a function ofapplied current for models with varying D. In the limit of small D the exponentapproaches the constant value β = 2 consistent with previous uniaxial macrospinresults [136, 155, 159]. For D > 0, the exponent β depends nonlinearly on theapplied current intensity. Only for values D ∼ D0 do we notice that in the limit ofsmall applied currents, β → 1 as suggested by similar energy diffusion studies fromthe literature [139,154]. For intermediate values D0 > D > 0 the low current limitof β can be obtained analytically (5.47). On the other hand, in the limit I → I1

C

the exponent β can be shown to diverge for all non-zero values of D.

non-linearly dependent on the applied current as already discussed in relation to

the simpler uniaxial macrospin model in the previous chapter [159].

5.3 OOP Dynamics

In this approach, we now consider damping, applied current and thermal noise

effects on an OOP ε > 0 orbit. First, we average (5.16) over conservative positive

energy trajectories (5.9) (see Appendix A). Due to the symmetry of such trajecto-

ries, most terms average to zero with the remaining nonzero terms leading to the

113

CEOA equation:

〈∂tε〉 = 2α[I(D − ε)(sinω cos2 ψ)〈mz〉 −D(D + 1)〈m2

z〉+ ε(1 + ε)]

+ h(ε)

+

√2αD(D + 1)

ξ

√〈m2

z〉 −ε(1 + ε)

D(D + 1)· Wε, (5.49)

where angular brackets 〈·〉 denote averaging over a constant-energy trajectory with

energy ε. The second drift term (following the square brackets) h(ε) is a result of

transforming (1.29) into its Ito representation before performing the average over

orbits (see Appendix B). As a result, the multiplicative noise terms appearing in

the averaged energy equation above are now interpreted in the Ito sense4.

We note that, as has been found for negative (IP) CEOA states [156], the

dynamics as a function of applied current for different spin-polarization tilts are

identical, the current is simply rescaled by sinω cos2 ψ (refer to Fig. 1.5). This al-

lows us to numerically verify the CEOA approach by checking that the macrospin’s

evolution over some (properly rescaled) applied current is exactly identical for dif-

ferent tilts of the spin-polarization axis.

The associated equation describing the stochastic evolution of the dynamical

phase can at once be written down also:

〈∂tχ〉 =2π

T (ε)+

√2αD(D + 1)

ξ

√〈m2

z〉 −ε(1 + ε)

D(D + 1)· Wχ, (5.50)

where, again, T (ε) is the period of the orbit at energy ε. The structure of the

phase dynamics will become important when we discuss different aspects of phase

noise in later sections.

4As usual, we distinguish equations written in Ito vs. Stratonovich form by writing themultiplicative noise as ‘· ˙Wε,φ’

114

To compute the averages 〈mz〉 and 〈m2z〉 explicitly, we note that the positive

energy trajectories can be geometrically parametrized as follows:

m0x(s) =

√ε sinh(s),

m0y(s) = ±

√1 + ε

√1− γ2 cosh2(s)

m0z(s) = ±

√ε

Dcosh(s)

γ2+ =

ε(D + 1)

D(ε+ 1), (5.51)

where the parameter s ranges from −acosh(1/γ) < s < acosh(1/γ). Upon com-

puting the averages explicitly (Appendix A), the CEOA equations for the positive

energy dynamics (0 < ε < D), expressed in terms of γ+, read 5

∂tε(γ+) =πα

η0(γ+)

D(D + 1)

[D(1− γ2+) + 1]3/2

×±I(1− γ2

+)− 2

π

√D(1− γ2

+) + 1

[η1(γ+)−

γ2+

(D(1− γ2+) + 1)

η0(γ+)

]+ h(ε)+√

2α

ξ

D(D + 1)

D(1− γ2+) + 1

(η1(γ+)

η0(γ+)− γ2

+

D(1− γ2+) + 1

)· Wε (5.52)

∂tχ(γ+) =π

2η0(γ+)

√D(D + 1)

D(1− γ2+) + 1

+√2α

ξ

D(D + 1)

D(1− γ2+) + 1

(η1(γ+)

η0(γ+)− γ2

+

D(1− γ2+) + 1

)· Wχ (5.53)

where η0(γ+) = K[1− γ2+] and η1(γ+) = E[1− γ2

+] are again expressed in terms of

5We can allow ourselves the freedom to switch between expressions involving γ+ and ε. γ2+ =ε(D+1)D(ε+1) is a monotonically increasing function of ε with the convenient property that ε = 0 →γ+ = 0 and ε = D → γ+ = 1. As such limits written in terms of γ+ and ε are equivalent.

115

complete elliptic integrals of the first and second kind. For notational simplicity,

the geometrical tilts have been absorbed into I ≡ I sinω cos2 ψ.6 It is important

to note the applied current acts either to positively or negatively dampen the

dynamics depending on which ε > 0 basin the magnetization is in (see Figure 1.1).

The second drift term h(ε) appearing in (5.52), is the drift correction due to our

change to Ito calculus. As discussed in Appendix B, the extra drift term results in a

negligible correction. The following analysis will, analogously to the IP case, hence

ignore its second order effects although they can be reintroduced straightforwardly

if higher quantitative accuracy is desired.7

In following the outlined procedure, we have again reduced the complexity of

the magnetization dynamics to a one-dimensional stochastic differential equation,

whose properties we will now show to be analytically tractable.

5.3.1 Stability Analysis

As seen from (5.52), in the absence of applied currents, the deterministic drift

portion (first term on the RHS) of the energy diffusion dynamics is globally neg-

ative, ∂tε < 0. The energy ε flows from positive to negative energy basins toward

its minimum value of −1. This is consistent with our physical notion of the ε > 0

basins being energetically unfavorable. Upon introducing an applied current, the

behavior remains unchanged as long as no tilt is present between easy and spin-

polarization axes (ω = 0). If a nonzero tilt is introduced into the system, the

symmetry of the two positive energy basins is broken. In particular, due to the

dependence on ±I (everything else inside the curly brackets is always negative),

6Note that in contrast to the in-plane precessional dynamics discussed in [156], the current isrescaled by sinω as opposed to cosω

7The Ito drift-diffusion correction becomes relevant for dynamics close to the ε = 0 separatrix.

116

a critical current will exist, corresponding to a fixed point in the energy dynamics

appearing in the positive z, ε > 0 basin. The presence of a fixed point in the energy

dynamics can then be expected to correspond to a stable precessional (limit cycle)

state of the magnetization dynamics. The dynamics in the negative z basin, on the

other hand, will continue to be globally dissipative. Physically this is explained by

the fact that the tilt ω biases the magnetic evolution away from one basin in favor

of the other.

The critical current at which a fixed point appears can be obtained by studying

the behavior of the energy dynamics in the limit ε = γ+ → 0. Requiring that

limε→0

T(ε)ε ∝ −2√D + 1 + πI = 0, (5.54)

we obtain

IOOP =2

π

√D + 1, (5.55)

as the current at which a stable fixed point appears at ε = 0. Increasing I further

will shift the fixed point to higher energies. Qualitatively, this will result in an

increase of frequency and decrease of amplitude of the limit cycle oscillations. The

maximum possible energy obtainable by the oscillator is ε = D. This is achieved

when 8

Imax = D +1

2. (5.56)

Increasing the current beyond Imax simply overdrives the magnetization. As we

will see later, the CEOA approximation breaks down beyond this point and stable

8One analogously seeks a null net drift of the energy dynamics at ε = D: limε→D

T(ε)ε = 0

117

oscillations disappear. Fig. 5.10 shows a sample of the drift field due to (5.52) for

I < IOOP, IOOP Imax. IOOP and Imax represent the lower

and upper threshold currents for the appearance of steady-state precessions in the

stable OOP basin due to the nonlinear character of the magnetization dynamics.

Comparing with the CEOA treatment of magnetic switching [156] discussed

in the previous section, we note that Iswitch9, the critical current for switching,

equals√DIOOP. As such, the minimal currents sustaining stable OOP precessional

states are generally smaller than the critical switching current. This results in the

prediction of a hysteretic dependence of IPOOP transitions on applied current,

which has been observed recently in experiment [130]. In detail, since Iswitch =

Iswitch cosω and IOOP = IOOP sinω cos2 ψ, one can see that the relation between

direct critical switching current and threshold current for sustainment of OOP

precessions is

IOOP =Iswitch√

D tanω cos2 ψ. (5.57)

5.3.2 CEOA Breakdown

For our approximations to be valid, the averaged energy flow (T (ε)|∂tε|) over

any given orbit must be small compared to the maximum allowable energy varia-

tions (0 < ε < D):

εmax T (ε)|∂tε| D. (5.58)

9We limit ourselves to considering samples with D > D0. As discussed in the section on IPdynamics, the switching current for such models is I0C = 2

π

√D(D + 1). From now on we will

refer to I0C as Iswitch in the text. We remind the reader that the tilde stands to signify a currentrescaled by cos(ω) as relevant to the IP dynamics: Iswitch = I0C = I0C cos(ω).

118

Figure 5.10: Three regimes of deterministic energy flow ε as a function of energy forD = 10. (blue-dashed) I < IOOP: Subcritical regime. Energy flows from positiveto negative energy basins due to dynamics being globally dissipative (overdamped).(red-dashdotted) I > Imax: Supercritical regime. Energy flows towards limitingstable value ε = D due to dynamics being overdriven by applied current. (green-dotted) IOOP < I < Imax: Oscillator regime. Energy flow will stabilize at a fixedpoint corresponding to a precessing oscillator state. In this regime, the fixed pointrepresents a constant energy trajectory where spin-torque and damping effectsbalance.

119

The analysis proceeds identically to the previous section on IP dynamics [147,156]

so we simply state the results for OOP dynamics. For CEOA to be applicable one

must have IOOP . I . Imax.

In Figure 5.11 we show a comparison between theory and numerical results by

plotting average energy 〈ε〉 as a function of applied current. Ensembles consisting

of 5120 macrospins were initialized antiparallel to the easy-axis and allowed to

relax subject to a steady applied current. Upon varying the angular tilt ω between

easy and spin-polarization axes, we notice that the data follow our theory down to

a minimum critical angle ωC . For angular tilts less than ωC , stable positive energy

steady states cease to be accessible regardless of the applied current. The origin

of this angular cutoff is geometrical in nature and corresponds to the necessity

for the spin-polarization axis to be pointing inside the positive energy basin. The

condition for this to happen can be seen from (5.1) by solving for the separatrix

of the energy basins. One obtains

ωC =π

2− arctan(

√D), (5.59)

which is in excellent agreement with numerical data. This geometrical intuition

can be seen from theory by determining the tilt for which the threshold current

for OOP precessions equals that for direct switching. Starting with (5.57), and

setting ψ = 0 for convenience, leads to (5.59).

For large currents I > Imax, numerical results seem to indicate a steady drop in

ensemble energy as the applied current is increased. In fact, contrary to the CEOA

description, the macrospin’s magnetization ceases to precess around the hard axis

and instead settles into a magnetic configuration where all static torques balance

and spin-torque effects compete with the magnetic anisotropies.

120

Figure 5.11: Steady-state ensemble energy as a function of dimensionless appliedcurrent I (rescaled by IOOP = (2/π)

√D + 1) for a model with D = 10, ξ = 80 and

α = 0.04. Red line shows an analytic fit to numerical data within the current limitsdefined by the theory (for reference Imax/IOOP ≈ 4.97). Insets show density plots inspherical coordinates of 5120 numerical trajectories for a sample with a 2.56ωC tiltbetween easy and spin-polarization axes, driven by a current of I/IOOP = 4 (top),and I/IOOP = 15 (bottom). The dotted line denotes the conservative trajectory.

121

5.3.3 Thermal Stability, Precession Linewidth, Phase, Am-

plitude and Power Fluctuations

So far, we have provided an analytical approach that enables the study of the

properties of OOP dynamics. Once the strength of the applied current I has

been chosen, and provided that the angular tilt of the spin-polarization vector is

sufficient (ω > ωC), the average energy ε0 = 〈ε〉 of the equilibrium steady state tra-

jectory can be obtained by solving for the fixed point of the energy dynamics (5.52)

for OOP and (5.28) for IP dynamics respectively. Due to the dependence of the

precessional period T(ε) on the energy of the orbit, the expected precessional fre-

quency can be inferred.

Thermal noise will, however, perturb the magnetization about the fixed point,

resulting in fluctuations of the macrospin’s energy around its average ε0 value and

diffusion of its phase χ along the relevant constant energy orbit. These deviations

are believed to be the source of the oscillator’s experimentally measured frequency,

linewidth and phase decoherence. We will now proceed to derive an estimate for

such linewidths.

The general stochastic energy evolution equation (5.52,5.28) can be written

concisely as

∂tε = f(ε, I) + h(ε) + g(ε) · Wε, (5.60)

where f(ε, I), h(ε) and g(ε) are, respectively, the deterministic drift, Ito drift-

diffusion correction and multiplicative noise. For definiteness, we will concentrate

on OOP steady-state precessions in this section. The methods and techniques can

however be extended to the study of steady-state IP precessional dynamics also.

122

Following Ref. [22], one can use the stochastic energy evolution equation to

compute the mean time one must wait to observe a thermal excitation out of an

OOP trajectory. The asymptotic dependence of such a mean escape time is then

log(〈τjump〉) ∝ 2

∫ ε0(I)

0

dxf(x, I)

g2(x)

= ξ

(ε0 −

I

IOOP

∫ ε0

0

dxD − x√

1 + x(Dη1(x)− xη0(x))

), (5.61)

where ε0 ≡ 〈ε〉 is the usual solution of the fixed point equation (dependent on I).

Due to the dependence of the equilibrium oscillator energy on the applied current

ε0(I), the thermal stability of the OOP precessional states will depend non-linearly

on the applied current I.

The Fokker-Planck (FP) equation is:

∂tρ = ∂ε

[f(ε, I)ρ− 1

2g2(ε)∂ερ

], (5.62)

whose solution describes the full evolution of the energy distribution ρ(ε, t) as

a function of time (Appendix B). At equilibrium (∂tρ = 0), the saddle point

approximation can be used to determine a steady state distribution

ρeq(ε) ∝ exp

[2

∫ ε

0

dxf(x, I)

g2(x)

]' exp

[f ′(ε0, I)

g2(ε0)(ε− ε0)2

], (5.63)

that is valid as long as I > IOOP. We can then write an expression for the amplitude

noise by computing the variance of the energy in an equilibrium OOP distribution:

〈(ε− ε0)2〉 ' g2(ε0)

2|f ′(ε0, I)|. (5.64)

123

Figure 5.12: Standard deviation of the energy distribution plotted as a func-tion of dimensionless applied current I (rescaled by IOOP = (2/π)

√D + 1) for

D = 10, ξ = 80 and α = 0.04. The solid blue line shows the theoretical predic-tion (5.64) calculated within the current limits defined by the theory (for referenceImax/IOOP ≈ 4.97).

In Figure 5.12 we compare the theoretical approximation resulting from (5.64)

with the equilibrium energy variance extracted from our numerical simulations.

Whereas the variance does not appear to rescale trivially with the spin-polarizer

tilt, all tilts seem to show a variance versus applied current curve that peaks

within the same general region predicted by our rough estimate. For currents

I ' IOOP, Imax the approximation breaks down due to failure of the CEOA ap-

proximation.

Using (5.63), we see that all energy-dependent stationary characteristics 〈Q〉 =∫Q(ε)ρeq(ε) of the oscillator can be computed via distribution averaging. However,

we employ our saddle point estimate (5.64) to study thermal fluctuations. The

relative fluctuation of a quantity Q(ε) at equilibrium will be given by δQ/Q =

124

(Q′(ε)/Q(ε))|ε=ε0√〈(ε− ε0)2〉.

As a first example, the experimentally observed oscillator power depends on

the square of the oscillator’s precession amplitude along the in-plane direction.

Having chosen a coordinate system with the reference magnetic layer aligned in-

plane, power fluctuations are directly proportional to fluctuations in the precession

amplitude of the oscillator as projected along the in-plane axial direction. From

our previously derived expression of the oscillation amplitude along the in-plane

direction (5.13), one has:

δP

P=δA2

x

A2x

'√〈(ε− ε0)2〉. (5.65)

Analogously, denoting the oscillation frequency by ν(ε) = 2π/T (ε), one finds

for the precession linewidth quality factor Q dependence on amplitude noise:

1

Q=δν

ν' T′(ε0)

T(ε0)

√〈(ε− ε0)2〉. (5.66)

Fig. 5.13 shows how the quality factor is a monotonically increasing function of

applied current. Overall, increasing the driving current reduces the linewidth of

the oscillator in line with classical oscillator theory which predicts a linewidth

scaling dependent on the ratio of the thermal and oscillator energy (kBT/ε). In

practice, however, at currents high enough for the breakdown of the macrospin

model, micromagnetic effects due to Oersted fields are expected to complicate the

physical picture in non-trivial ways.

One may proceed further and ask whether the CEOA formalism is capable

of shedding light on the phase noise and, more generally, the phase decoherence

driving a magnetic system. The assumption that “sufficiently weak” noise drives

125

Figure 5.13: Inverse quality factor (5.66) vs. applied current for D = 10 set atroom temperature (ξ = 80). Red dashed line denotes the upper bound of thevalidly of the CEOA formalism: Imax/IOOP ≈ 4.97 for the parameters chosen.

diffusion from one energy orbit to another does not impose any limit on how

strong the noise driving the phase of the actual constant energy oscillation can

be. Both phase noise due to thermal diffusion along a given constant energy

orbit and amplitude noise can drive phase decoherence in a magnetic system. As

such, the relative intensity of both effects must be determined to understand phase

decoherence.

To do so, we consider how energy fluctuations about the ε0 equilibrium fixed

point influence the phase dynamics described in (5.52). Let ε(t) ≡ ε0 + δε(t) and

expand (5.60) in powers of δε. Denoting F (ε) ≡ f(ε)+h(ε), the resultant stochastic

differential equation can be formally integrated to give:

δε(t) = eF′(ε0)t

[c+ g(ε0)

∫ t

0

dt′e−F′(ε0)t′ · Wε

], (5.67)

126

where primes represent differentiation with respect to energy (F ′(ε0) ≡ ∂εF |ε=ε0),

and c is an (unimportant) initial condition. |F ′| represents the relaxation rate of

amplitude fluctuations to the ε0 baseline. Given the explicit dependence of the

phase χ on the energy evolution, such energy fluctuations are expected to play a

crucial role in the thermally driven phase dynamics.

Expanding the phase dynamics about ε0 to lowest order, we have:

∂tχ =2π

T (ε0)− 2πT ′(ε0)

T 2(ε0)δε(t) + g(ε0) · Wχ. (5.68)

Substituting (5.67) into (5.68) and recalling that Wε and Wχ are uncorrelated

stochastic processes, the expected phase variance at equilibrium can be evaluated

to give (we suppress the dependence on ε0):

〈∆χ2〉(t) =g2

[1 +

(2πT ′

F ′T 2

)2]|t|

+g2

2F ′

(2πT ′

F ′T 2

)2 [4(

1− eF ′|t|)−(

1− e2F ′|t|)]. (5.69)

which closely resembles the more general prediction from oscillator theory. [146,

161] Since the power spectrum can be written as a Fourier transformation of the

correlation function 〈exp[i(χ(t) − χ(t))]〉 ≈ exp[i〈χ(t) − χ(t)〉] exp[−〈∆χ2〉(t)/2],

the linewidth can be predicted [160] by inspecting (5.69).

The temporal dependence of the phase variance is responsible for the decoher-

ence of the magnetic ensemble over time. We interpret the decoherence time τdec as

the timescale necessary for the ensemble to homogeneously distribute itself along a

given constant energy orbit similarly to what is shown in Fig. 5. We quantify τdec

127

by asking on what timescale the width of the phase distribution begins to encom-

pass the entire constant energy orbit: 〈∆χ2〉(τdec) = 4π2. Although the temporal

dependence is generally quite complicated, two limiting regimes can be explored.

For low enough temperatures, the phase decoherence time τdec will be larger than

the relaxation timescale of the amplitude fluctuations τdec 1/|F ′|. Decoherence

can then be expected to mostly take place due to the differences in orbital evo-

lution at the different energies explored by the amplitude fluctuations. This will

eventually lead the spin ensemble to decohere and thermalize to a homogenous

distribution of phases relative to the referential ε0 orbit. The dominant amplitude

fluctuations driving such a low temperature regime result in a linear dependence

of the phase variance.

〈∆χ2(t)〉 ≈ g2

[1 +

(2πT ′

F ′T 2

)2]|t|. (5.70)

Due to the dependence of the multiplicative noise term in (5.52) on temperature

(g(ε) ∝√T ), the decoherence time τdec ∝ T−1/2 ∝

√ξ can be predicted to depend

on the inverse square root of temperature. Furthermore, a linear dependence on

time will imply a Lorentzian power spectrum with linewidth ∆νL = (g2/2π)(1+µ2)

(µ = 2πT ′/F ′T 2).

In a high temperature limit, pure phase noise will compete with the amplitude

noise effects by decohering the ensemble on a timescale smaller than the amplitude

fluctuation relaxation rate τdec 1/|F ′|. The exponential contributions in (5.69)

cease to be negligible and the approximate temporal dependence of the phase

variance can be written to second order in time as:

128

〈∆χ2(t)〉 ≈ g2

[|t|+ 2

(2πT ′

F ′T 2

)2

|F ′||t|2]. (5.71)

If (2πT ′/√|F ′|T 2)2 1 (typically the case when ε0 D), the term linear in

time can be dropped altogether resulting in a purely quadratic dependence of the

phase variance on time. In such a scenario, the decoherence time can be expected

to scale linearly with the inverse temperature τdec ∝ T−1 ∝ ξ. A phase variance

scaling quadratically in time will in turn lead to a gaussian power spectrum with

linewidth ∆νL =√

2gµ2F ′/2π.

We explore these predictions by studying switching probability curves of a

macrospin ensemble at varying temperatures for applied current intensities and

effective spin-polarization axial tilt consistent with an OOP precessional behav-

ior. Upon switching the current off, the phase of the oscillator will select the

macrospin’s relaxation outcome (either parallel or antiparallel to the easy axis of

the magnetic film) with high probability. In the absence of thermal noise, a cur-

rent pulse of fixed duration will lead to either a parallel or antiparallel relaxed

state after the pulse terminates (see Fig. 5.14) with absolute certainty. At nonzero

temperatures, however, oscillator ensemble phase decoherence is expected due to

thermal noise. As a result, long spin-current pulse times will lead to equally likely

parallel (antiparallel) relaxation due to ensemble thermalization along the OOP

constant energy orbit. In Fig. 5.15 we find good qualitative agreement between

such an understanding of phase decoherence behavior and numerical simulations.

The equilbrium probability bias for higher P switching is due to some of the states

thermally equilibrating into the IP energy basin before the current pulse is switched

off.

129

Figure 5.14: Switching probability vs. spin-current pulse length for a macrospinmodel with D = 10, ω = 2.12ωC) driven by a spin-current intensity of I =2.75 IOOP in the absence of thermal noise. Times are shown in units of (s · T )where T stands for Tesla: real time is obtained upon division by µ0HK . Before thecurrent pulse is switched on, the magnetic ensemble is taken to be antiparallel tothe easy-axis of the magnetic film. Switching probability is defined as the ensemblefraction that relaxes into a parallel configuration upon switching the current pulseoff. The right-hand vertical axis plots the evolution of the average 〈mz〉 component.In the absence of thermal noise the oscillator remains coherent at all times andits periodic motion is clearly seen. Due to the deterministic nature of the zero-temperature dynamics, the macrospin will deterministically switch either into theparallel or antiparallel state at all times.

130

Figure 5.15: Switching probability vs. spin-current pulse length for a macrospinmodel with D = 30, ω = 3ωC driven by a spin-current intensity of I = 5 IOOP

in the presence of thermal noise corresponding to ξ = 80 (left) and ξ = 1200(right). Times are shown in units of (s · T ) where T stands for Tesla: real timeis obtained upon division by µ0HK . Before the current pulse is switched on, themagnetic ensemble is taken to be antiparallel to the easy-axis of the magnetic film.Switching probability is defined as the fraction of the ensemble that relaxes into aparallel configuration upon switching the current pulse off. For long pulse timesthe switching probability converges to a value indicating that the phase of theOOP precession has decohered. The red dashed lines are a qualitative graphicalrepresentation of the decoherence time.

131

Figure 5.16: Log-log plot of ensemble decoherence time vs. energy barrier heightto thermal energy ratio ξ for a macrospin model with D = 30, ω = 3ωC drivenby a spin-current intensity of I = 1.5 Iswitch. Times are shown in units of (s · T )where T stands for Tesla: real time is obtained upon division by µ0HK . Linearregression (solid lines) of data points demonstrates a transition between a phasenoise dominated regime τdec ∝ 1/T below a certain critical inverse temperatureξ < ξC . Above ξ > ξC (T < TC), both amplitude and phase noise contribute toensemble decoherence.

The switching probability curves can be employed to numerically extract the

decoherence time at different temperatures. Fig. 5.16 shows a log-log plot of τdec

on ξ for a D = 30 model with a ω = 3ωC tilt, driven by a I = 1.5 Iswitch applied

current. Linear regression to numerical data shows an inverse proportionality

τdec ∝ 1/T ∝ ξ between decoherence time and temperature for temperatures

larger than a certain critical temperature. For T < TC , however, both amplitude

and phase noise seem to contribute to ensemble decoherence thus not allowing us

to probe the pure amplitude noise decoherence mechanism previously discussed.

132

Overall, the theoretical techniques developed in this chapter allow for a very de-

tailed characterization of the most relevant dynamical regimes that the macrospin

can be subject to via spin-torque induced excitations. The constant energy aver-

aged theory has allowed for very precise predictions to be made regarding when

a magnetic element can be expected to behave as a switching element or a spin-

torque nano-oscillator. In the next section we will briefly explore experimental

evidence for the OOP precessionary orbits.

5.4 Experimental Verification of OOP orbits

A polarizing layer that is magnetized perpendicularly to the free layer can signif-

icantly improve write speed and energy efficiency of spin-transfer torque magnetic

random access memories [81–83,168–171] by providing a large initial spin-transfer

torque. An orthogonal spin-torque OST device, of this kind (refer to Fig. 1.4)

was shown to also function as a microwave oscillator in the previous section, since

the polarizer can produce precessional magnetization dynamics, with the free layer

precessing OOP [42, 78, 79]. This precessional motion can also be used for ultra-

fast magnetization switching. For instance, sub-nanosecond switching has been

observed to be bipolar and to induce magnetization precession [81, 82, 170]. Fur-

ther, the effect of applied fields on these curretn-induced switching thresholds has

not yet been reported or considered in model studies. In this section we report

on the experimental observation of bipolar switching in OST spin valve devices as

discussed in [130]. Hysteretic transitions into intermediate resistance (IR) states

at large current are found to exist, with the IR state persisting to currents less

than the threshold currents for P to AP and AP to P switching.

133

5.4.1 Experiment

The layer stack studied consists of a perpendicularly magnetized spin-polarizing

layer, a non-magnetic metallic spacer layer, a free magnetic layer followed by an-

other non-magnetic spacer layer and a reference magnetic layer as illustrated in

the inset of Figure 5.17. The polarizer consists of a Co/Pd and Co/Ni multilayer,

with the Co/Ni multilayer closest to the free layer (FL), providing a highly spin-

polarized current [81]. The FL is a 3 nm thick CoFeB layer. The full layer stack is

6.2 [Co/Pd][Co/Ni]/10 Cu/3 CoFeB/10 Cu/12 CoFeB, with the layer thicknesses

indicated in nanometers. The stack was patterned into nanopillar devices with var-

ious shapes and sizes using e-beam lithography and ion-milling. Here we present

results on 50 nm × 100 nm devices in the shape of an ellipse. The magnetic easy

axis of the free layer is int he film plane along the long axis of the ellipse due to

magnetic shape anisotropy. Shape anisotropy also sets the magnetization direction

of the 12 nm thick CoFeB reference layer (RL).

Figure 5.17 shows measurements of the differential resistance (dV/dI) as a

function of applied field alogn the easy axis. The measurements are made with a

lock-in amplifier using an ac current of 200µA at a frequency of 473 Hz. A field

sweep from -200 mT to 200 mT (major hysteresis loop) shows steps in resistance

of 0.1 Ω indicative of switching of the FL from P to AP relative to the RL. The

coercive field of the RL is about 150 mT. A minor loop (-50 mT to 140 mT) shows

the switching of only the FL. the change in resistance between P and AP states is

∆RAP−P = 0.1 Ω. The coercive field for P to AP FL transitions is H+C = 59 mT

and the coercive field for P to AP FL transitions is H−C = 23 mT. The minor loop

is centered at H0 = (H+C + H−C )/2 = 41 mT due to dipolar coupling between the

Fla nd RL. Thus an external field of H0 corresponds (on average) to zero effective

134

Figure 5.17: Resistance versus in-plane applied field hysteresis loops. The majorloop (black curve) shows the switching of both the free and reference layers. Theminor loop (blue curve) shows the response of just the free layer. The loop iscentered at 41 mT due to dipolar interactions between the reference and free layer.Inset: Schematic of the spin-valve’s layer stack showing the out-of-plane magne-tized polarizing layer (OP), in-plane magnetized free layer (FL) and reference layer(RL).

field applied to the FL.

Current induced switching was characterized by measuring the differential resis-

tance as a function of current for a series of easy axis applied fields. The magnetic

state (P or AP) is first set by applying a large magentic field 200 mT and then a

lower field (larger than 25 mT for the P state, less than 58 mT for the AP state).

Then the current Idc was slowly ramped (' 0.1mA/s) from 0 to ±5 mA and then

ramped back to 0 mA, with dV/dI versus I recorded at each measuring field. Pos-

itive current corresponds to electron flow from polarizing to the reference layer

(from bottom to the top of the layer stack represented in the inset of Figure 5.17).

In this case the spin-torque associated with the RL favors an AP state for positive

currents. Representative measurement results starting from the P state are shown

135

in Figure 5.18. Similar results were found in measurements starting from the AP

state, which are discussed below.

In Figure 5.18(a), as the magnitude of the current is increased (black curves),

there is a discrete increase in differential resistance of ∆RAP−P = 0.1 Ω, associated

with a P to Ap transition. This is seen to occur for both polarities of the current.

On further increasing the current there is a change in resistance of about half of

∆RAP−P , i.e. a transition into an intermediate resistance state (IR). On decreasing

the current (red curves) the resistance eventually returns to that of the device’s AP

state. Near zero effective applied field (H ' H0), P to AP switching is only seen at

positive current polarity (Figure 5.18(b) and (c)). Whereas, for negative current,

only P to IR transitions occur as the current is increased. When the magnitude

of the current is decreased, there is a transition from an IR state to an AP state

for applied magnetic fields smaller than H0 (= 41 mT) and to a P state for fields

larger than H0.

This seemingly complex switching behavior can be summarized by plotting

the threshold currents for switching between resistance states in a current-applied

field state diagram (Figure 5.19). Each symbol in this diagram corresponds in a

discrete change in the resistance. The solid symbols represent resistance changes

of ∆RAP−P corresponding to transitions between P and AP states. They form

a diamond-shaped central zone within which both P and AP states are possible.

When the current is greater than 2.9 mA or is less than -1.4 mA, the step change

in resistance is less than ∆RAP−P . The boundaries are labeled by open symbols

and correspond to P and AP to IR transitions. These boundaries meet and join

the P to AP transition boundaries. Further, they define two triangular zones that

encompass IR states at high current magnitudes (both for positive and negative

136

Figure 5.18: Differential resistance versus current at various easy axis applied fieldsstarting from the P state. The magnitude of the current |Idc| is increased (blackcurves) and then decreased (red curves). (a) The black curve shows switching fromP to AP at 1.5 mA and also -1.2 mA, i.e. the switching occurs for both polaritiesof the current. At larger positive and negative current the resistance change isintermediate of that of the P to AP transitions. On reducing the current thereis a transition from the intermediate resistance (IR) state into an AP state. (b)At 40 mT switching from P to AP only occurs for positive polarity current andon reducing the current there is an IR to AP state transition for |Idc| ≤ 1 mA.(c) At 42 mT switching from P to AP again only occurs for positive polaritycurrent. However, on reducing the applied field the transition is from IR to P for|Idc| ≤ 1 mA.

137

Figure 5.19: Current swept state diagram of an OST spin valve device showing thethreshold currents for switching as a function of applied easy axis field. IP−APC andIAP−IRC are labeled by solid and open blue symbols. IAP−PC and IP−IRC are labeledby solid and open red symbols. The green curves indicate the IIR−APC (crosses)and IIR−PC (dashes), showing the bistability range of the IR states.

current polarities). As the current is swept back to zero, two parabolic shaped

curves (green) show the IR to P or AP transitions.

The general features of the state diagram of an OST spin valve device are

the following: (1) For magnetic fields near the FL coercive fields (H+C and H−C ),

current induced switching is bipolar. For fields close to but less than H+C , AP to

P transitions occur for both positive and negative currents and for fields near but

greater than H−C , P to AP transitions occur for both current polarities. (2) Near

H0, the center of the FL’s hysteresis loop, the switching occurs for only one current

polarity, positive current for the P to AP transition and negative current for the

AP to P current. (3) At large currents, transitions into an IR state are observed

138

Figure 5.20: (a) Representative FL minor hysteresis loops measured with a slowlyswept field at several fixed currents. The scale bar shows ∆RAP−P = 0.1 Ω, theresistance difference between the AP and P states. (b) State diagram constructedfrom dV/dI|H hysteresis loops. The color represents ∆R, the resistance differencebetween field increasing and field decreasing measurements. The central zone (or-ange color) corresponds to the AP/P bistable zone. Black dashed curves trace theboundaries between P, AP and IR states.

and this state persists even as the current is reduced well below the threshold

current for P/AP transitions. These features were seen in all ten 50 nm × 100 nm

ellipse devices that were studied.

The device states can also be determined by measuring the differential resis-

tance as a function of field at constant current. This is shown in Figure 5.20.

Figure 5.20(a) shows representative hysteresis loops at several currents. At zero

dc current the coercive field (HC = (H+C − H−C )/2) of the FL is largest and the

coercive field decreases as the current magnitude increases. For currents greater

than 2.9 mA or less than -1.4 mA, a plateau at a resistance between that of the

P and AP state resistances is seen, with the field range of the plateau increasing

with current magnitude.

A field swept state diagram is constructed as follows. The resistance measured

with decreasing field is subtracted from the resistance measured with increasing

139

field. The resistance difference ∆R is then plotted on a color scale versus current

and field (Figure 5.20(b)). ∆R is nonzero only in field ranges in which the device

response is hysteretic. The boundaries between zero and and non-zero ∆R regimes

are the boundaries between the P, AP and IR states. Thus the same general switch-

ing characteristics are observed in both current and field swept measurements.

5.4.2 Modeling

The device characteristics just discussed can be understood by employing the

biaxial macrospin model outlined in the first chapter of this thesis, with energy

landscape U(m) = K(Dm2z − m2

x). Spin-torque contributions due to both the

polarizer (magnetized out-of-the film plane, along z) and RL (magnetized in the

film plane along x) can be described in terms of effective spin-polarization direction

that is tilted with respect to the plane:

ΓS = Im× (m× nS),

nS =ηR

1− λRmx

x +ηP

1− λPmz

z. (5.72)

Here ηR,P and λR,P are the spin polarizations and spin-torque asymmetry pa-

rameters for the RL and polarizer layer respectively. I = (~/2e)I/(µ0M2s V ) is a

normalized applied current. Therefore, the combined spin-torque acting on the FL

magnetization can be written as:

ΓS = I

√1 + tan2(ωeff)

1− λRmx

m× (m× n) (5.73)

140

where the orientation of the effective spin-polarization axis n is tilted with respect

to the in-plane (IP) direction by an angle ωeff with

tan(ωeff) =ηRηP

(1− λRmx

1− λPmz

)= tan(ω)

1− λRmx

1− λPmz

. (5.74)

Naturally, in the case of ηP = 0 (no out-of-plane (OOP) polarizer), ΓS will reduce

to the conventional collinear spin-torque expression and n will lie in plane.

A qualitative understanding of central zone of the state-diagram can be seen

from the form of the spin-torque in (5.72). The torque associated with the reference

layer is initially collinear with the damping torque. It thus leads to switching

via the antidamping mechanism, typical of spin-transfer devices with collinear

magnetizations. However, the spin-transfer torque from the polarizer (∝ m ×

(m× z)) is equivalent to an effective field in the direction m× z, which is initially

in the direction of the FL’s medium axis y. Such a field reduces the FL’s easy axis

coercive field (for both current polarities) as is the case in the Stoner-Wohlfarth

model with a medium axis magnetic field. In the Stoner-Wohlfarth model the result

is an astroid shaped switching boundary, which resembles the diamond shaped

bistable central zone of the state diagram.

More quantitatively, the spin-torque asymmetry parameters λR,P lead to torques

that depend on the magnetization state of the FL. For example, different current

magnitudes are typically necessary for AP to P and P to AP switching [3]. Gen-

erally, larger currents are needed for P to AP switching at the same field, as seen

in the state diagram (see Figure 5.19). For simplicity in our model, we consider

λP = 0, as mz is typically small during the switching process. Therefore, λR ac-

counts for the main asymmetries we observe in spin-torque switching. We study

the switching by simulating an ensemble of 5000 macrospins under the influence of

141

spin-torque, thermal noise and an applied field. The ensemble is initially taken to

be thermalized in one of its (two possible) equilibrium easy-axis states before a cur-

rent is applied. After a current is applied the ensemble reaches a new steady-state

configuration over the course of a microsecond. The simulation is then repeated at

a higher current in incremental steps using the steady-state ensemble of the previ-

ous current as the initial condition. Upon reaching the limit of our current range,

we incrementally reduce the current to reproduce the procedure in the current

swept experiments.

Simulations results are plotted in Figure 5.21 both for current ramped-up (a),

and current ramped-down (b) cases, with parameters determined as follows. D is

governed by magnetic shape anisotropy and is calculated based on the FL’s shape to

be 17. The spin-torque asymmetry is taken to reproduce the measured ratio of pos-

itive I+C to negative I−C switching currents at effective zero field I+

C /I−C = 2, giving

λR = 0.5. K = 12µ0MsHKV is estimated to be 3.3×10−19 J (i.e. ξ = K/kBT = 80,

with T = 300 K), taking µ0Ms = 1.5T and µ0HK = 35 mT. Simulations were run

with spin-torque ratios ηP/ηR = 0, 0.24, 0.51, 0.68 and damping α = 0.04. The

results are shown as a colormap where red and green represent in-plane AP/P

configurations and blue corresponds to IR states. Figure 5.21 shows results for

ηP/ηR = 0.68. (Results for the other spin-torque ratios studies are shown in the

Supplementary Materials of [130].) Depending on the proportions in which the

ensemble is partitioned between the three available states, bistability regions arise

and are represented by a superposition of the colors, an example being the dark

yellow (green+red) P/AP bistability region at the center of the state diagram. The

magnetic field is normalized to HK and current is normalized by the positive criti-

cal current I+C at zero temperature. The normalized critical current (I+

C /IC ,I−C /IC)

142

and coercive field (H+C /IC ,H−C /IC) are smaller than 1 because of thermal fluctua-

tions.

The simulation captures the main switching features observed in the experi-

ment both for current ramping up and down. First, we observe a distorted di-

amond shaped central P/AP bistable central zone (Figure 5.21(a)), which shows

bipolar switching near the layer’s coercive field. The distortion (e.g. the lower

switching current for AP to P transitions) is associated with the non-zero spin-

torque asymmetry parameter λR. Second, there are transitions into an out-of-plane

precessional state which we associate with the IR state we observe experimentally.

We find that the threshold current for P/AP to IR transitions is higher when

the current is increasing than the IR to P/AP transitions when the current is

decreasing (Figure 5.21(b)), as observed in experiment. The vertical boundaries

in Figure 5.21(b) are artifacts that result from not having applied large enough

currents to realize an IR state when the current was increasing. In this case, when

decreasing the current, the ensemble appears to not transition out of its initial

AP or P configuration. Deviations between the experiment and macrospin model

appear at large currents. For example, the curvature of the AP to IR switching

boundary is positive in the simulation but negative in the experimental data. This

indicates that more sophisticated models, such as micromagnetic models, of the

magnetization dynamics may be needed to explain the large current dynamics.

The hysteretic transitions to the IR state can thus be understood under the frame-

work of the analytic theory presented in this thesis, which examined the influence

of the spin-torque ratio (ηP/ηR) on the magnetization dynamics.

143

Figure 5.21: Simulations of an ensemble of 5000 macrospins represented as a state-diagram with current increasing (a) and current decreasing (b). The three relevantstates AP, P and IR, are color coded as red, green and blue respectively. Eachdata point is represented by a RGB color that is determined by the proportion ofthe ensemble populating each corresponding state. Currents are shown in unitsof switching positive current at zero field and room temperature. Applied fieldsare shown in units of anisotropy field. The parameters used in the simulation aredescribed in the main text.

144

Chapter 6

Structure of Escape Trajectories

In this final chapter, work we will attempt to construct a general overview

of the effective energy barriers characterizing thermally driven systems. Since

gradient systems exhibit energy barriers which can be derived directly from the

energy landscape of the physical system, we will be particularly interested in non-

gradient systems (see the final section of Chapter 2). Let us then consider an

n-dimensional Langevin equation of motion with multiplicative noise understood

in the Ito sense.

x = F(x) +√εH(x) · W (6.1)

where F(x) and H(x) are the model’s given drift field and space-dependent vari-

ance, with ε the noise strength. At any given moment in time, the stochastic

contribution results from the addition of multiple gaussian processes with different

variances. These can, in turn, all be condensed into one single gaussian process

˙W with effective variance given by square summation:∑

j HijWj = gi˙W where

g2i =

∑j H

2ij.

145

6.1 Friedlin-Wentzell (FW) theory

Since the probability of witnessing a specific dynamical trajectory depends on

the realization of the noise ˙W in the system, the assumed gaussianity of the Wiener

process allows us to state that, in the limit of weak noise ε, the probability of the

system evolving from an initial position xA to a final state xB will be given by:

PA→B ∝ exp

[−1

2

∫ B

A

W 2dt

]= exp

[− 1

2ε

∫ B

A

(∑i

(xi − Fi)2

g2i

)dt

]. (6.2)

In the limit of small noise ε, the most likely path between xA and xB will

then be the one that, in maximizing the probability of realization, minimizes the

effective Lagrangian LFW =∑

i(xi−Fi)2

2g2iwhere the suffix FW stands for Friedlin

and Wentzell, who systematically defined this approach [172]. We will call the

minimizing trajectory the most probable escape path (MPEP) and the action cor-

responding to its transition between the equilibrium and unstable fixed points of

F will represent the effective energy barrier of the escape.

Denoting by G the diagonal matrix with elements Gii = g2i and the associ-

ated drift vector with elements fi = Fi/g2i , the Friedlin-Wentzell Lagrangian and

associated Hamiltonian can then be conveniently written as:

LFW =1

2(x− F) · G−1 · (x− F) (6.3)

HFW =1

2

[(p + f) · G · (p + f)− f · G · f

], (6.4)

where p is the canonical momentum with elements pi = ∂xiLFW . Upon inspection

146

of (6.3), it is at once apparent that escape dynamics are equivalent to those of

an electrically charged particle moving on a Riemannian manifold with positive

definite metric G under the action of a vector potential A = −f and electric scalar

potential Φ = −f · G · f = −|f |2G

,1 whose equations of motion can be obtained at

once using Hamilton’s equations:

x = G · p + F = G · (p + f) (6.5)

p = −∇xHFW , (6.6)

where optimal escape trajectories are those for which the Hamiltonian vanishes

identically. Setting HFW = 0, for any given state vector x, the set of momenta

satisfying the zero energy condition defines an n-dimensional ellipse in momentum

space centered in p = −f , having axes ai =√

f · G · f/g2i .

Limiting ourselves to a 2-dimensional system for simplicity2, the momentum

ellipse can be conveniently parametrized as:

px(γ) =|f |Ggx

cos γ − fx (6.7)

py(γ) =|f |Ggy

sin γ − fy. (6.8)

For any given position in configuration space, this momentum ellipse defines all

possible least-action motions accessible by an escape trajectory traveling through x.

1Given a Riemannian metric Q, we will denot the inner product of two vectors a and b as〈a,b〉Q. Analogously, the inner product on Q of a vector with itself will be written 〈a,a〉Q ≡ |a|2Q.

2The procedure being presented can be straightforwardly generalized to higher dimensions byconstructing the parametrization of a n-dimensional ellipse.

147

Figure 6.1: Diagram of momentum ellipse parametrized by γ. γ0 and π + γ0

correspond to instanton and anti-instanton solutions respectively.

Anti-instanton trajectories (p = 0) characterizing noiseless relaxation correspond

to γ = γ0 ≡ −atan( fyfx

gygx

) and reverse-drift motion (p = −2f) correponds to

γ = π + γ0. Figure 6.1 shows a typical diagram of such a momentum ellipse.

Since γ parametrizes momentum at each point in configuration space, we can

substitute (6.7) into the Hamiltonian equations (6.5) obtaining the following set

of equations satisfied by x solely as a function of x and γ:

x = gx|f |G cos γ (6.9)

y = gy|f |G sin γ. (6.10)

The importance of γ as characterizing the direction of escape is at once apparent.

148

In fact, constructing y/x one obtains that the slope of the escape trajectory is

∂xy = (gy/gx) tan γ. In presence of solely additive, or isotropic noise, G ≡ 1,

(6.9) shows that thermally driven and deterministic dynamics will evolve at the

same rate, |x| = |f |. This is trivial for instanton x = −f and anti-instanton

x = f trajectories, and could have also been obtained directly by writing down the

effective Lorentz dynamics

x = ∇|f |2 − x× (∇× f) (6.11)

and noting that upon multiplying by x one obtains ∂t|x|2 = ∂t|f |2 which again

implies that the dynamical speed is equal to the norm of the drift field. This notion

has been employed in the literature [173, 174] to construct an efficient numerical

scheme (the String method) capable of computing transition pathways, free energy

barriers and transition rates in complex systems. The notion that the thermal

escape speed is identical to the modulus of the drift field at each point in the

configuration space of the system further allows us to write the Lorentz equation

of motion in the general form:

(S + A) · x = 2∇|f |, (6.12)

where S and A are skew-symmetric and antisymmetric matrices respectively. Dy-

namical equations of this form have been extensively studied in recent litera-

ture [175, 176] where the effective potential 2|f | appearing on the right side of

(6.12) is argued to be a suitable Lyapunov function for the dynamics of x.

This result can be argued more generally, even for systems with multiplicative

noise by inspection of the Friedlin-Wentzel Lagrangian:

149

LFW =1

2|x− F|2

G−1 =

1

2

[|x|2

G−1 + |F|2

G−1 − 2x · G−1 · F

]= |f |2

G− x · f

= |f |2G

(1− cos Ψ) (6.13)

where we have employed the following identities:

F = G · f (6.14)

x · G−1 · x = |f |2G, (6.15)

and defined

Ψ ≡ acos

(x · f|f |2

G

)(6.16)

as the angle between the instantaneous escape velocity x and the field orientation

f at x. We find that, as long as Ψ does not vary too much over the course of the

escape trajectory, the effective action U(x) =∫ x

x0LFWdt will be dominated by the

behavior of |f |G which is in agreement with the statement that the norm of the

drift field coarsely captures the structure of the system’s action.

Often, non-gradient dynamics depending on a tunable set of parameters can

exhibit radically different thermally driven behavior as soon as a critical parameter

threshold is crossed [100,103]. This is reminiscent of phase transitions and has been

explored extensively in the literature. Let us then look at a sample system under

the lens developed up to now. We consider the following 2-dimensional stochastic

150

model taken from Maier and Stein [177,178]:

x = x(1− x2 − αy2) +√εW (6.17)

y = −y(1 + x2) +√εW, (6.18)

where noise is isotropic and the drift field is non-gradient (∇ × f 6= 0) for all

α > 1 (the model’s tunable parameter). For dynamics constrained on the x ≥ 0

half-plane, the point xS = (1, 0) is a stable fixed point of the dynamics whereas

xU = (0, 0) is an unstable saddle point. One is typically interested in the most

probable escape path (MPEP) leading from xS to xU . The system is known to

follow the x-axis along a soliton trajectory that counters the flow for all α < 4. At

α = 4, the MPEP bifurcates into two trajectories symmetric with respect to the

x-axis, with all other unsuccessful escape trajectories forming caustics with focus

point along the x-axis.

A simple analysis of the norm of the drift field unveils that in transitioning

across the critical threshold αC = 4, the structure of the extrema of f changes

sharply. If for α < 4, the norm exhibits two global minima at the fixed points

along with a saddle along the x-axis, for α > 4 two symmetric local minima

appear off the x-axis transforming the previous saddle into a local maxima. These

local minima are responsible for lowering the effective action of escape trajectories

transiting off the x-axis (see Figures 6.2).

151

Figure 6.2: Contour plot of the norm of the drift field taken from the Maier-Steinmodel [177]. On the left, for α = 3, two global minima are present at the unstable(0, 0) and stable (1, 0) equilibria respectively along with a saddle along the x-axis.On the right, for α = 5, the previous saddle has become a local maxima due tothe appearance of two new local minima off the x-axis.

152

6.2 Macrospin Escape Paths

The results of the previous section briefly allow us to make an important ob-

servation regarding the MPEPs expected to be followed by a macrospin subject to

spin-torques and thermal noise. For any non-zero applied current, the drift dynam-

ics (1.28) are non-gradient due to the non-conservative character of spin transfer

torque effects. One might then wonder whether the structure of the escape dy-

namics might be expected to change drastically as the dynamical parameters are

altered.

In the collinear uniaxial limit, the Hamiltonian dynamics arising from the min-

imization of the Friedlin-Wentzell action can be solved analytically. The F-W

Lagrangian for the thermally activated dynamics of the macrospin, expressed in

spherical coordinates (θ,φ), are:

LFW =1

2

[(θ − fθ)2 + sin2(θ)(φ− fφ)2

]. (6.19)

Passing to the Hamiltonian formalism we have:

HFW =1

2

[p2θ +

p2φ

sin2(θ)

]+ pθfθ + pφfφ, (6.20)

where pθ and pφ are the conjugate momenta of θ and φ.

Hamilton’s equations are then

153

θ = pθ + fθ (6.21)

φ =pφ

sin2(θ)pφ + fφ (6.22)

pθ =cos(θ)

sin3(θ)p2φ − pθ∂θfθ − pφ∂θfφ (6.23)

pφ = −pθ∂φfθ − pφ∂φfφ, (6.24)

where, up to this point, the setup is completely general and extensible to models

of any complexity (anisotropy ratio D 6= 0, positive tilts between axes, etc...). For

a uniaxial macrospin with collinear easy-/spin-current axes, the drift vector field

of the θ,φ dynamics can be obtained by rewriting (1.28) in spherical coordinates.

One then obtains

fθ = −α(j + cos(θ)) sin(θ) (6.25)

fφ = − cos(θ), (6.26)

where, in addition, j is the normalized spin-torque intensity.

We immediately notice that φ is a cyclic variable for this model and, as a result,

we expect pφ = c constant. Imposing our interest in a zero energy trajectory

starting from the hamiltonian HFW we can at once solve for pθ:

pθ = −fθ

[1±

√1− c

(2fφfθ

+αc

ξfθ sin2(θ)

)]. (6.27)

This solution is physically valid as long as the radicand is greater than or equal

154

to zero for all values of θ in the range of interest. This necessarily implies that

pφ = c = 0 which, in turn, is expected to remain constant for the whole switching

dynamics. The azimuthal coordinate φ can hence be expected to evolve following

the deterministic drift field even during thermal escape: φ = fφ. As a consequence,

pθ = −fθ[1 ± 1]. This is consistent with our knowledge of the gradient character

of the dynamics. In fact, pθ = 0 corresponds to standard relaxation to equilibrium

and pθ = −2fθ corresponds to an against-gradient instanton escape of the θ coor-

dinate. It is interesting to note that in the latter case, the precessional dynamics

of φ remain unaltered and, as such, escape from the stable well does not merely

correspond to a reflection in time.

Having been able to solve for the general momenta in terms of the spherical

coordinates, we can analytically derive the relation between φ and θ along the

escape trjectories. Dividing φ by θ from Hamilton’s equations, one has:

φ(θ) =1

α

∫ θ

π/2

cos(θ)

(j + cos(θ)) sin(θ)

=1

α(1− j2)log [tan(θ/2)]− j log

[−sin(θ/2) cos(θ/2)

j + cos(θ)

]− j log(2j)

+ const (6.28)

where the integration constant depends on the choice of the initial φ value at

the singular θ = π pole. Figure 6.3 compares this analytical result to the escape

trajectories obtained by numerically solving Hamilton’s equtions using a standard

shooting method. Various initial orientations of the escape trajectory at the θ = π

pole are simulated and, for each, the FW action is computed. Upon correcting for

the additive constant at the end of equation (6.28) we obtain very good matching

155

Figure 6.3: Escape trajectory for a uniaxial macrospin model with j = 0.3 andω = 0. Blue line shows the least-action result of numerical integration of FWdynamics. Dashed red line is the analytical result shown in equation (6.28).

between theory and numerics.

The inclusion of a small tilt ω between easy- and spin polarization axes does

not alter the escape dynamics. It simply rescales the current magnitude appearing

in 6.28 by the tilt-dependent critical switching current (j → j cosω) fits the data

very well (see Figure 6.4).

Upon passing to a biaxial macrospin model, it is not possible to solve the FW

dynamics analytically and no straightforward way exists to modify our collinear

uniaxial escape trajectory in a satisfactory way. We can however look at the norm

of the macrospin drift field and check wither its topological structure is stable

as the main parameters of the theory (α and j) are tuned. Without diving into

any calculations, we limit ourselves to state that the smallness of the damping

parameter α precludes any drastic alterations to the set of maxima and minima

156

Figure 6.4: Escape trajectory for a uniaxial macrospin model with j = 0.3 andω = 0.2π. Blue line shows the least-action result of numerical integration of FWdynamics. Dashed red line is the analytical result shown in equation (6.28).

appearing in the norm of the macrospin’s drift. The contribution of the precessional

terms will in fact dominate for all α ' 10 and lower. This should not surprise the

reader as this property of the macrospin dynamics is precisely what allowed the

development of the CEOA theory.

157

Conclusion

This thesis work has constructed and studied the theory underlying the dynam-

ics of a biaxial macrospin subject to thermal noise and the presence of spin-torque

due to both a perpendicularly magnetized polarizer and an in-plane magnetized

reference layer. Their combined spin-torque effects leads to an effective tilt ω

between the easy- and spin-polarization axes. The full stochastic magnetization

dynamics were solved employing the parallelization properties of graphics process-

ing units (GPUs) to repeatedly integrate the macrospin’s Langevin dynamics in an

effort to reconstruct its non-equilibrium ensemble properties. The technique has

allowed us to explore thermally activated behavior out to microsecond timescales.

We have also presented a theory capable of reducing the complexity of the

3D macrospin dynamics, under the action of both spin-torque and thermal noise,

to a 1D stochastic differential equation in the energy space of the macrospin. This

was achieved by averaging the sLLGS dynamics over constant energy trajectories

that were derived analytically.

Under such an approximation, the resulting theory predicts that the geometries

involved influence the respective dynamics in very precise ways. Particularly, we

found that the angular tilt between spin polarization and the easy-axes factors

158

into the dynamics only as a trivial rescaling of the applied current. Similarly, the

relative orientation of the hard axis is predicted to play no role in the switch-

ing dynamics under conditions where the constant-energy orbit averaging (CEOA)

assumption is a valid. We employed the theory to study the macrospin’s behav-

ior both within in-plane (IP) and out-of-plane (OOP) energetic basins. In both

scenarios, the main parameter characterizing the different dynamical regimes was

found to be the ratio D between the hard and easy axis anisotropies.

For IP dynamics, two critical currents I1C = (D + 2)/(2 cosω) and I0

C =

(2/π)√D(D + 1)/ cosω were found to exist: the relative magnitude of these crit-

ical currents depending nonlinearly on D. For D > D0 ' 5.09, we showed that

stable limit cycle magnetization precessions appear in well-defined current ranges;

transitions between these stable limit cycles proceeds through thermal activation.

When D < D0, limit cycles generally do not appear and the switching dynamics

become qualitatively similar to those of a uniaxial macrospin model.

Friedlin-Wentzell theory was employed to analytically study the exponential

scaling dependence between mean switching time and applied current in thermally

activated scenarios. The exact analytical scaling was reduced to quadratures and

an analytical approximation was suggested. The resulting analytical scaling de-

pendence to the standard form log(τ) ∝ ξ(1− I/IC)β and the current dependence

of the exponent β was studied (5.45). The exponent β was found to depend non-

linearly on the applied current intensity, similar to [147]. In the uniaxial macrsopin

limit D → 0, the constant β = 2 result was recovered. For D 6= 0, β was found to

depend on both the applied current intensity and the precise value of D.

For OOP dynamics, applied currents greater than IOOP = (2/π)√D + 1/ sinω,

were found to give rise to stable fixed points in the macrospin’s energy dynamics.

159

This is consistent with the description of a stable limit cycle, interpreted as an

OOP precessional state. We predict that stable OOP precessions are possible only

in one of the two out-of-plane directions, selected by the direction of the applied

current. Furthermore, by comparing our results to those obtained via CEOA

methods to study the threshold currents for magnetic switching, we predict the

occurence of hysteretic transitions between IP and OOP stable states for effective

tilts larger than a critical tilt ωC = arctan(1/√D), which has been observed in very

recent experiments [130]. For tilts ω < ωC , we predict that magnetic switching

will take place since the threshold current for onset of stable OOP precessionary

states is expected to be larger than that required for destabilization of the initial

IP basin. Overall, this leads to a very simple condition that spin-valves or MTJs

must satisfy to behave like a spin-torque oscillator (STO): ηref/ηpol <√D. Our

theory agrees with numerical results and represents a starting point for testing

how well the macrospin approximation captures the magnetization dynamics in

real world devices.

Upon exploring the thermal contribution to oscillator linewidth broadening,

we observe the existence of a critical temperature TC separating a regime where

phase noise dominates decoherence and one where decoherence is the result of both

phase and amplitude noise. The former cannot be accounted for by our CEOA

theory and is a result of the full complexity of the LLG dynamics. This is in

agreement with self-oscillator theory, where a transition temperature is predicted

to exist between a phase noise dominated regime at large temperatures and one

limited by thermal deflections about the equilibrium magnetic trajectory at low

temperatures [18, 161].

Our methodology is similar to that proposed by Slavin, Tiberkevich and Kim [12,

160

146, 148]. However, instead of approaching the multiscaling analysis by study-

ing the complex oscillatory amplitude of the macrospin’s dynamics using a self-

oscillator equation, we focused on the macrospin’s diffusion over its energy land-

scape. The loss of generality in doing so is compensated by new insights into the

macrospin’s dynamical characteristics capable of describing parameter spaces of

the macrospin model previously considered analytically unobtainable.

Last but not least, in the final chapter we have shown how to bypass the

limits of the constant energy orbit averaged theory and numerically reconstruct

the thermal escape properties of the macrospin dynamics. In doing so, we have

framed the sLLGS dynamics within the more general context of noisy dynamical

systems driven by nonconservative force fields. The most probable escape paths

MPEPs of such systems are in fact known to often exhibit sudden transitions due

to parametric instabilities. We have looked into whether the macrospin model may

exhibit such behavior and have found that the small size of the Landau damping

constant guarantees that such parametric instabilities will not occur.

Overall, the sLLGS dynamics studied throughout this thesis represent a first

approximation of the micromagnetic dynamics exhibited by thin (< 10 nm) mag-

netic layers . The choice of driving the system via DC currents was paramount

towards the numerical stability of the stochastic integrators employed. Upon in-

troducing other dynamical degrees of freedom (i.e. AC currents, non-constant MS,

etc.) the dynamics will likely become chaotic adding an entire layer of difficulty

to their analytical treatment. This should, however, not discourage the curious re-

searcher. The exploration of chaotic regimes to the macrospin dynamics might in

fact allow for more efficient techniques for destabilizing the IP basins allowing for

either faster switching transients or, conversely, more energy efficient destabiliza-

161

tion. It remains to be understood how far the macrospin model can truly be taken

as a proxy for the much richer micromagnetic dynamics. The true dynamics of a

magnetic element might in fact transition between regimes where the macrospin

model is applicable only on short timescales. Ultimately, the development of effi-

cient integration methods capable of evolving non-coherent magnetic states subject

to the thermal noise will be crucial towards answering such problematics.

162

Appendix A

Constant Energy Orbit Averaging

of sLLGS Equation

The procedure outlined in the main text considers the total time differntial of

the macrospin’s reference landscape energy:

ε = 2 (Dmzmz −mxmx) , (A.1)

where mx and mz are to be substituted in by employing the sLLGS dynamical

equations (1.28) which we rewrite for convenience.

163

mx = Dmzmy

+ α[(Inx +mx)(1−m2

x) + Inzmx(√

1− n2mz − nmy) +Dmxm2z

]my = −(D + 1)mxmz

+ α[mymz(Inz

√1− n2 +Dmz)−mxmy(Inx +mx) + Inzn(1−m2

y)]

mz = mxmy

− α[(Inz√

1− n2 +Dmz)(1−m2z) +mxmz(Inx +mx) + Inznmzmy

].

(A.2)

Plugging the sLLGS equation into the time differential for the macropin’s en-

ergy then results in:

ε = −2α

[Inz√

1− n2(Dmz −Dm3

z +m2xmz

)+ Inx

(Dmxm

2z +mx −m3

x

)+ Inznmy

(Dm2

z −m2x

)+(D2m2

z −D2m4z +m2

x −m4x + 2Dm2

xm2z

) ], (A.3)

which, collecting all terms Dm2z −m2

x = ε, simplifies to:

ε = −2α[I cos(ω)(1 + ε)mx + I sin(ω) cos(ψ)(D − ε)mz

+ I cos(ω) sin(ψ)εmy +D2m2z +m2

x − ε2] (A.4)

We remind the reader that our choice of notation was such that nx = cos θ,

164

nz = sin θ and n = sinψ represent the various polar and azimuthal tilts of the net

effective spin-polarization axis’ orientation with respect to the macropin’s easy-

and hard-axes.

Deriving now the energy evolution equation corresponding to IP (5.22) and

OOP (5.49) dynamics is a straightforward manner. In particular, the symmetry

of constant energy trajectories (see Figure 1.1) immediately makes sure that the

third term appearing in (A.4) is zero.

We now proceed to show how to compute the averages appearing in (5.22) and

(5.49) explicitely.

A.1 IP dynamics: 〈mx〉 and 〈m2x〉

The next step is to show how the necessary averages can be computed explicitly.

As an example, consider 〈mx〉; its average is given by:

〈mx〉 =1

T (ε)

∫ T

0

dtm0x(t), (A.5)

where the integration is over time and one uses the expression for m0x in terms

of its Jacobi elliptic function. We can express the same integral in terms of the

geometrical parametrization (5.21). In fact

〈mx〉 =1

T (ε)

∫ T

0

dtm0x(t) =

1

T (ε)

∮dw

dm0z/dw

mz0

m0x(w). (A.6)

165

Putting together the expression for mz0 from (5.2) and, again, employing the

geometrical parametrizations from (5.21), we obtain

〈mx〉 = − 4

T (ε)√D + 1

∫ acosh(1/γ−)

0

dwcosh(s)√

1− γ2− cosh2(s)

= − 2π

T (ε)√D + 1

= −π2

√D

D + 1− γ2−

1

K(1− γ2−). (A.7)

Repeating the procedure for the 〈m2x〉 term in (5.22),

〈m2x〉(ε) =

1

T(ε)

∫ T

0

dtm2x(t) =

4

T(ε)

∫ acosh(1/γ−)

0

ds|∂sm0z

m0x

|(m0x)

2

=4

T (ε)γ2−

√D

D + 1(D + 1− γ2

−)

∫ acosh(1/γ−)

0

dscosh(s)√

1− γ2− cosh2(s)

=4

T (ε)√D + 1

√D

D + 1− γ2−η1

=D

D + 1− γ2−

E(1− γ2−)

K(1− γ2−), (A.8)

where we have used:

T (ε) = 4

√D + 1− γ2

−

D(D + 1)η0(γ−) = 4

√D + 1− γ2

−

D(D + 1)K(1− γ2

−) (A.9)

A.2 OOP dynamics: 〈mz〉 and 〈m2z〉

To compute the constant energy orbit averages in (5.49), we write the integrals

using the geometric parametrizations (5.51):

166

〈mz〉m0 =±

T(ε)

∫ T

0

dtmz(t) =±4

T(ε)

∫ acosh(1/γ+)

0

ds|∂sm0z

m0z

|m0z

=±4

T(γ+)

γ+√D(D + 1)

∫ acosh(1/γ+)

0

dscosh(s)√

1− γ2+ cosh2(s)

=±π

2√D(1− γ2

+) + 1

1

K[1− γ2+].

(A.10)

Proceeding analogously for 〈m2z〉:

〈m2z〉m0 =

1

T(ε)

∫ T

0

dtm2z(t) =

4

T(ε)

∫ acosh(1/γ+)

0

ds|∂sm0z

m0z

|(m0z)

2

=4

T(γ+)

γ2+√

D(D + 1)√

1 +D(1− γ2+)

∫ acosh(1/γ+)

0

dscosh2(s)√

1− γ2+ cosh2(s)

=1

1 +D(1− γ2+)

E[1− γ2+]

K[1− γ2+],

(A.11)

where, as stated in the main text, E[x] is the complete elliptic integral of the second

kind.

In both derivations we have taken advantage of eqns. (15) and (24) to write

the period as a function of γ+. Written explicitly, the period reads:

T(ε) =4√

D(1 + ε)K[

D − εD(1 + ε)

] = 4

√1 +D(1− γ2

+)

D(D + 1)K[1− γ2

+]. (A.12)

167

A.3 Proof of Special Elliptic Integral Identities

In this section we’d like to prove a non-trivial elliptic integral identity used in

simplifying the expressions for (5.28) and (5.52).

e(x) ≡ xE(1− 1

x2) = E(1− x2) (A.13)

k(x) ≡ 1

xK(1− 1

x2) = K(1− x2) (A.14)

can be immediately proven by considering the defining differential equation satis-

fied by K(x), namely [179]:

∂xK(x) =E(x)− (1− x)K(x)

2x(1− x). (A.15)

Computing then explicitely the derivative of k(x) using definition (A1) and

rearranging, one finds that:

∂xk(x) =e(x)− (1− x)k(x)

2x(1− x), (A.16)

in other words they satisfy the same differential relation. The identical procedure

can be performed on e(x) thus proving the assertions made.

168

Appendix B

Orbit averaging of a Stratonovich

Equation

There are several advantages in adopting a Stratonovich convention when writ-

ing the dynamical equations. First, it is the most natural way of modeling a

physical process where the Gaussian noise represents the short correlation time

limit of a colored noise process: by the Wong-Zakai theorem [92], such a limit

of multiplicative noise converges to Statonovich calculus. Second, a Stratonovich

interpretation follows the conventional rules of calculus in dealing with functions

of a stochastic variable. Third, many conventional numerical schemes used to sim-

ulate Langevin equations (such as the Heun scheme adopted for this work) evolve

towards the Stratonovich solution.

The Stratonovich formulation of a stochastic differential equation (SDE), how-

ever, fails to accurately represent the correlation between multiplicative terms and

the specific noise realization [88]. To average the multiplicative noise terms over

constant energy orbits, we take advantage of the fact that sums of Gaussian ran-

169

dom variables∑

i µixi (where xi are standard 0 mean and variance 1 Gaussian vari-

ables) behave like a single Gaussian variable x with variance given by the square

sum of the individual variances µ2 =∑

i µ2i . Since the multiplicative noise terms

B(m) W appearing in our sLLGS equations are state-dependent, the Gaussian

variable summation cannot be employed due to the temporal correlation between

the state-dependent variances B2(m) and the specific noise realization W.

This problem can be avoided by converting the sLLGS equations into their Ito

representation. The multplicative noise terms of (5.49) become (DmzBxj−mxBzj)·

Wj (with summation over repeated indices). The state-dependent variances are

now uncorrelated with respect to the noise realization, and so a summation of

Gaussian random variables can now be employed. Averaging over constant energy

orbits then leads, after a bit of algebra, to the noise term appearing in (5.49).

Altering the multiplicative noise convention can generally alter the qualitative

nature of the solution to the stochastic differential equation. To maintain con-

sistency between Ito and Stratonovich models, the drift term must be modified

to ensure that Boltzmann equilibrium is obtained at long times in the absence of

non-conservative forces (in our case, the applied current). The fundamental rea-

son is that the SDE is simply a model of the underlying dynamics subject to two

constraints: the chosen form of the thermal noise and the steady-state equilbrium

Boltzmann distribution [164, 184]. In the absence of applied currents, (5.49) can

be written more concisely as:

〈∂tε〉 = [−αf(ε) + h(ε)] +

√2α

ξf(ε) · W (B.1)

170

with

f(ε) = 2[D(D + 1)〈m2

z〉+ ε(1 + ε)], (B.2)

where h(ε) represents the extra modification necessary in the drift term to retain

all physically relevant Boltzmann relaxation properties. Deriving the Ito Fokker-

Planck equation relative to such a dynamic then gives:

∂tρ = ∂ε

[(αf(ε)− h(ε) +

α

ξ∂εf(ε))ρ+

α

ξf(ε)∂ερ

]. (B.3)

Upon imposing h(ε) ≡ αξ∂εf(ε), the steady-state solution reduces to the simple

form ρeq(ε) ∝ exp[−ξ ε] as expected.

Employing the previously derived expression for 〈m2z〉 from Appendix A, h(ε)

is found to be (in terms of the auxiliary variable γ):

h(ε) =α

ξ

D(1− γ2+) + 1

1− γ2+

×[1−

(D(1− γ2

+) + 2

D(1− γ2+) + 1

)E[1− γ2

+]

K[1− γ2+]

+1

γ2+(2− γ2

+)

(E[1− γ2

+]

K[1− γ2+]

)2]

+α

ξ

D(1 + γ2+) + 1

D(1− γ2+) + 1

, (B.4)

which can be shown to lead to a negligible correction of the drift dynamics (≈

0.1α/ξ ≈ 10−5 since typical parameter values are α ∼ 0.01 and ξ ∼ 100).

171

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