Theoretical Investigations of Skyrmions in Chiral Magnets

Dissertation

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

James Rowland, B.S.

Graduate Program in Physics

The Ohio State University

2019

Dissertation Committee:

Professor Mohit Randeria, Advisor Professor Nandini Trivedi Professor Samir D. Mathur Professor Fengyuan Yang c Copyright by

James Rowland

2019 Abstract

Magnetic skyrmions have attracted substantial interest due to their potential for use in spintronics devices and next-generation information storage, and due to the novel phenomena that result from the topological winding of skyrmions. In this dissertation we theoretically study the properties of magnetic skyrmions in chiral magnets.

We focus on two key parameters that must be optimized to achieve maximum device performance: skyrmion size, and skyrmion stability. Skyrmions are stabilized in materials where inversion symmetry is broken. We show that the skyrmion crystal phase is more stable in systems with broken mirror inversion symmetry compared with systems where only bulk inversion symmetry is broken. To understand this effect we study a system where both mirror and bulk inversion symmetry are broken.

We show that broken bulk inversion symmetry tends to stabilize a conical phase, and this phase becomes progressively less stable when broken mirror inversion symmetry is introduced. The phase diagram reveals a large region of skyrmion crystal stability, as well as a stable elliptic cone phase, and a square skyrmion crystal phase.

In addition to crystal structures with broken inversion symmetry, the presence of an interface in magnetic ﬁlms introduces a source of broken mirror symmetry.

We show that this added source of symmetry breaking enhances the stability of the skyrmion crystal phase. Films surfaces and interfaces also stabilize a novel phase of matter called a chiral bobber crystal. This phase can be uniquely identiﬁed by

ii the presence of a two-dimensional lattice of singular points in the magnetization ﬁeld called Bloch points. We present experimental evidence for the observation of a chiral bobber crystal using magnetization data.

Skyrmions can also be found outside the skyrmion crystal phase as metastable, particle-like excitations with a ﬁnite lifetime. We show that the lifetime and size of skyrmions have a strong interdependence. Putting limits on the lifetime of a skyrmion also restricts the range of skyrmion sizes. For a given lifetime, the smallest skyrmion size is determined by competition between the exchange interaction and the Dzyaloshinskii-Moriya interaction. We also show that the lifetime of a skyrmion depends essentially on two parameters: the energy of a skyrmion, and the energy of a Bloch point. In two-dimensional systems (magnetic ﬁlms and interfaces), the

Bloch point forms in space-time; in three-dimensional systems, the Bloch point forms totally in space and has the structure of a chiral bobber. Finally, we show that entropic corrections to the lifetime are typically small, except near a special point in the phase diagram where skyrmion size diverges, and soft-modes contribute to a divergence in the entropy of a skyrmion.

iii Acknowledgments

During my time at OSU, I have had tremendous support from friends, family, colleagues, and mentors. Out of these, I would ﬁrst like to thank my advisor, Professor

Mohit Randeria. His ability to conceptualize and attack complex problems is uncanny and infectious; and, without a doubt, I can say that following his guidance has always brought me closer to my goals.

I am also deeply grateful for the mentorship I received from my dissertation committee: Professors Nandini Trivedi, Fengyuan Yang, and Samir Mathur, as well as from Professors Yuan-Ming Lu, Chris Hammel, Ilya Gruzberg, Ciriyam Jayaprakash,

Richard Furnstahl, and Yuri Kovchegov. They are superior role models and teachers, and their lessons are entangled in my dissertation.

OSU is home to a herculean experimental effort to study magnetic skyrmions that includes Professors Chris Hammel, Fengyuan Yang, Roland Kawakami, Jay Gupta, and David McComb. They have influenced my growth as a scientist through our numerous meetings and scientific discussions. I worked closely with their graduate students and postdocs, especially Brendan McCullian, Bryan Esser, Aidan Lee,

Joseph Corbett, Jacob Repicky, and Adam Ahmed. These are brilliant scientists, and their experiments played a large role in the formation of my dissertation. I am proud to call them friends and colleagues. Chapter 4 is based primarily on a project borne out of the experiments of Adam Ahmed.

iv I would like to thank my peers in condensed matter theory. The graduate students who came before me (Onur Erten, Will Cole, Eric Duchon, Mason Swanson, and Nganba Meetei) oﬀered me tutorials in CMT, and advice about the pitfalls and windfalls of graduate school. My contemporaries (Chris Svoboda, Tim McCormick,

Hassan Khan, Noah Charles, Jiaxin Wu, Cheng Li, Joseph Szabo, Franz Utermohlen,

David Ronquillo, Wenjuan Zhang, Alex Davis, Bowen Shi, Waylon Chen, Tamaghna

Hazra, Po-Kuan Wu Nishchhal Verma, Robert Ivancic, Ian Osborne, and Michael

Ferrarelli) helped me deepen my understanding of physics in a way that can only be achieved through the shared experience of learning. Po-Kuan Wu has been in- dispensable in helping me complete Chapter 5 of my dissertation. I received a great deal of support and companionship from the CMT postdocs at OSU (Sumilan Baner- jee, Biao Huang, David Nozadze, Mehdi Kargarian, Kyusung Hwang, Desmond Yin,

Kyungmin Lee, Nirav Patel, Subhasree Pradhan, Alex Rasmussen, and Xin Dai). I am especially grateful for the time I spent with Sumilan Banerjee. Sumilan is a world class physicist, and a selﬂess mentor. The foundation of my dissertation, Chapter 2 and Chapter 3, was built under his guidance.

My friends and family contributed to my dissertation by oﬀering unwavering emo- tional support. I thank Sara Mueller, Sierra O’Bryan, Chris Pierce, Blythe More- land, Carola Purser, Anne Benjamin, Khalida Hendricks, Oindree Banerjee, and

Mark Schillaci; together, we successfully completed our ﬁrst year courses, and we somehow managed to ﬁnd time and energy for great conversations. I thank my dear friends Stephen Smith, Michelle Smith, Christian Stackhouse, Juliana Stackhouse,

Alex Mauney, Morgan Carter, and Brian Grose, for encouraging me to pursue my dreams, and for keeping in touch even when it is inconvenient. I thank my wife,

v Amanda Thompson, for her love and support, and for listening to long didactics about whatever physics topic has caught my fancy. I also thank my parents, grand- parents, and my sister, for cultivating my sense of optimism and my passion for discovery.

Last, but not least, I would like to thank the National Science Foundation for funding my research through the Graduate Research Fellowship Program under Grant

No. DGE-1343012, and through the Center For Emergent materials, an NSF MRSEC, under Grant No. DMR-1420451. I would also like to thank the Defense Advanced Re- search Projects Agency for funding my research through the Topological Excitations in Electronics program under Grant No. D18AP00008.

vi Vita

2013 ...... B.S. Physics

Publications

Research Publications

Size and lifetime of metastable skyrmions, J. Rowland, P. K. Wu, M. Randeria. in preparation Spin-Hall Topological Hall Effect in Highly Tunable Pt/Ferrimagnetic-Insulator Bilayers, A. S. Ahmed, A. J. Lee, N. Bagués,B. A. McCullian, A. Thabt, A. Perrine, P. K. Wu, J. Rowland, M. Randeria, P. C. Hammel, C. W. McComb, F. Yang. Nano Lett. forthcoming A Novel Assay for Profiling GBM Cancer Model Heterogeneity and Drug Screen- ing, C. T. Stackhouse, J. Rowland, R. S. Shevin, R. Singh, G. Y. Gillespie, C. D. Willey. Cells 8 (7), 702 (2019) Chiral bobbers and skyrmions in epitaxial FeGe/Si (111) films, A. S. Ahmed, J. Rowland, B. D. Esser, S. R. Dunsiger, D. W. McComb, M. Randeria, R. K. Kawakami. Phys. Rev. M 2 (4), 041401 (2018) Molecular beam epitaxy growth of [CrGe/MnGe/FeGe] superlattices: Toward ar- tificial B20 skyrmion materials with tunable interactions, A. S. Ahmed, B. D. Esser, J. Rowland, D. W. McComb, R. K. Kawakami. J. Cryst. Growth 467, 38 (2017) Skyrmions in chiral magnets with Rashba and Dresselhaus spin-orbit coupling, J. Rowland, S. Banerjee, M. Randeria. Phys. Rev. B 93 (2), 020404 (2016) Enhanced stability of skyrmions in two-dimensional chiral magnets with Rashba spin-orbit coupling, S. Banerjee, J. Rowland, O. Erten, M. Randeria. Phys. Rev. X 4 (3), 031045 (2014)

vii Fields of Study

Major Field: Physics

viii Table of Contents

Page

Abstract ...... ii

Acknowledgments ...... iv

Vita...... vii

List of Figures ...... xii

1. Introduction ...... 1

2. Skyrmions with Rashba SOC ...... 5

2.1 Ginzburg-Landau Theory ...... 8 2.2 Phase Diagram ...... 10 2.2.1 Easy-Plane vs. Easy-Axis Anisotropy ...... 11 2.2.2 Phase Transitions ...... 13 2.3 Discussion ...... 14 2.4 Conclusions ...... 15 2.5 Technical Details of Calculations ...... 16 2.5.1 Variational Calculation ...... 16 2.5.2 2D Minimization ...... 18 2.5.3 Skyrmion Cell Size and Core Radius ...... 22 2.5.4 Phase Transition From SkX to Easy-Axis FM ...... 23

3. Skyrmions With Mixed DMI ...... 26

3.1 Summary of Results ...... 27 3.2 Free Energy ...... 30 3.3 Phase Diagram ...... 31 3.4 Helicity and Ferrotoroidic Moment ...... 32

ix 3.5 Rashba Limit ...... 33 3.6 Spin Textures and Topological Charge ...... 34 3.7 Conclusions ...... 35 3.8 Technical Details of Calculations ...... 36 3.8.1 Continuum Free Energy ...... 36 3.8.2 Ferromagnetic and Cone Phases ...... 40 3.8.3 Numerical Methods ...... 42 3.8.4 Variational Solution ...... 48 3.8.5 Phase Transitions ...... 51 3.8.6 Rashba Limit Phase Diagram ...... 53 3.8.7 Magnetic Anisotropy ...... 55

4. Chiral bobbers ...... 57

4.1 Model ...... 59 4.2 Estimate for Anisotropy of FeGe Thin-Films ...... 61 4.3 Experimental Evidence for Chiral Bobbers ...... 62 4.4 Conclusions ...... 65

5. Skyrmion Lifetimes ...... 67

5.1 Energy Barrier for Skyrmion Collapse ...... 69 5.1.1 Numerical Results ...... 70 5.1.2 Variational Results ...... 72 5.2 Entropic Corrections to Skyrmion Lifetimes ...... 78 5.2.1 Harmonic Transition State Theory for Spin Systems . . . . 79 5.2.2 Numerical Results ...... 82 5.2.3 Variational Results ...... 85 5.3 Skyrmion Collapse in 3D ...... 92 5.3.1 Numerical Results ...... 93 5.3.2 Variational Results ...... 95 5.3.3 Entropy of a Skyrmion Tube ...... 96 5.3.4 Entropy Barrier in 3D ...... 97 5.4 Conclusions ...... 98 5.5 Technical Details of Calculations ...... 100 5.5.1 Novel Method for Finding Saddle Points ...... 100

5.5.2 Dimensionless Integrals IX (α)...... 101 5.5.3 Viariational Methods for Entropy Calculations ...... 104 5.5.4 Bose Statistics ...... 105 5.5.5 Zero Field Spiral to Ferromagnet Phase Transition . . . . . 105 5.5.6 Lattice Spin Models in TensorFlow and PyTorch ...... 107

x Bibliography ...... 109

xi List of Figures

Figure Page

1.1 Illustration of a map from (a) real-space to (b) spin-space that has

skyrmion number Nsk = −1. To calculate the skyrmion number, every triangle on the real-space lattice is mapped to a spherical triangle on the unit sphere. Spherical triangles that preserve the orientation of the real space triangle (indicated by black arrows) correspond to positive skyrmion density, while spherical triangles with opposite orientation, e.g., the highlighted magenta triangle, correspond to negative skyrmion density. The area of the spherical triangle determines the magnitude of skyrmion density. The skyrmion number is the signed skyrmion density summed over all plaquettes. This deﬁnition corresponds to equation (1.1) in the continuum limit...... 2

2.1 (a) A skyrmion configuration. (b) The anisotropy-field phase diagram with ferromagnetic (FM), spiral and skyrmion crystal (SkX) phases 2 for D/J = 0.01 and AcJ/D = 1/2 with A = Ac + As. Double lines denote first order transitions, while the single line is an unusual first order transition with a divergent length scale; see text. The dashed line H = 2A separates the out-of-plane FM from the tilted FM. mz for the FM is shown in the color bar. Results are obtained from a circular-cell variational calculation...... 6

xii 2.2 Skyrmion core structure from circular-cell calculation with D/J = 0.01 2 and AcJ/D = 1/2. Here LD = (J/D)a where a is the microscopic lattice spacing. (a, b): False color plots of mz (shown in color bar). (c, d): Angle-averaged topological charge density |2πrχ(r)| and mz(r) (right axes). Left panels (a) and (c) correspond to easy-axis anisotropy AJ/D2 = −0.5 and HJ/D2 = 0.28. The skyrmion core is conventional with a single peak in the topological charge density. Right panels (b), (d) are for easy-plane anisotropy AJ/D2 = 1.35 and HJ/D2 = 1.96. Here the core has a large ‘transition’ region (yellow-orange) from down (center) to up (boundary) in m leading to an unusual two-peak structure for |2πrχ|...... 12

2.3 The phase diagram shown here is obtained as a result of a full 2D variational calculation, as distinct from the eﬀectively 1D variational calculation shown in Fig. 2.1(b). The symbols and parameters used are exactly the same as described in the caption for Fig. 2.1(b). Note that the 2D square cell calculation and the 1D variational calculation, although quite diﬀerent in their computational complexity, nevertheless lead to essentially identical results for the overall phase diagram. The dotted boundaries shown here are obtained from the simplest ‘linear’ variational ansatz for SkX described in text...... 19

2 2.4 Skyrmion core structure with D/J = 0.01 and AcJ/D = 1/2 obtained from a full 2D variational calculation, which should be compared with circular-cell results shown in Fig. 2.2. (a, b): False color plots of mz (shown in color bar). (c, d): Angle-averaged topological charge density |2πrχ(r)| and mz(r) (right axes). Left panels (a) and (c) correspond to easy-axis anisotropy AJ/D2 = −0.5 and HJ/D2 = 0.3. Right panels (b), (d) are for easy-plane anisotropy AJ/D2 = 1.2 and HJ/D2 = 1.1. Note that the parameters used here are slightly diﬀerent from those used in Fig. 2.2, however the nontrivial structure of the skyrmion core in the easy-axis case is qualitatively similar to that in the circular cell calculations...... 20

xiii 2.5 Plots of the H-dependence of the optimal skyrmion cell radius R∗ and the core radii defined by the location of the maxima of |dmz/dr|. For the ansatz of eq. (2.5) in the text, dmz/dr = |2πrχ(r)|. (a) In the ∗ easy-axis region, both R and the core radius Rc are finite at the spiral- ∗ to-SkX phase boundary, but R diverges while Rc remains finite at the SkX-to-FM transition. The vertical dashed lines indicate phase transitions from the SkX to the spiral state (at small H) and to the FM (at large H). (b) In the easy-plane region, there are two core radii corresponding to the two maxima in |2πrχ(r)|. These inner and outer ∗ core radii Rc1 and Rc2, and the cell radius R , all remain finite at the two phase transitions out of the SkX phase. Here the vertical dashed lines indicate SkX-FM transitions...... 21

3.1 Phase diagrams as a function of AJ/D2 and HJ/D2 for four values of

D⊥/Dk. Easy-axis anisotropy corresponds to A < 0 while easy-plane to A > 0. The cone, elliptic cone, and tilted FM phases are shown schematically, with the Q-vector shown in red and the texture traced out by spins shown in black. The color bar on the right indicates

mz for the elliptic cone and tilted FM phases in the Dk = 0 panel. Insets: Unit cell in the hexagonal (Hex) skyrmion crystal (SkX) phase with white arrows indicating the projection of magnetization on the x-y plane. The colors indicates the magnitude and direction of the spin projection following the convention of ref. [1] indicated in the color wheel. Thick lines denote continuous transitions, while thin lines indicate ﬁrst-order phase transitions...... 28

3.2 Evolution of the spin texture m (top row) and the topological charge density χ (bottom row) for four values of AJ/D2 at ﬁxed HJ/D2 = 0.7

in the Rashba limit (Dk = 0). White arrows indicate the projection of m into the x-y plane. The colors also indicates the magnitude and direction of the spin projection following the convention of ref. [1] indicated in the color wheel. The development of nontrivial spatial variation in χ(r) is discussed in the text. Note, however, that in each case integral over a single unit cell R d2r χ(r) = −1...... 29

3.3 Illustration of the vertical cone phase, with q-vector (red arrow) along the z-axis, and the elliptic cone phase. In the elliptic cone phase the magnetization traces out an elliptic cone, i.e., the cross section is an ellipse rather than a circle. The elliptic cone phase shown here is for

the Dk = 0 limit. If the cone height is decreased so that the spins lie in the x-z plane the conﬁguration becomes a cycloid (Neel-like spiral). 43

xiv 3.4 Illustration of square (left) and hexagonal (right) unit cells. The dark lines indicate the region of independent spins (indicated by red dots).

Spins in the dashed region (gray dots) are determined by C4 symmetry for the square system and C3 symmetry for the hexagonal system. . . 48

3.5 Phase diagram obtained from variational Ansatz (3.23) for Dk = 0. Thick lines denote continuous transitions and thin lines denote ﬁrst order transitions. The variational state does not allow for skyrmion phases so we do not expect them in this phase diagram. The main result is that the phase boundaries at H = 2A and AJ/D2 = 2 agree to arbitrary precision with the numerical results, i.e., the polarized FM- elliptic cone and tilted FM-elliptic cone phase boundaries are identical

to the boundaries in the Dk = 0 limit of Fig. 1 of the main text. . . . 51

3.6 Free energy relative to the tilted FM (top) as a function of AJ/D2 at fixed HJ/D2 = 0.8 and the derivative dF/dA of the free energy (bottom) for the hexagonal skyrmion crystal (red), square skyrmion crystal (blue), elliptic cone (black) and tilted FM (green) phases. Phase transitions are marked by cyan lines. In the plot of dF/dA it is easy to see which phase transitions are continuous (elliptic cone-tilted FM) and which are first order (hexagonal-square skyrmion crystal and square skyrmion crystal-elliptic cone) by examining the jump discontinuities in the derivative of the free energy. The stable phase is indicated by a darker line in the bottom figure...... 54

4.1 Illustration of (a) a skyrmion lattice and (b) a chiral bobber lattice. The insets show a cutout of (a) a single skyrmion and (b) a single chiral bobber. Color and opacity indicates the direction of spin as shown in the colorwheel...... 58

4.2 Phase diagram for a thin film B20 magnet with thickness L, anisotropy A (A < 0 is easy-axis anisotropy), and external field H. There is interfacial DMI that stabilizes a chiral bobber phase and a stacked spiral phase near one film surface. The phases are the chiral bobber lattice (ChB), skyrmion crystal (SkX), a helical spiral phase, the stacked spiral phase (SS), and the cone phase. The fraction of the system that is cone phase is indicated by blending colors...... 61

4.3 Saturation ﬁeld Hc2 as a function of ﬁlm thickness L and anisotropy A. This can be used to determine A if Hc2 and L are known. . . . . 63

xv 4.4 (a) Magnetic susceptibility curves χ(H) for FeGe films with varying thickness, and (b) a scatter plot of |dχ/dH| along with a model fit assuming chiral bobbers (red line). The dashed red line is a fit to the 1/L behavior for thick films. The plot clearly shows intrinsic scaling (no L dependence) for films with thickness less than 40nm and extrinsic scaling for films with thickness greater than 40nm, indicative of a surface magnetic texture with thickness 40nm...... 66

5.1 Numerical results for log skyrmion lifetime as a function of anisotropy (AJ/D2, A < 0 is easy-axis), and applied ﬁeld (HJ/D2), for two different values of temperature. At low temperatures the lifetime is dominated by ∆E (shown in Figure 5.3) which is large near the phase boundary between skyrmion crystal and ferromagnet (polarized), and also near the phase boundary between the spiral and ferromagnet. At higher temperature the lifetime is dominated by ∆S (shown in Fig- ure 5.7) which diverges near the phase boundary between spiral and ferromagnet...... 68

5.2 Illustration of the minimum energy path in a two-dimensional energy landscape. (a) Contour plot of the energy with the minimum energy path overlayed. Color indicates the value of the “reaction coordinate”. The black cross is a saddle point. (b) Line plot of energy along the minimum energy path. The endpoints of the path are local minima, and the maximum along the path is a saddle point of the energy. . . 71

5.3 Numerical results for the energy barrier (∆E = Esp −Esk) for skyrmion collapse as a function of anisotropy (AJ/D2, A < 0 is easy-axis), and applied ﬁeld (HJ/D2). The system is two-dimensional and the lattice spacing is a = J/D/32. The energy barrier is largest near the transition to the polarized phase, and decays to zero deep in the polarized phase. The lifetime of the skyrmion is exponentially sensitive to the energy barrier (∆E ∝ ln τ)...... 73

5.4 Numerical results for the energy of a skyrmion (blue) and the saddle point (cyan) as a function of dimensionless lattice spacing a×D/J. The skyrmion state converges exponentially fast to the continuum value. The saddle point is a singular skyrmion state, so the energy decays to the continuum value with a power law form as discussed in Section 5.1.2. The dashed lines represent the a → 0 limit...... 74

xvi 5.5 Illustration of domain wall skyrmion with radius R and wall width w. For the skyrmion in this ﬁgure α = R/w = 3. (a) shows a line cut through the center of the skyrmion. (b) shows a 2D contour plot of the skyrmion...... 74

5.6 Parametric plot of skyrmion radius vs skyrmion energy barrier as a function of applied ﬁeld for several values of anisotropy (AJ/D2). The energy of the saddle point is 4π. The curves are bounded above and

below. The upper bound for Rsk corresponds to zero applied ﬁeld. The lower bound for Rsk corresponds to zero anisotorpy. The dashed line at ∆E = 4π indicates the point where the ferromagnetic phase becomes unstable...... 76

5.7 Numerical results for the entropy barrier (−∆S = Ssk − Ssp) for skyrmion collapse as a function of anisotropy (AJ/D2, A < 0 is easy- axis), and applied ﬁeld (h = HJ/D2). The system is two-dimensional and the lattice spacing is a = (J/D)/32. The entropy barrier diverges near the phase transition at AJ/D2 = −π2/8 and H = 0. Near the phase transition the skyrmion radius diverges and there are low energy bound states of the skyrmion. The entropy barrier becomes the dominant contribution to τ at high temperature when −∆S β∆E. 83

5.8 Low energy eigenmodes of a skyrmion with Rsk = 2J/D, and the corresponding saddle point (a singular skyrmion with Rsk ≈ a). The images show the eigenmodes projected onto the local θˆ direction. The eigenmodes of the saddle point are scattering states of spin-waves, except for the bound state n = 1, and the Goldstone mode n = 2 (not shown). The large skyrmion captures a series of low energy bound states, and the energy of the bound states vanish near the zero field spiral to ferromagnet transition. The modes have additional symmetry of a square due to finite size effects. The goldstone mode n = 1 is not included as it cancels the n = 2 mode of the saddle point...... 84

5.9 Analytic estimate (5.55) of the entropy barrier for skyrmion collapse as a function of anisotropy AJ/D2 (A < 0 is easy-axis), and applied ﬁeld HJ/D2. The entropy barrier diverges near the zero ﬁeld spiral to ferromagnetic transition. The analytic results are only valid near

the phase transition where Rsk → ∞. The entropy of the skyrmion is dominated by vibrational modes that become soft when Rsk → ∞. In contrast, the entropy of the saddle point is gapped...... 86

xvii 5.10 Illustration of the collapse path for a 3D. The reaction coordinate is increasing from left to right. The surface represents the skyrmion ra-

dius defined by the condition mz = 0. In thick films, a Bloch point forms on one film surface; then, the Bloch point propagates through the film and leaves the other film surface...... 93

5.11 Numerical results for energy barrier for skyrmion collapse in a film geometry as a function of film thickness for AJ/D2 = −0.53, HJ/D2 = 0.37, and J/D = 64a. For thin films with d J/D the barrier increases proportional to d. In contrast, in thicker films the barrier saturates to the bulk value. The dashed gray lines represent the bulk and thin-film limits...... 94

xviii Chapter 1: Introduction

Magnets have a rich variety of structures and patterns at the nanoscale. The

patterns are described in terms of the magnetization ﬁeld m(r), and they are com-

monly referred to as spin-textures. In magnetic thin ﬁlms and interfaces (where r is

a two-dimensional vector), we can characterize m(r) by a topological invariant called

the skyrmion number Z 2 Nsk = d rmˆ · (∂xmˆ × ∂ymˆ ). (1.1)

This number counts the times mˆ (r) wraps the unit sphere as real space is traversed.

A spin-texture with skyrmion number -1 is illustrated in ﬁgure 1.1. Two spin-textures

with diﬀerent Nsk are said to be in diﬀerent topological sectors. It is impossible to

change Nsk by making smooth changes to m(r).

Skyrmions were ﬁrst considered in the study of hadrons in high energy physics [2],

but they have proved to be central in the study of chiral magnets [1, 3, 4], in addition

to a variety of other condensed matter systems, including the quantum Hall eﬀect [5,

6, 7] and ultra cold atoms [8, 9, 10]. Skyrmions give rise to novel phenomena that

result from their topological wrapping of the unit sphere. When itinerant electrons are

ferromagnetically coupled to skyrmions they rotate in the direction of m(r) as they move. The rotation of the electron spin introduces a Berry curvature that appears as an emergent magnetic ﬁeld [11, 12]. In metals, the emergent magnetic ﬁeld of

1 (a) (b)

Figure 1.1: Illustration of a map from (a) real-space to (b) spin-space that has skyrmion number Nsk = −1. To calculate the skyrmion number, every triangle on the real-space lattice is mapped to a spherical triangle on the unit sphere. Spher- ical triangles that preserve the orientation of the real space triangle (indicated by black arrows) correspond to positive skyrmion density, while spherical triangles with opposite orientation, e.g., the highlighted magenta triangle, correspond to negative skyrmion density. The area of the spherical triangle determines the magnitude of skyrmion density. The skyrmion number is the signed skyrmion density summed over all plaquettes. This deﬁnition corresponds to equation (1.1) in the continuum limit.

skyrmions manifests itself through the topological Hall eﬀect [13, 14, 15, 16, 17].

Skyrmions may also be involved in non-Fermi liquid behavior [18, 19, 20].

Besides their exciting fundamental properties, skyrmions have attracted interest for their exciting prospects for applications in spintronics [1, 21, 22, 23]. The ability to write and erase individual skyrmions [24], along with their topological stability, small size, and low depinning current density [21], paves the way for potential information storage and processing applications. There has been tremendous progress in estab- lishing skyrmion crystal (SkX) phases, using neutrons [25] and Lorentz transmission electron microscopy [26], and in magnetic materials that lack bulk inversion symmetry, ranging from metallic helimagnets like MnSi and FeGe [1, 25, 27] to insulating multi- ferroics [28]. Early experiments focused on non-centrosymmetric crystals with broken

2 bulk inversion symmetry: metals like MnSi, FeGe and insulators like Cu2OSeO3. In these materials, the skyrmion crystal (SkX) phase is stable only in a very limited region of the magnetic ﬁeld (H), temperature (T ) phase diagram [25, 29, 28, 30, 31].

On the other hand, the skyrmion phase is found to be stable over a much wider region of (T,H) in thin ﬁlms of the same materials [32, 33, 29, 34], even extending down to

T =0 in some cases [34, 32].

In Chapter 2 we discuss the stability of skyrmions in thin-ﬁlms and interfaces, where Rashba spin-orbit coupling and broken mirror symmetry are responsible for the

Dzyaloshinskii-Moriya interaction (DMI). We show that skyrmions in broken mirror systems are stable over a wider range of parameter space than their counterparts in systems with broken bulk inversion symmetry. Skyrmions in broken mirror systems are especially stable in the presence of easy-plane anisotropy [35], in contrast to broken bulk inversion systems, where easy-axis anisotropy is needed to stabilize skyrmions [31].

In Chapter 3 we expand on Chapter 2 by including two sources of DMI, corresponding to a system with both mirror and bulk inversion symmetry broken. Con- sidering both types of DMI allows us to understand why skyrmions are more stable in systems with only broken mirror symmetry. We also show that new magnetic textures, the elliptic cone and square skyrmion crystal, arise in the presence of easy-plane anisotropy. In addition to helping us understand the limiting cases (only broken mirror or bulk inversion symmetry), including both sources of DMI is relevant for future materials, like heterostructures of B20 materials [27].

In Chapter 2 and Chapter 3 we assume that magnetic textures are uniform along the z-axis, even when the systems we are interested in have ﬁnite thickness. In

3 Chapter 4 we relax this assumption, and we ﬁnd an entirely new class of topological spin textures called chiral bobbers [36, 37, 38]. A chiral bobber is distinguished by the formation in real space of a magnetic singularity called a Bloch point. We show that a chiral bobber crystal is stable in systems with a mixture of broken mirror and bulk inversion symmetry [38]. The mechanism for stabilizing the chiral bobber cyrstal has a natural explanation in terms of the results in Chapter 3.

For memory applications where a classical bit is stored in the presence or absence of a skyrmion, one needs to look at the properties of metastable skyrmions in a FM background. In Chapter 5 we leave the realm of ground state calculations to study the lifetime of metastable skyrmions at ﬁnite temperature. Finding skyrmions with long lifetimes is a key engineering constraint that is important for applications. Chapter

5 contains a bevy of results that provide deep insight about skyrmion lifetimes. We show that in most cases skyrmion lifetimes can be calculated when only the energy of a metastable skyrmion is known; other factors can be lumped into a single phenomenological parameter. We show that skyrmions are stable near the phase transition from skyrmion crystal to ferromagnet. This puts restrictions on the skyrmion radius, and we ﬁnd that J/D sets the scale for skyrmion size. We show that the entropy of a skyrmion diverges in a small region near the zero ﬁeld spiral to ferromagnet phase transition. In that region the skyrmion radius diverges and we exploit this fact to calculate the entropy barrier analytically. Finally, we study skyrmion lifetimes for a

ﬁlm geometry with thickness d. We discover an interesting crossover from d J/D to d J/D. We show that in thick ﬁlms the path for skyrmion collapse occurs through the formation of a real space Bloch point.

4 Chapter 2: Skyrmions with Rashba SOC

Spin-orbit coupling (SOC) in magnetic systems without inversion gives rise to the

chiral Dzyaloshinskii-Moriya (DM) [39, 40] interaction Dij ·(Si × Sj). This competes with the usual Si · Sj exchange to produce spatially modulated states like spirals and SkX. The 2D case is particularly interesting. Even in materials that break bulk inversion, thin ﬁlms show enhanced stability [29, 41] of skyrmion phases, persisting down to lower temperatures. Inversion is necessarily broken in 2D systems on a substrate or at an interface, and this too may lead to textures arising from DM interactions. Spin-polarized STM [42, 24] has observed such textures on magnetic monolayers deposited on non-magnetic metals with large SOC.

Recently, there have been tantalizing hints of magnetism at oxide interfaces like

LaAlO3/SrTiO3 [43, 44, 45, 46] and GdTiO3/SrTiO3 [47]. The 2D electron gas at the

interface between two insulating oxides has a large and gate-tunable Rashba SOC [48].

Broken surface inversion and Rashba SOC has been proposed for these systems [49],

giving rise to chiral magnetic interactions and phases with spin textures [49, 50].

This work is based in part on the publication Enhanced stability of skyrmions in two-dimensional chiral magnets with Rashba spin-orbit coupling, S. Banerjee, J. Rowland, O. Erten, M. Randeria. Phys. Rev. X 4 (3), 031045 (2014).

5 a

b 2.5 Easy-Axis Easy-Plane

2. mz 1.0 0.8 0.6 1.5

2 0.4 D

0.2 J

---604246 0 H 1. SkX FM FM

0.5

Spiral 0. -1.5 -1. -0.5 0. 0.5 1. 1.5 AJ D2

Figure 2.1: (a) A skyrmion configuration. (b) The anisotropy-field phase diagram with ferromagnetic (FM), spiral and skyrmion crystal (SkX) phases for D/J = 0.01 2 and AcJ/D = 1/2 with A = Ac + As. Double lines denote first order transitions, while the single line is an unusual first order transition with a divergent length scale; see text. The dashed line H = 2A separates the out-of-plane FM from the tilted FM. mz for the FM is shown in the color bar. Results are obtained from a circular-cell variational calculation.

6 With this motivation we investigate 2D chiral magnets with broken inversion in

the z-direction. Microscopically, this leads to Rashba SOC. General symmetry con-

siderations imply that the form of the free energy for broken surface inversion is quite

diﬀerent from that in the usually studied case of non-centrosymmetric materials with

broken bulk inversion.

Our results are summarized in the T = 0 phase diagram in Fig. 2.1 as a function of

perpendicular magnetic field H and anisotropy A. For easy-axis anisotropy (A < 0), our 2D results with broken z-inversion turn out to be essentially the same as those for the 3D problem with broken bulk inversion [3, 41]. The easy-plane regime (A > 0) in the 2D Rashba case leads to a surprise: we find an unexpectedly large stable SkX phase. Skyrmions not only gain DM energy, but are also an excellent compromise between the field and easy-plane anisotropy. Moreover, we show that the skyrmions have a nontrivial spatial variation of their topological charge density (see Fig. 2.2) for A > 0.

Analysis of three exchange mechanisms – superexchange in Mott insulators, and double exchange and Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction in metals

– shows that the same SOC that gives rise to the DM interaction D also leads to an easy-plane compass anisotropy Ac [49]. The compass term is usually ignored since it is higher order in SOC than DM; however, its contribution to the energy is comparable

2 to that of DM, with Ac|J|/D ' 1/2 for all three mechanisms, where J is the exchange

coupling.

Our results should serve as a guide for material parameters of 2D chiral magnets

such that a large SkX region can be probed experimentally. These results are of

particular relevance to magnetism at oxide interfaces as discussed above. We should

7 also emphasize that our 2D results are not necessarily restricted to monolayers. We discuss the case of quasi-2D materials in Section 2.3.

2.1 Ginzburg-Landau Theory

The continuum free-energy functional F [m] = R d2rF(m) for the local magnetization m(r) of a 2D chiral magnet in an applied ﬁeld H is given by

F = Fiso(m) + FDM(m) + Faniso(m) − H.m. (2.1)

The isotropic term (α = x, y, z)

X α 2 Fiso = F0(m) + (J/2) (∇m ) (2.2) α consists of F0 that determines the magnitude of m and a stiﬀness J that controls the gradient energy. Microscopically the stiﬀness is determined by the ferromagnetic

2 exchange coupling. At T = 0 we replace F0 with the constraint m (r) = 1. Rashba

SOC, arising from broken z-inversion, leads to the DM term

z x x z FDM = −D[(m ∂xm − m ∂xm )

y z z y −(m ∂ym − m ∂ym )]. (2.3)

We will see below that this leads to “hedgehog”-like skyrmions (Fig. 2.1(a)). This form of FDM is dictated by the DM vector Dij ∼ bz×brij for Rashba SOC. In contrast, broken bulk inversion with Dij ∼ brij gives rise to the more familiar DM term m · (∇×m) that leads to “vortex”-like skyrmions.

We note that the m · (∇ × m) DM interaction can be transformed to FDM of eq. (2.3) by a global π/2-rotation of m about the z axis. We will not exploit this transformation here since we focus only on the Rashba SOC in this paper. See,

8 however, Section 2.3, where we comment on the case where both surface and bulk

inversion is broken.

Rashba SOC also leads to the anisotropy term

y 2 x 2 Faniso = (Ac/2)[(∂xm ) + (∂ym ) ]

x 2 y 2 z 2 − Ac[(m ) + (m ) ] + As(m ) . (2.4)

The Ac > 0 “compass” terms give rise to easy-plane anisotropy, while the single-ion

As term can be either easy-axis (As < 0) or easy-plane (As > 0). We deﬁne length in

units of lattice spacing a so that J, D, Ac and As all have dimensions of energy.

While the form of the free energy (2.1) follows from symmetry, microscopic analysis

gives insight into the relative strengths of various terms [35]. The origin of the DM

and compass terms lies in Rashba SOC, whose strength λ t, the hopping, in

materials of interest. Thus we obtain a hierarchy of scales with the exchange J

2 D ∼ J(λ/t) Ac ∼ J(λ/t) . Naively one might expect the compass term to be

unimportant, however its contribution to the energy O(Ac) is comparable to that of

the DM term O(D2/J). While the DM term is linear in the wave-vector q of a spin conﬁguration, its energy must be O(q2). Thus compass anisotropy, usually ignored in the literature, must be taken into account whenever the DM term is important.

2 It can be shown that AcJ/D ' 1/2 for a wide variety of exchange mechanisms

independent of whether the system is a metal or an insulator [49, 35]. Note that the

eﬀective anisotropy in model (2.1) is governed by A = Ac + As, which is easy-axis for

A<0 and easy-plane for A>0.

9 2.2 Phase Diagram

We begin by examining the T = 0 phase diagram for ﬁxed D J as function of magnetic ﬁeld H = Hzˆ and the dimensionless anisotropy AJ/D2, which we explore

2 by varying As with AcJ/D = 1/2. We look for variational solutions using analytical and numerical approaches. Here we focus on the SkX phase; the ferromagnetic (FM) and spiral phases are discussed in Appendix 2.5.1.

A skyrmion is a spin-texture with a quantized topological charge q = (4π)−1 R d2r mˆ ·

(∂xmˆ ×∂ymˆ ), which is restricted to be an integer. For example, the q = −1 skyrmion in Fig. 2.1(a) is a smooth spin conﬁguration with the topological constraint that the central spin points down while all the spins at the boundary point up.

The SkX state is a periodic array of skyrmions, often described by multiple-Q spiral condensation [51, 52]. We use an ‘optimal unit-cell’ approach, similar to ref. [3], where we impose the topological constraint for the center and boundary spins within a unit cell. We then ﬁnd the optimal conﬁguration within a single cell, whose size R is also determined variationally.

We describe the results from a ‘circular-cell’ ansatz, which leads to an eﬀec- tively 1D (radial) problem. This is computationally much simpler than the full 2D conjugate-gradient minimization of (2.1). The 2D and 1D methods lead to essentially identical phase diagrams; see Appendix 2.5.2. Here, we take a skyrmion conﬁguration

mskyrmion(r) = sin θ(r)ˆr + cos θ(r)zˆ (2.5)

in a circular cell of radius R, with the topological constraint θ(0) = π and θ(R) =

0. We minimize the energy (2.1) with θ(r) and the cell radius R as variational

10 parameters. We construct the SkX by an hexagonal packing of the optimal circular cells and recalculate the energy with up spins ﬁlling the space between the circles.

As a ﬁrst step, we use the linear ansatz [3] θ(r) = π(1 − r/R) with skyrmion size

R, a simple approximation that has the great virtue of being analytically tractable.

The resulting phase diagram is shown in the Appendix (see dotted lines in Fig. 2.3) rather than in the main text, so as not to clutter up Fig. 2.1(b). We note here that this very simple approximation already gives us our ﬁrst glimpse of the large SkX phase for easy-plane anisotropy, despite the fact that it greatly underestimates the stability of the SkX phase.

Next we obtain the phase diagram in Fig. 2.1(b) by numerical minimization using the more general form of eq. (2.5) and discretizing θ(r) on a 1D grid. This conﬁrms the qualitative observations from the linear approximation and yields an even larger SkX phase on the easy-plane side. Our 2D square cell calculations essentially reproduce the same phase diagram (see Fig. 2.3).

2.2.1 Easy-Plane vs. Easy-Axis Anisotropy

Our results for the 2D phase diagram in the easy-axis region (A < 0) is much the same as previous 3D studies [3, 41]. One might have thought that the perpendicular

ﬁeld H and easy-axis anisotropy would both favor a skyrmion, all of whose spins are pointing up far from the center, but then the FM state is even more favorable.

The remarkable result in Fig. 2.1(b) is that the SkX phase is much more robust for easy-plane anisotropy (A > 0). We can understand this as follows. The twisted spins in the skyrmion lower the DM contribution to the free energy as compared to a ferromagnetic conﬁguration. Furthermore, the skyrmion is a better compromise

11 m a 4 b z 5 2 0.5 D 0 0 0 y/L −2 −0.5 −5 −4 −5 0 5 −5 0 5 x/LD x/LD c 1 d 0.02 1 0.02 0.5 0.5 | | χ χ z z r r 0 0.01 0 m π m π 2 2 | | −0.5 −0.5

0 −1 0 −1 0 2 4 0 2 4 6 8 r/LD r/LD

Figure 2.2: Skyrmion core structure from circular-cell calculation with D/J = 0.01 2 and AcJ/D = 1/2. Here LD = (J/D)a where a is the microscopic lattice spacing. (a, b): False color plots of mz (shown in color bar). (c, d): Angle-averaged topological charge density |2πrχ(r)| and mz(r) (right axes). Left panels (a) and (c) correspond to easy-axis anisotropy AJ/D2 = −0.5 and HJ/D2 = 0.28. The skyrmion core is conventional with a single peak in the topological charge density. Right panels (b), (d) are for easy-plane anisotropy AJ/D2 = 1.35 and HJ/D2 = 1.96. Here the core has a large ‘transition’ region (yellow-orange) from down (center) to up (boundary) in m leading to an unusual two-peak structure for |2πrχ|.

12 between easy-plane anisotropy and a ﬁeld alongz ˆ than is a spiral conﬁguration. Thus, the large SkX region in the phase diagram is more or less oriented around H = 2A, the dashed line in Fig. 2.1(b) that separates the ‘tilted FM’ from easy-axis FM.

The internal structure of a skyrmion gives further insight into the stability of the SkX phase. In Fig. 2.2 we plot mz(r) and the (angular averaged) topological charge density |2πrχ(r)|, where χ(r) = [m·∂xm×∂ym]/4π. For the easy-axis case

the skyrmion core shows a conventional structure with a single peak in |2πrχ| in

Fig. 2.2(a,c). In contrast, easy-plane anisotropy can lead to a non-trivial core with

a double peak in |2πrχ(r)|; see Fig. 2.2(b,d). As the spins twist from down at the center (θ(0)=π) to all up (θ(R)=0) at the boundary, it is energetically favorable to

have an extended region where θ(r) ' θtilt (see Appendix 2.5.3), the best compromise between the ﬁeld and easy-plane anisotropy. As a result, |2πrχ| shows a two-peak structure in the topological charge density.

2.2.2 Phase Transitions

We next describe the various phase transitions within our variational framework.

The transitions between the spiral state and FM or SkX states are ﬁrst order, with a crossing of energy levels, as is the SkX to tilted FM transition for H < 2A. These are all denoted by double lines in Fig. 2.1. The SkX to easy-axis FM transition for

H > 2A (denoted by a single line) is also ﬁrst order in our numerics, but with the unusual feature that the optimal SkX unit cell size diverges at this transition; see

Appendix 2.5.3. Another interesting feature of Fig. 2.1 are the reentrant transitions from FM → SkX → FM for AJ/D2 ∼> 1.

13 2.3 Discussion

We now discuss two important questions: (a) the applicability of our results with

Rashba SOC to quasi-2D systems or films with finite thickness, and (b) the differences

between the broken surface or z-inversion, which has been our primary focus here,

and broken bulk inversion.

First, let us consider quasi-2D systems made of materials that do not break bulk- inversion. Chiral interactions then arise only from Rashba SOC. It might seem, at first sight, that the effects of surface-inversion breaking would be restricted to very thin, possibly monolayer, samples. However, it is known in the semiconductor literature that Rashba SOC can be very strong even in films of thickness of order a micron due to strain effects [53]. Thus we believe that the 2D results described in this paper are not restricted as such to monolayer materials. The spatial variation of the local magnetization m(r) will be translationally invariant in the z-direction, and the SkX phase will continue to show the large region of stability for in-plane anisotropy shown in Fig. 2.1.

In systems with broken bulk inversion, a cone phase [31] overwhelms both the

SkX and FM in the easy-plane anisotropy regime. The cone phase, with spin texture varying along the ﬁeld axis, gains energy due to a DM term with Di,i+zˆ k zˆ. Such a term does not exist in 2D or even in quasi-2D systems with Rashba SOC, where the

DM vector Dij ∼ bz × brij lies in the xy-plane. Our phase diagram is thus completely diﬀerent from that of ref. [31] for the case of in-plane anisotropy. Ref. [31] considers bulk-inversion breaking with a m · (∇×m)

DM term and ﬁnds a stable cone-phase for A > 0. We, on the other hand, consider

14 surface-inversion breaking with Rashba SOC leading to the DM term of eq. (2.3) and

ﬁnd a large region where the SkX is stable for A > 0.

An interesting question arises for a quasi-2D system, such as a thin ﬁlm, made

of a material that breaks bulk inversion. Now one has to take into account both

Dresselhaus and Rashba terms arising from bulk and surface inversion breaking, re-

spectively. In the next chapter we show that by tuning the relative strengths of

Rashba to Dresselhaus SOC one can continuously interpolate between the results

presented here (only surface inversion broken) and those of ref. [31] (only bulk in-

version broken) with interesting evolution of skyrmion chirality from hedgehog-like

to vortex-like. Interestingly, the data in Fig. 2.1 of ref. [54] show a SkX phase in

epitaxial MnSi thin ﬁlms for thickness . (J/D)a.

2.4 Conclusions

We have shown enhanced stability of skyrmions in 2D for Rashba SOC when the

eﬀective anisotropy is easy-plane. The compass term Ac is intrinsically easy-plane

and we suggest that experiments should look for 2D systems with suitable single-ion

anisotropies As, or ways to tune it, e.g., using strain, so as to enhance the SkX region.

In the future, it would be interesting to study the ﬁnite temperature phase diagram for 2D systems with easy-plane anisotropy, and to understand electronic properties, like the topological Hall eﬀect and non-Fermi liquid behavior in this regime.

15 2.5 Technical Details of Calculations

2.5.1 Variational Calculation

We consider the FM, spiral and SkX phases in turn. We use A = Ac + As as the

eﬀective anisotropy, and omit additive constants in the energy, which are common to

all phases.

z 2 z FM: The energy for the FM state evaluated from eq. (2.1) is FFM = A(m ) −Hm

2 with minimum FFM = −H /4A for H ≤ 2A and FFM = A − H for H > 2A The corresponding magnetizations are mz = H/2A and mz = 1 respectively.

For the A > 0 FM state, the easy-plane anisotropy competes with the ﬁeld along

−1 zˆ so that the magnetization points at an angle θtilt = cos (H/2A) with respect to

the z-axis for H ≤ 2A and eventually aligns with the ﬁeld for H > 2A. We denote

the FM state for H ≤ 2A as the ‘tilted FM’. The dashed line H = 2A in Fig. 2.1(b)

separates the ﬁeld-aligned FM from the tilted FM.

Spiral: The simplest zero-ﬁeld variational ansatz [49] yields a FM ground state

for |A|J/D2 > 1. When |A|J/D2 < 1, the H =0 ground state is a coplanar spiral with ˆ spins lying in a plane perpendicular to the xy-plane: m(r) = sin(Q0.r)Q0+cos(Q0.r)ˆz

with Q0 = (D/J)(cos ϕxˆ + sin ϕyˆ).

We extend the simple spiral above to incorporate more general 1D modulation ˆ described by mspiral(r) = sin[θ(Q0.r)]Q0 + cos[θ(Q0.r)]zˆ, where θ varies only along ˆ Q0, chosen to bex ˆ without loss of generality. In contrast to the linear variation in

the simplest ansatz, here θ(x) is an arbitrary function with m(x + R) = m(x) where

R is the period. We numerically minimize (2.1) with the variational parameters θ(x)

16 and R. This more general 1D periodic modulation stabilizes the spiral relative to FM

beyond |A|J/D2 = 1 to ' 1.25 at H = 0; see Fig. 2.1(b).

With the general 1D periodic modulation, m(x) = sin θ(x)ˆx + cos θ(x)ˆz, the free

energy is

Z R 1 2 Fsp = dx (J/2) (∂xθ) − D∂xθ R 0 +A cos2 θ − H cos θ , (2.6)

where ∂xθ = (∂θ/∂x). We use conjugate gradient minimization with respect to the size R and the function θ(x) which is discretized on a 1D grid. We use the periodic boundary condition θ(R) = θ(0) + 2πn where n is an integer. This form allows for a spiral solution with a net magnetization mz in the presence of a perpendicular magnetic ﬁeld.

The more restrictive (linear) variational ansatz θ(x) = 2π(x/R) is equivalent to the previously studied case [49] and is analytically tractable. In this case the energy of the spiral can be easily evaluated by minimizing with respect to R. This gives the

2 spiral pitch R = Rsp = 2π(J/D) and the energy Fsp = −D /2J + A/2.

Skyrmion crystal: We have discussed in the main paper the method used to construct a hexagonal SkX solution using the circular cell ansatz with rotationally symmetric form of eq. (2.5) in the text.

To qualitatively understand the stability of SkX over FM and spiral states, one can use a simple linear ansatz θ(r) = π(1−r/R) and minimize the energy by choosing an optimal R. This leads to the solution Rsk ≈ πJ/D for the optimal skyrmion cell

17 size, with the energy given by

−π2 D2 A 4 F = + − H sk 2[π2 + γ + log(2π)−Ci(2π)] J 2 π2 D2 A 4 ' −0.4009 + − H (2.7) J 2 π2

R ∞ Here Ci(x) = − x dt cos t/t is the cosine integral and γ is the Euler constant. The result for Fsk makes it clear that SkX gains energy from both DM and Zeeman terms.

For the more general θ(r) variation within the circular cell ansatz, we need to numerically minimize

2 Z R Fsk = 2 rdr [eJ + eD + eC + eS − H cos θ] . (2.8) R 0

with " # J ∂θ2 sin2 θ e = + J 2 ∂r r2 ∂θ sin 2θ e = −D + D ∂r 2r A ∂θ sin θ2 e = A cos2 θ + c cos θ − C c 8 ∂r r 2 eS = As cos θ

We need to ﬁnd the optimal cell size R and optimal values of θ(r), which we discretize

on a 1D grid in the radial direction. We have carried out 1D conjugate gradient

minimization using Mathematica on a laptop, using grids of up to 250 points.

2.5.2 2D Minimization

To check the validity of the circular cell ansatz, we have also performed a full 2D

minimization by discretizing the GL functional (1) over a square grid. For the 2D

calculation, we used up to 100×100 grids with polar and azimuthal angles (θ(r), φ(r))

18 2.5 Easy-Axis Easy-Plane

2. mz: 1.0 0.8 0.6 1.5 0.4 2 D

0.2 J

---604246 0 H 1. FM SkX FM

0.5

Spiral 0. -1.5 -1. -0.5 0. 0.5 1. 1.5 AJ D2

Figure 2.3: The phase diagram shown here is obtained as a result of a full 2D variational calculation, as distinct from the eﬀectively 1D variational calculation shown in Fig. 2.1(b). The symbols and parameters used are exactly the same as described in the caption for Fig. 2.1(b). Note that the 2D square cell calculation and the 1D variational calculation, although quite diﬀerent in their computational complexity, nevertheless lead to essentially identical results for the overall phase diagram. The dotted boundaries shown here are obtained from the simplest ‘linear’ variational ansatz for SkX described in text.

19 a 4 b mz 5 2 0.5 D 0 0 0 y/L −2 −0.5 −5 −4 −1 −4 −2 0 2 4 −5 0 5 x/LD x/LD c0.03 1 d0.015 1

0.5 | |

0.02 z 0.01 χ χ z r m 0 r 0 m π π 2 2

| 0.01 0.005 | −0.5

0 −1 0 −1 0 2 4 6 0 5 10 r/LD r/LD

2 Figure 2.4: Skyrmion core structure with D/J = 0.01 and AcJ/D = 1/2 obtained from a full 2D variational calculation, which should be compared with circular-cell results shown in Fig. 2.2. (a, b): False color plots of mz (shown in color bar). (c, d): Angle-averaged topological charge density |2πrχ(r)| and mz(r) (right axes). Left panels (a) and (c) correspond to easy-axis anisotropy AJ/D2 = −0.5 and HJ/D2 = 0.3. Right panels (b), (d) are for easy-plane anisotropy AJ/D2 = 1.2 and HJ/D2 = 1.1. Note that the parameters used here are slightly diﬀerent from those used in Fig. 2.2, however the nontrivial structure of the skyrmion core in the easy-axis case is qualitatively similar to that in the circular cell calculations.

20 a 10. AJ D2=-0.50 8. J

6. R

RD 4. 2. R 0. c 0.2 0.25 0.3 0.35 0.4 HJ D2 b 10. AJ D2=1.35 R 8. J

6. Rc2

RD 4. 2. Rc1 0. 1.8 1.9 2. 2.1 HJ D2

Figure 2.5: Plots of the H-dependence of the optimal skyrmion cell radius R∗ and the core radii defined by the location of the maxima of |dmz/dr|. For the ansatz of eq. (2.5) in the text, dmz/dr = |2πrχ(r)|. (a) In the easy-axis region, both R∗ and the ∗ core radius Rc are finite at the spiral-to-SkX phase boundary, but R diverges while Rc remains finite at the SkX-to-FM transition. The vertical dashed lines indicate phase transitions from the SkX to the spiral state (at small H) and to the FM (at large H). (b) In the easy-plane region, there are two core radii corresponding to the two maxima in |2πrχ(r)|. These inner and outer core radii Rc1 and Rc2, and the cell radius R∗, all remain finite at the two phase transitions out of the SkX phase. Here the vertical dashed lines indicate SkX-FM transitions.

21 of m(r) at each grid point as variational parameters. The 2D conjugate gradient

calculations are done using a Numerical Recipes [55] subroutine in C on a local cluster

of computers. This 2D minimization is much more computationally intensive than

the 1D calculation for the circular cell ansatz.

The 2D square cell result shown in Fig. 2.3 for the phase diagram is essentially

the same as that obtained from the circular cell calculation; see Fig. 2.1(b). We

show in Fig. 2.4 the internal structure of the skyrmion as calculated from the full

2D square-cell minimization. This ﬁgure should be compared with the results from a

circular cell calculation in Fig. 2.2. Note that the parameters used here are slightly

diﬀerent from those used in Fig. 2.2, however the nontrivial structure of the skyrmion

core in the easy-axis case – the two-peak structure in the topological charge density

|2πrχ(r)| – is qualitatively similar to that in the circular cell calculations.

2.5.3 Skyrmion Cell Size and Core Radius

It is conventional to deﬁne the ‘core radius’ of a skyrmion from the maximum of

|dmz/dr|. For the rotationally symmetric ansatz, eq. (2.5) in the main text, dmz/dr =

|2πrχ(r)|.

In Fig. 2.5 we show the optimal skyrmion cell size R∗ and core radii as a function of ﬁeld for (a) easy-axis anisotropy with AJ/D2 = −0.5 and (b) easy-plane anisotropy with AJ/D2 = 1.35. As described in the main paper, and shown in Fig. 2.2, there is only one length scale associated with skyrmion core size for the easy axis case, where as two-length scales appear for the easy-plane side, near the re-entrant region of the

SkX phase diagram.

22 The inner core radius, denoted Rc and Rc1 in Fig.2.5 (a) and (b) respectively, is found to be essentially constant as a function of H; it is ﬁxed by the competition between ferromagnetic and DM terms to a value of order J/D. On the other hand, the optimal skyrmion cell size R∗ can have non-trivial variation with H. For instance, in Fig.2.5(a) we see that R∗ diverges at the phase boundary between the SkX and out-of-plane FM in the easy-axis case. In this case the skyrmion spins change from

∗ down to up on the length scale Rc, and then remain up out to R . We next discuss the implications of the divergence of R∗ for the nature of the phase transition.

2.5.4 Phase Transition From SkX to Easy-Axis FM

The dark black phase boundary in Figure 1 between the hexagonal skyrmion crystal (SkX) and the polarized ferromagnet (FM) was erroneously identiﬁed as an

“unusual ﬁrst-order transition with a divergent length scale”. This phase transition is, in fact, a continuous (second order) transition in the calculation reported in the paper [35].

The numerically computed ground state energies Esk of the SkX and Efm of the

FM as functions of the external ﬁeld H, meet at the phase transition Hc with no change in slope. Hence the magnetization M = ∂E/∂H changes continuously at Hc.

We next present a simple analytical argument that permits us to see the continuous nature of the transition, This corrects an error in the argument presented in Appendix

D of the paper. Let us denote all energy densities by E = E/L2. We look at the diﬀerence in the energy densities between the SkX and FM given by

∆E = Esk − Efm (2.9)

23 and focus on its behavior as a function of h = (H − Hc)/Hc to understand the nature

of the phase transition at h = 0.

We can write " # R 2 R 2 E = E c + E 1 − c . (2.10) sk c R∗ fm R∗

Here the ﬁrst term corresponds to the skyrmion “core” of size Rc with an energy

∗ density Ec. The SkX has an inter-skyrmion spacing R , so that the region beyond Rc

is simply a fully polarized FM, which is the second term above. Using this, we ﬁnd

that R 2 ∆E = [E − E ] c . (2.11) c fm R∗

Let us look at the small-h dependence of each of the quantities in eq. (3). First,

[Ec − Efm] ∼ h since for h > 0 the ground state is a FM while for h < 0 it is a SkX.

Next, our numerical results (see Figure (a)) show that for h < 0, the skyrmion size

Rc is a weak function h which remains ﬁnite at h = 0, the skyrmion spacing diverges

R∗ ∼ |h|−ν∗ as h → 0−. (2.12)

The precise value of ν∗ > 0 is not important for our purposes. Combining all this,

we have

∆E = h1+2ν∗ f(h) (2.13) for h < 0, where f(h) is a smooth, positive function of h. (Note that the extra power

of h was erroneously omitted in our Appendix D). Therefore, we get

∂∆E ∗ ∗ = h1+2ν f 0(h) + (1 + 2ν∗)h2ν f(h) (2.14) ∂h

which clearly shows that ∂∆E/∂h → 0 as h → 0−. Thus the SkX and FM en-

ergy densities approach each other with zero relative slope indicating a continuous

24 (second-order) transition. Thus even though the divergent skyrmion spacing is not a correlation length, we do get a continuous transition.

25 Chapter 3: Skyrmions With Mixed DMI

Early on in the quest to ﬁnd skyrmions experiments focused primarily on non- centrosymmetric crystals with broken bulk inversion symmetry: metals like MnSi,

FeGe and insulators like Cu2OSeO3. In these materials, the skyrmion crystal (SkX) phase is stable only in a very limited region of the magnetic ﬁeld (H), temperature

(T ) phase diagram [25, 29, 28, 30, 31]. On the other hand, the skyrmion phase is found to be stable over a much wider region of (T,H) in thin ﬁlms of the same materials [32, 33, 29, 34], even extending down to T =0 in some cases [34, 32].

A key question that we address in this Chapter is: How can we enhance the domain of stability of skyrmion spin textures? We are motivated in part by the thin

ﬁlm experiments, and also by the possibility of chiral magnetism in new 2D systems like oxide interfaces [49, 35, 50]. We show how the SkX becomes progressively more stable over ever larger regions in parameter space of ﬁeld H and magnetic anisotropy

A, as the eﬀects of broken surface inversion dominate over those of broken bulk inversion.

This work is based on the publication Skyrmions in chiral magnets with Rashba and Dresselhaus spin-orbit coupling, J. Rowland, S. Banerjee, M. Randeria. Phys. Rev. B 93 (2), 020404 (2016).

26 3.1 Summary of Results

We begin by summarizing our main results, which requires us to introduce some terminology. We focus on magnets in which spin textures arise from the interplay between ferromagnetic exchange J and the chiral Dzyaloshinskii-Moriya (DM) interaction Dij ·(Si ×Sj). Spin-orbit coupling (SOC) determines the magnitude of the

D vector, while symmetry dictates its direction. Broken bulk inversion symmetry

(r → −r) leads to the Dresselhaus DM term with Dij = Dk brij, where brij = rij/|rij| with rij = (ri −rj). On the other hand, broken surface inversion or mirror symmetry

(z → −z) leads to the Rashba DM term with Dij = D⊥(bz × brij). In the limit of

2 2 1/2 weak SOC, D/J 1, where D = (Dk + D⊥) , the length scale of spin textures is (J/D)a a (the microscopic lattice spacing) and we can work with a continuum “Ginzburg-Landau” ﬁeld theory. This is adequate to describe long length scale textures, but not the atomic-scale skyrmions [56] that arise from competing local interactions.

We show in Fig. 1 the evolution of the T = 0 phase diagram going from the pure Dresselhaus limit to the pure Rashba limit. Each phase diagram is plotted as a function of the (dimensionless) ﬁeld HJ/D2 and anisotropy AJ/D2. Here A > 0

(A < 0) corresponds to easy-plane (easy-axis) anisotropy. Our main results are:

(1) As the Rashba D⊥ is increased relative to the Dresselhaus Dk, the spiral and skyrmion phases become increasingly more stable relative to the vertical cone phase, and penetrate into the easy-plane anisotropy side of the phase diagram.

(2) With increasing D⊥/Dk, the textures change continuously from a Bloch-like spiral to a Neel-like spiral. Correspondingly, the skyrmion helicity evolves with a

27 D⟂=0 D⟂=0.5D∥ D⟂=2D∥ D∥=0

m⟂ mFM,z 4 4 0.9 0.7 0.5 3 3 0.3 Tilted FM

2 2 0.1 Polarized FM Polarized FM Elliptic cone HJ/D 2 2 HJ/D 〈mEC,z〉 Cone Square SkX 0.9 0.7 1 Hex SkX Hex SkX Hex SkX Hex SkX 1 0.5 Spiral Spiral 0.3 0.1 0 0 -1 0 -1 0 -1 0 1 -1 0 1 2 AJ/D2 AJ/D2 AJ/D2 AJ/D2

Figure 3.1: Phase diagrams as a function of AJ/D2 and HJ/D2 for four values of D⊥/Dk. Easy-axis anisotropy corresponds to A < 0 while easy-plane to A > 0. The cone, elliptic cone, and tilted FM phases are shown schematically, with the Q-vector shown in red and the texture traced out by spins shown in black. The color bar on the right indicates mz for the elliptic cone and tilted FM phases in the Dk = 0 panel. Insets: Unit cell in the hexagonal (Hex) skyrmion crystal (SkX) phase with white arrows indicating the projection of magnetization on the x-y plane. The colors indicates the magnitude and direction of the spin projection following the convention of ref. [1] indicated in the color wheel. Thick lines denote continuous transitions, while thin lines indicate ﬁrst-order phase transitions.

vortex-like structure in the Dresselhaus limit to a hedgehog in the Rashba limit, which is shown to impact the ferrotoroidic moment.

(3) In the pure Rashba limit, we ﬁnd the largest domain of stability for the hexagonal skyrmion crystal. In addition we also ﬁnd a small sliver of stability for a square skyrmion lattice, together with an elliptic cone phase, distinct from the well-known vertical cone phase in the Dresselhaus limit.

(4) We see in Fig. 2 that in the Rashba limit the spin texture of the skyrmion and their topological charge density χ(r) begins to show non-trivial spatial variations

R 2 as one changes anisotropy, but the total topological charge Nsk = d r χ(r) in each

28 unit cell remains quantized, even when χ(r) seems to “fractionalize” with positive

and negative contributions within a unit cell.

(5) For H > 2A, one can have isolated skyrmions in a ferromagnetic (FM) back-

ground, and their topological charge Nsk is quantized, as usual, by the homotopy

2 group π2(S ) = Z. For H < 2A, we ﬁnd that skyrmions cannot exist as isolated

objects, and Nsk must now be deﬁned by the Z Chern number classifying maps from the SkX unit cell, a two-torus T 2 to S2, the unit sphere in spin-space, a deﬁnition

that works for all values of H/2A.

AJ/D2=0.0 AJ/D2=0.6 AJ/D2=1.2 AJ/D2=1.4

m⟂

χ 0.01

-0.01

-0.02

-0.03

Figure 3.2: Evolution of the spin texture m (top row) and the topological charge density χ (bottom row) for four values of AJ/D2 at ﬁxed HJ/D2 = 0.7 in the Rashba limit (Dk = 0). White arrows indicate the projection of m into the x-y plane. The colors also indicates the magnitude and direction of the spin projection following the convention of ref. [1] indicated in the color wheel. The development of nontrivial spatial variation in χ(r) is discussed in the text. Note, however, that in each case integral over a single unit cell R d2r χ(r) = −1.

29 3.2 Free Energy

We consider a continuum (free) energy functional F [m] = R d3rF(m) with

F = FJ + FDM + FA − Hmz (3.1)

P 2 whose form is dictated by symmetry. The isotropic exchange term FJ = (J/2) α(∇mα) (α = x, y, z) controls the gradient energy through stiﬀness J. The DM contribution in the continuum

FDM = D cos β m · (∇ × m) + D sin β m · [(zˆ × ∇) × m] (3.2)

is the sum of the two terms discussed above. The Dk = D cos β term arises from

Dresselhaus SOC in the absence of bulk inversion and D⊥ = D sin β from Rashba

2 SOC with broken surface inversion. The anisotropy term FA = Amz can be either easy-axis (A < 0) or easy-plane (A > 0). Several diﬀerent mechanisms contribute to A, including single-ion and dipolar shape anisotropies. In addition, Rashba SOC

2 naturally leads to an easy-plane, compass anisotropy A⊥ ' D⊥/2J, which is energetically comparable to the DM term [49, 35]. We treat A as a free, phenomenological parameter.

We focus on T = 0 where the local magnetization is constrained to have a ﬁxed length m2(r) = 1, and it should be hardest to stabilize skyrmions; once |m(r)| can become smaller due to thermal ﬂuctuations, skyrmions should be easier to stabilize.

It is convenient to scale all distances by the natural length scale J/D (setting the microscopic a = 1) and scale the energy F by D2/J. All our results will be presented in terms the three dimensionless parameters that describe F, namely ﬁeld HJ/D2,

2 anisotropy AJ/D , and tan β = D⊥/Dk.

30 3.3 Phase Diagram

In Fig. 1, we show the evolution of the (A, H) phase diagram as a function of tan β = D⊥/Dk, increasing from left to right. These results were obtained by min-

2 imizing the energy functional FDM subject to m (r) = 1. The energies of the fully polarized ferromagnet (FM), tilted FM, and vertical cone states can be determined analytically, while the energies for the spiral, the skyrmion crystals and the elliptic cone state were found by a numerical, conjugate gradient minimization approach. In all cases, the results were checked by semi-analytical variational calculations. Details of the methodology are described in the Supplementary Materials; here we focus on the results.

We begin with well known [31] results in the Dresselhaus limit (left panel of

Fig. 1), where the hexagonal SkX and spiral phases are stable only in a small region with easy-axis anisotropy (A≤0). The A>0 region is dominated by the vertical cone phase, where mcone(z) = (cos ϕ(z) sin θ0, sin ϕ(z) sin θ0, cos θ0) with ϕ(z) = Dkz/J

2 and cos θ0 = H/[2A + Dk/J]. The phase boundary between the vertical cone and

2 polarized FM is given by H = 2A + Dk/J. We note a change of variables that greatly simplifies the analysis of skyrmion crystal and spiral phases. This transformation is useful when m = m(x, y) has no z-dependence (along the field). Using a rotation Rz(−β) by an angle −β about the z-axis, we define n(x, y) = Rz(−β)m(x, y). It is then easy to show that (3.2) simplifies to a pure Dresselhaus form FDM = Dn · (∇ × n), while the other terms in (3.1) are invariant under m→n.

We choose, without loss of generality, xˆ as the propagation direction for the spiral of period L, so that nsp(x) = (0, sin θ(x), cos θ(x)) with nsp(x + L) = nsp(x). (Note

31 that this is not in general a single-q spiral). We minimize F to ﬁnd the optimal L and optimal function θ(x), which is a 1D minimization problem. For the SkX we pick a unit cell (hexagonal or square) and ﬁnd its optimal size and optimal texture n =

(cos ϕ sin θ, sin ϕ sin θ, cos θ), by solving a 2D minimization problem. We determine

ϕ(x, y) and θ(x, y) within a unit cell subject to periodic boundary conditions.

With increasing D⊥/Dk, we see that the SkX and spiral phases become more stable relative to the vertical cone, and their region of stability extends into the easy-plane regime. To understand this, consider increasing the Rashba D⊥ keeping Dk ﬁxed.

The energy of cone m(z) depends only on Dk, and is unchanged as D⊥ increases. In

2 2 1/2 contrast, the SkX and spiral, with m = m(x, y), utilize the full D = (Dk + D⊥) to lower their energy.

3.4 Helicity and Ferrotoroidic Moment

We ﬁnd that the spin textures smoothly evolve as a function of D⊥/Dk. The spiral continuously changes from a Bloch-like helix in the Dresselhaus limit to a Neel-like cycloid in the Rashba limit. In between, the spins tumble around an axis at an angle

−1 β = tan (D⊥/Dk) to the q-vector of the spiral. Similarly the skyrmions smoothly evolve from vortex-like textures in the Dresselhaus limit to hedgehogs in the Rashba limit; see Fig. 1. In fact, γ = π/2 − β is the “helicity” [1] of the skyrmions.

Our results imply that D⊥/Dk controls the helicity γ, where Rashba D⊥ could be tunable by electric field at an interface or by strain in a thin film. The ability to tune γ could be important in several ways. There is a recent proposal to use helicity to manip- ulate the Josephson effect in a superconductor/magnetic-skyrmion/superconductor junctions [57]. Another interesting phenomenon that depends on the helicity of is the

32 “ferrotoroidic moment” t = (1/2) R d3r[r × m(r)] [58, 59]. We will show elsewhere that t = t0 sin γ bz for the SkX.

3.5 Rashba Limit

Next we turn to the Dk = 0 results in the right panel of Fig. 1, where one has the maximum regime of the stability for the spiral and the hexagonal SkX, in addition to a small region with a square lattice SkX (ﬁrst predicted in ref. [60]), an elliptic cone phase and a tilted FM. This phase diagram improves upon all previous works [35, 50, 60]; see Supplementary Material.

The tilted FM, which spontaneously breaks the U(1) symmetry of F (in a ﬁeld),

2 has mz = H/2A and exists in the regime 2A > H and AJ/D⊥ > 2 for Dk = 0. We also see a new phase where the spins trace out a cone with an elliptic cross-section.

−1 The elliptic cone axis makes an angle θ0 = cos (H/2A) with zˆ, and the spatial variation of m is along a q-vector in the x-y plane.

In Fig. 3.1, thick lines denote continuous while thin lines denote ﬁrst-order transitions; for details see Supplementary Materials. (A = 2,H = 4)J/D2 is a Lifshitz point [61] at which a state without broken symmetry (polarized FM) meets a broken symmetry (tilted FM) and a modulated (elliptic cone) phase.

Let us next consider deviations from pure Rashba limit to see how the extreme right panel of Fig. 1 evolves into the Dk 6= 0 phase diagrams. As soon as one breaks bulk inversion, an inﬁnitesimal Dk leads to the tilted FM being overwhelmed by the vertical cone, which gains Dresselhaus DM energy. On the other hand, the elliptic and vertical cone states compete for Dk 6= 0 and for some small, but ﬁnite, Dk the vertical cone wins.

33 3.6 Spin Textures and Topological Charge

There are interesting diﬀerences between the skyrmions for H < 2A and H > 2A

(H = 2A is marked as a dashed line in the phase diagrams of Fig. 1). For SkX with

H > 2A, the focus of all the past work, the spins at the boundary of the unit cell

(u.c.) are all up, parallel to the ﬁeld. Hence one can make an isolated skyrmion in

a fully polarized FM background; see Fig. 2 left-most panels. The identiﬁcation of

the point at inﬁnity in real space for an isolated skyrmion lets us deﬁne a map from

2 2 2 S → S and use the homotopy group π2(S ) = Z to characterize the topological charge or skyrmion number Nsk.

In contrast, when H < 2A, the spins at the boundary are not all pointing up and the only constraint is periodic boundary conditions on the u.c.; see Fig. 2. There is no way to isolate this spin texture in a FM background. We must now consider the map r → m(r) from the u.c., which is a 2-torus T 2 to S2 in spin space (such maps are well known when T 2 represents a Brillouin zone in k-space, but the mathematics is

R 2 identical). This map is characterized by an integer Chern number Nsk = u.c. d r χ(r), where χ(r) = m · (∂xm × ∂ym)/4π is the topological charge density. In fact, one can use this deﬁnition of Nsk for all values of H/2A.

From the A = 0 panel on the left side of Fig. 2, we see that χ(r) is concentrated near the center of the u.c. and it is always of the same sign, as it is for all H > 2A.

With increasing A, once H < 2A, we see that χ(r) begins to spread out and even changes sign within the u.c. In the square SkX phase χ is again concentrated, but this time in regions near the center and the edges of each u.c. along with regions of opposite sign at the u.c. corners. For H ∼< 2A, the spin textures in the SkX phases are essentially composed of vortices and anti-vortices. Nevertheless, the Chern number

34 argument shows that the total topological charge in each u.c. is an integer; Nsk = −1 in all the panels of Fig. 2.

3.7 Conclusions

Previous theories on understanding the increased stability on skyrmions in thin

ﬁlms of non-centrosymmetric materials [32, 33, 29, 34] focussed primarily on the changes in uniaxial magnetocrystalline anisotropy [62, 31, 63] with thickness, or on

ﬁnite-size eﬀects [64, 36]. In fact, the latter can give rise to spin-textures more complicated than skyrmion crystals, with variations in all three directions. However, none of these theories take into account the role of broken surface inversion and

Rashba SOC. As we have shown here, a non-zero Rashba D⊥ leads to a greatly

enhanced stability of the SkX phase, particularly for easy-plane anisotropy, while at

the same time giving a handle on the helicity of skyrmions with interesting internal

structure.

We note that the phase diagrams in Fig. 3.1 apply to all systems with broken

mirror symmetry, with or without bulk inversion. Mirror symmetry can be broken by

certain crystal structures in bulk materials, by strain [65] in thin ﬁlms, or by electric

ﬁelds at interfaces. For Dk = 0 the vertical cone phase, which dominates much of

the phase diagram for Dk 6= 0, simply does not exist. After our paper was written,

we became aware of the very recent observation [66] of hedgehog-like skyrmions in

GaV4S8, a polar magnetic semiconductor with broken mirror symmetry, dominated by Rashba SOC. Skyrmions are, however, stabilized in this material only at ﬁnite temperature due to the large easy-axis anisotropy.

35 In conclusion, we have made a comprehensive study of the T =0 phases of chiral

magnets with two distinct DM terms. D⊥, arises from Rashba SOC and broken surface inversion, while Dk comes from Dresselhaus SOC and broken bulk inversion symmetry.

We predict that increasing the Rashba SOC and tuning magnetic anisotropy towards the easy-plane side will greatly help stabilize skyrmion phases. Our results are very general, based on a continuum “Ginzburg-Landau” energy functional whose form is dictated by symmetry. We hope that it will motivate ab-initio density functional theory calculations of the relevant phenomenological parameters entering our theory and experimental investigations of skyrmions in Rashba systems.

3.8 Technical Details of Calculations

3.8.1 Continuum Free Energy

In this appendix we ﬁrst introduce the continuum free energy functional that we use to model the spin textures in a chiral magnet. The free energy for a magnetic system with broken bulk inversion and mirror symmetries is F [m(r)] = R d3rF(m(r))

where

F(m(r)) = (J/2)(∇m)2 (3.3)

+Dk m · (∇ × m)

+D⊥ m · ((zˆ × ∇) × m)

2 +Amz − Hmz.

2 P 2 and (∇m) is shorthand for i,α(∂imα) . The z-axis is the axis of broken mirror

symmetry. Here J is the exchange stiﬀness, Dij = Dkˆrij + D⊥zˆ × ˆrij is the DM

vector, A is the magnetic anisotropy, and H is the ﬁeld. We measure all the lengths

36 in units of microscopic lattice spacing a, which we set to unity, so that J, Dk, D⊥,

A and H all have units of energy. We are interested in low temperature behavior so we ignore ﬂuctuations in the magnitude of the local magnetization m and impose the

constraint that |m(r)| = 1.

In our free energy functional we take the normal to the plane in which mirror symmetry is broken and the easy-axis direction to be the same, namely zˆ. For simplicity,

we also choose the external ﬁeld to be along the same direction. One can, of course,

imagine more general situations in which these directions are not all the same, but

the “simple” case treated here has suﬃcient complexity that it must be thoroughly

investigated ﬁrst.

Next we deﬁne parameters D and β such that

Dk = D cos β and D⊥ = D sin β. (3.4)

It is convenient to rewrite (3.3) using the natural energy and length scales in the

problem. We measure energies in units of D2/J and lengths in units of J/D. In

scaled variables the free energy density is given by

F(m(r)) = (1/2)(∇m)2 (3.5)

+m · ([cos β ∇ + sin β zˆ × ∇] × m)

2 +Amz − Hmz,

2 2 −1 which depends on three dimensionless variables AJ/D , HJ/D and β = tan (D⊥/Dk).

In the main paper, we explicitly write the anisotropy and ﬁeld as AJ/D2 and HJ/D2,

but in the Appendices we simplify notation and denote them as just A and H. (The

total energy F depends on an inconsequential overall factor of (J/D)3 coming from

the integration over the volume.)

37 Next we brieﬂy discuss the phases that we ﬁnd as a function of A, H and β.

The two ferromagnetic (FM) phases – fully polarized and tilted – are states with no spatial variations. The (vertical) cone phase is a non-coplanar state which has only z-variations, along the magnetic ﬁeld direction, so that m = m(z). These states can

be treated analytically, as discussed in Appendix 3.8.2.

The spiral and skyrmion phases have a local magnetization of the form m =

m(x, y), as does the elliptic cone phase. Speciﬁcally, the spiral has spatial variation

along a single direction, say x, in the plane perpendicular to the ﬁeld. The SkX phases have magnetic texture varying in both x and y. We use numerical methods for the

analysis of phases with m = m(x, y) as described in Appendix 3.8.3. Note that we do

not consider states where m has non-trivial variations in all three coordinates; see,

e.g., ref. [36].

Rotation: We next give the details of a transformation (introduced in the main

text) that greatly simpliﬁes the analysis for states where m = m(x, y). We make the

rotation  cos β sin β 0  n =  − sin β cos β 0  m ≡ Rz(−β) m. (3.6) 0 0 1

−1 where β = tan (D⊥/Dk). Expressed in terms of m the free energy density (3.5)

simpliﬁes to

2 2 F = (1/2)(∇n) + n · (∇ × n) + Anz − Hnz, (3.7)

provided that m, and thus n, depends only on x and y, but not on z. This result (3.7)

has the same form as the free energy density in the pure Dresselhaus limit. After

solving the problem in the n representation, we must transform back to m to ﬁnd

the actual spin texture.

38 It is easy to see that the the exchange term is invariant under m → n, i.e.,

(∇m)2 → (∇n)2, as are the anisotropy term and Zeeman term coupling to H. The only term that has a nontrivial transformation is DM term in (3.5). We symbolically write the DM term as as m · (D × m) where

D ≡ cos β ∇ + sin β (zˆ × ∇)   cos β∂x − sin β∂y =  sin β∂x + cos β∂y  . (3.8) 0

Here we set Dz = cos β∂z → 0, because we focus on textures that have no z-variation,

as already stated above. A straightforward calculation, using m = Rz(β) n, then

allows us to derive

m · (D × m) = n · (∇ × n) (3.9)

which in turn leads to eq. (3.7).

There is a slick way to obtain this same result by noting that the Free energy is

invariant under a combined rotation in spin-space and in real space about the z-axis.

(A combined spin and spatial rotation is needed because of SOC. and the z-axis is

singled out by broken mirror symmetry). In fact, it is simple to proceed with the

general case where we retain Dz = cos β∂z. The transformation Rz(−β) of eq. (3.6)

acts only in spin-space. Thus to write the free energy in terms of n, we need to also

rotate ∇ in real space, so that D transforms to   cos β[cos β∂x − sin β∂y] + sin β[sin β∂x + cos β∂y]  − sin β[cos β∂x − sin β∂y] + cos β[sin β∂x + cos β∂y]  . cos β∂z

This can be simply written as

D → [∇ − (1 − cos β)∂zzˆ] . (3.10)

39 Thus, for any magnetic texture m(x, y, z), the DM term can we written in general as

+n · (∇ × n) − (1 − cos β)n · (∂z(zˆ × n)) (3.11)

For a magnetic texture m(x, y) that does not vary along the z-axis the z-derivative terms vanish and this result simpliﬁes to (3.7) derived above.

3.8.2 Ferromagnetic and Cone Phases

In this appendix (3.8.2) we consider phases which can be treated analytically: the polarized ferromagnet (FM), tilted FM and the vertical cone phase.

Ferromagnets: The free energy density for a FM state is

2 F = Amz − Hmz. (3.12)

Minimizing the free energy is easy in this case; we need to solve

∂F 0 = = 2Amz − H (3.13) ∂mz along with the constraint |m|2 = 1. The solution is ∗ 1 H ≥ 2A ∗ A − HH ≥ 2A mz = H ⇒ F = H2 . (3.14) 2A H < 2A − 4A H < 2A ∗ We call the solution with mz = 1 a polarized FM since the magnetization is aligned

∗ with the magnetic ﬁeld. The solution with mz < 1 we call tilted FM since the magnetization is tilted away from the magnetic ﬁeld. Unlike the polarized FM, the tilted FM spontaneously breaks the U(1) symmetry of spin rotation around z axis in

3.12.

Cone: Another simple class of magnetic states are textures m(z) that do not break translational symmetry in the x-y plane. We will ﬁnd that the optimum con-

ﬁguration is a cone texture. This texture is called a cone because the magnetic moments trace out a cone as a function of z; see the illustration in ﬁgure 3.3.

40 For textures m(z) with translation symmetry in the x-y plane the Rashba term in (3.5), with strength sin β, does not contribute to the free energy. To implement the constraint |m(z)| = 1 we deﬁne angular variables θ(z) and φ(z) such that

m(z) = (cos φ sin θ, sin φ sin θ, cos θ). (3.15)

In terms of θ(z) and φ(z) the free energy density is

1 1 F(θ(z), φ(z)) = (θ0)2 + (φ0)2 sin2 θ − φ0 cos β sin2 θ 2 2 +A cos2 θ − H cos θ. (3.16)

The Euler-Lagrange equation for φ(z),

∂ ∂F ∂ 0 = = (φ0 sin2 θ − cos β sin2 θ), (3.17) ∂z ∂φ0 ∂z

can be integrated with the result

φ0 sin2 θ − cos β sin2 θ = C. (3.18)

Thus the free energy density can be written

1 F(θ(z)) = (θ0)2 + f(θ) (3.19) 2

where f does not depend on θ0. If θ∗ is a minimum of f then f(θ(z)) ≥ f(θ∗) for

all z. It is clear that θ(z) = θ∗ is an extremum for the free energy obtained from

(3.16). Given that a constant θ(z) = θ∗ minimizes the free energy it is easy to ﬁnd the optimum value θ∗ and the extremal function φ which is

φ∗(z) = z cos β. (3.20)

There are two solutions for θ∗. The ﬁrst solution is θ∗ = 0 which is a ferromagnetic

solution with energy F = A − H. The second solution, with

θ∗ = cos−1(H/(2A + cos2 β)), (3.21)

41 is called a cone texture. To make contact with the main text, we must recall

that lengths are measured in units of J/D, so that eq. (3.20) becomes φ∗(z) →

2 ∗ z(D/J)(Dk/D) = zDk/J, and energies A and H in units of D /J, so that θ →

−1 2 cos [H/(2A + Dk/J)]. We will occasionally refer to the cone as a vertical cone to distinguish it from the elliptic cone found in Appendices 3.8.3 and 3.8.4. The elliptic cone varies in the x-y

plane while the vertical cone varies along the z-axis. An illustration of the vertical

and elliptic cone textures is given in ﬁgure 3.3. In the Rashba limit, Dk = 0, the cone

phase is not stable for any values of H and A. For ﬁnite Dk the tilted FM phase is

not stable anywhere and the vertical cone takes its place in the phase diagram.

3.8.3 Numerical Methods

In appendix 3.8.3 we consider phases that cannot be treated analytically: the

spiral, elliptic cone, square skyrmion crystal and hexagonal skyrmion crystal phases.

For these phases the free energy density needs to be integrated and minimized nu-

merically; we use conjugate gradient minimization to achieve this. All of these phases

are of the form m(x, y) so we use the transformed free energy density (3.7) and we

will refer to n(x, y) as the texture in the spin rotated frame. To implement numerical

integration of the free energy some boundary conditions need to be speciﬁed. We use

three boundary conditions: periodic along the x-axis, periodic with square symmetry

and periodic with hexagonal symmetry.

For each boundary condition the minimization procedure is very similar: convert

the function n(x, y) to a vector s(i, j), write the integral as a sum, then minimize to

ﬁnd the optimum vector. We discuss this procedure in detail only for the simplest

42 Figure 3.3: Illustration of the vertical cone phase, with q-vector (red arrow) along the z-axis, and the elliptic cone phase. In the elliptic cone phase the magnetization traces out an elliptic cone, i.e., the cross section is an ellipse rather than a circle. The elliptic cone phase shown here is for the Dk = 0 limit. If the cone height is decreased so that the spins lie in the x-z plane the conﬁguration becomes a cycloid (Neel-like spiral).

43 case (periodic along the x-axis) and we discuss the important diﬀerences for the other

two cases.

Spiral and elliptic cone: First we consider textures which are periodic along a single axis; here we choose the x axis without loss of generality. The periodic boundary condition means that the texture n(x) satisﬁes n(x) = n(x + L). Since the free energy density is uniform along the y and z axes the free energy is just a one-dimensional integral along the x axis.

To evaluate this integral numerically we can convert it to a sum. This involves discretizing the function n(x); let s(i) = n((i − 1)∆x) where ∆x = L/(N − 1). Then

s(i + N) = s(i) is the periodic boundary condition. Next we replace all derivatives in

the free energy density with finite differences, i.e., ∂xn(x) → (s(i+1)−s(i))/∆x for the R L PN point x = (i − 1)∆x. Lastly we replace the integral with a sum ( 0 dx → ∆x i=1). At this point the free energy can be computed given a configuration s(i) and this will

be a good approximation to the true free energy when N is large, i.e., F (s) ≈ F [n] is

a good approximation when N is large.

We also have the constraint |n(x)| = 1. In terms of s the constraint is |s(i)| =

1. We impose the constraint by introducing angular variables, θ(i) and φ(i), at

each site so that s(i) = (cos φ sin θ, sin φ sin θ, cos θ). Now we can think of F as a

function of the 2N component vector {θ(1), ..., θ(N), φ(1), ..., φ(N)} = {θ, φ} = p.

We can minimize the vector function F (p) using conjugate gradient minimization.

Conjugate gradient minimization accepts as its input a function, in this case F ,

the gradient of that function ∇pF , and an initial vector, call it p0; the output of

conjugate gradient minimization is a local minimum p∗. We choose single-q spirals

m(x) = (cos qx, sin qx, 0) for our initial condition where q = 2π/L. This choice

44 of initial condition does not limit the output of conjugate gradient minimization to

single-q spirals; the output has a more complicated Fourier space structure in general.

Note that the function F (p) must be a real valued function of p. In particular we need to specify A, H and the periodicity L for F to be a real valued function.

Note that L is also a variational parameter in this case. In order to find the optimum value of L we include L in the conjugate gradient minimization procedure, i.e., we think of F (p,L) as a function of the 2N + 1 component vector {p,L}. We then sweep across the phase diagram and find the optimum configuration for each value of

−1.6 < A < 3.1 and 0 < H < 4.1 in steps of 0.1 along both axes.

The results of this minimization procedure are incorporated into the phase dia-

gram in Fig. 1 of the main text. The phases that we ﬁnd are spiral phases and elliptic

cone phases. Since we minimize the simpliﬁed free energy (3.7) the spirals we ﬁnd are

all Bloch-like spirals. These textures can be expressed in terms of a single function

θ(x) where n(x) = (0, sin θ(x), cos θ(x)). The textures obtained in the lab frame are

related to these texture by a rotation by β about the z-axis as discussed in Appendix

3.8.1, i.e., m(x) = (sin β sin θ, cos β sin θ, cos θ).

The other phase we ﬁnd is the elliptic cone. This phase is more complicated than

the spiral phase. Where the spiral can be described in terms of a single function,

θ(x), the elliptic cone requires two functions in general, i.e., θ(x) and φ(x). As a

consequence of the φ(x) dependence all components of the magnetic moment vary as

a function of x, not just the y and z-components like the spiral. An illustration of

the elliptic cone is given in Fig. 3.3 and in Appendix 3.8.4 we discuss a variational

state that captures the physics of the elliptic cone.

45 Square skyrmion crystal: Next we consider textures that are periodic along both axes, i.e., n(x, y) = n(x + L, y) = n(x, y + L). We further restrict ourselves to textures with C4 symmetry and we use this symmetry to reduce the number of points in the unit cell by a factor of 4 (see Fig. 3.4 for an illustration). We convert the free energy integral over the square unit cell to a sum and we convert derivatives to finite differences. Here we need a 2N × N component vector p = {θ(1, 1), ..., θ(1,N), ..., θ(N,N), φ(1, 1), ..., φ(N,N)} to describe a given magnetic texture n(x, y). The unit cell has 8N × N sites. We need to specify the function, F , its gradient, ∇pF , and an initial vector p0. We use a single-q vortex-like skyrmion as our initial configuration, i.e.,

n(x, y) = [−(y/r)xˆ + (x/r)yˆ] sin(qr) + zˆ cos(qr), (3.22) where r = px2 + y2 and we use q = π/L. Note that this conﬁguration is actually a square skyrmion crystal since we are applying square periodic boundary conditions.

As we have already stressed, the simple form of the initial conﬁguration does not limit the form of the output of conjugate gradient minimization.

To eliminate discretization error and approach the continuum limit, we extrapolate our results to the N → ∞ limit to obtain the phase diagram in Fig. 1 of the main text. We use a polynomial fit of the energy vs ∆x for N = 40, ..., 60 and we find that the ∆x = 0 limit is the same whether we fit to a polynomial of degree 3, 4 or

5. This gives us conﬁdence that we are suﬃciently close to the N → ∞ limit that extrapolation is valid.

The optimum conﬁgurations we ﬁnd using conjugate gradient minimization are square skyrmion crystals. The topological charge density n · (∂xn × ∂yn) is not concentrated at the center of each unit cell like it is for skyrmions found in the

46 Dresselhaus limit. The topological charge density is concentrated at four points in

the unit cell: the center, corners and edges of the unit cell (see Fig. 2 in the main

text) with opposite sign at the corners. Such conﬁgurations have been referred to as

meron crystals [60]; nevertheless, the skyrmion number for this phase is quantized

in integer units by the Chern number argument given in the main text. The square

skyrmion crystal phase is stable in a small pocket of the Dk = 0 phase diagram.

Hexagonal skyrmion crystal Lastly we consider textures with hexagonal sym-

metry, i.e., x(x, y) = m(x + a1, y + a2) = m(x + b1, y + b2) where (a1, a2) and (b1, b2)

are lattice vectors for a regular triangular lattice with lattice spacing L and we also

assume C3 symmetry. In ﬁgure 3.4 we illustrate how we use the C3 symmetry to

reduce the number of points in our simulation by a factor of 3. Besides the symmetry

of the problem, the minimization follows exactly as for the square symmetry case.

The optimum conﬁgurations we ﬁnd using conjugate gradient minimization are

hexagonal skyrmion crystals. The hexagonal crystal has a much large range of sta-

bility than the square crystal. Above the line H = 2A the hexagonal crystal has

topological charge density concentrated in the center of each unit cell and the topo-

logical charge density is essentially zero throughout the unit cell. All skyrmions found

in the Dresselhaus limit are of this type. Below the H = 2A line the topological charge density becomes less dense at the center of the unit cell and the topological charge is smeared across the unit cell. Close to the square crystal phase the topological charge density gathers at the center, corners and edges of the unit cell, with the skyrmion number still remaining quantized.

47 Figure 3.4: Illustration of square (left) and hexagonal (right) unit cells. The dark lines indicate the region of independent spins (indicated by red dots). Spins in the dashed region (gray dots) are determined by C4 symmetry for the square system and C3 symmetry for the hexagonal system.

3.8.4 Variational Solution

In appendix 3.8.4 we discuss a variational state that can be integrated exactly and captures the physics of the polarized FM, tilted FM, spiral, elliptic cone and vertical cone phases. The main point of this variational state is to gain analytical insight into the new elliptic cone phase; in particular, this variational state conﬁrms the phase boundary between the elliptic cone and the tilted FM is A = 2 and the phase boundary between the elliptic cone and the polarized FM is H = 2A.

Consider the spin-texture

m(r) = eˆ1m1 cos q · r + eˆ2m2 sin q · r + eˆ3m3(r) (3.23)

where eˆ1, eˆ2 and eˆ3 are orthonormal vectors. The function

p 2 2 m3(r) = 1 − (m1 cos q · r) − (m2 sin q · r)

48 is deﬁned to enforce the normalization |m(r)| = 1 at every point. We must restrict

m1 and m2 to the range −1 < m1, m2 < 1 to make sure m3 is a real number. Notice

that this state varies along the qˆ axis and is uniform perpendicular to this axis.

Limits: Here we discuss various limits of the variational state. The simplest case

is the limit m1 = m2 = 0 where the variational state reduces to m = eˆ3 which is just

a ferromagnetic conﬁguration, either polarized (eˆ3 = zˆ) or tilted.

Another simple limit is m1 = m2 = 1. In this case the variational state becomes

m = eˆ1 cos q · r + eˆ2 sin q · r which is a spiral state. Notice that this is a very

restricted state which is a delta function in Fourier space. In general spiral textures

have complicated Fourier structure so we do not expect this state to capture the spiral

phase boundaries quantitatively, only qualitatively. There is a special point in the

phase diagram (H = 0 and A = 0) where the spiral actually has this simple form.

When 0 < m1 = m2 < 1 the state is a circular cone, i.e., the magnetic moments trace out a cone in spin space as a function of z when q = qzˆ. The circular cone

∗ found in Appendix 3.8.2 is exactly of this form with q = cos βzˆ, m1 = sin θ , eˆ1 = xˆ

and eˆ2 = zˆ.

Finally we discuss the most complicated limiting case of this variational state.

When 0 < m1 < m2 < 1 the state (3.23) is an elliptic cone, i.e., the magnetic

moments trace out an elliptic cone as a function of x, for example, when qˆ = xˆ.

An elliptic cone has a base and a tip, similar to a cone, but the cross sections are ellipses rather than circles. The actual elliptic cone found using numerics is not as simple as the variational state. In particular the numerical solution does not have simple Fourier structure along the eˆ1 and eˆ2 axes, in contrast to the variational state

3.23. However, near the polarized FM-elliptic cone and tilted FM-elliptic cone phase

49 boundaries the elliptic cone becomes more and more like the variational state 3.23, so the phase boundary found using this variational state is quantitatively correct.

Exact integration: It is convenient to trade the eˆi for Euler angles. This can be achieved by writing

eˆi = Rz(θ3)Ry(θ2)Rz(θ1)xˆi (3.24)

where xˆ1 = xˆ, xˆ2 = yˆ and xˆ3 = zˆ. This state (3.23) has 8 variational parameters: θ1,

θ2, θ3, m1, m2, and q. Without loss of generality we can choose q to lie in the x-z plane leaving 7 variational parameters. The free energy in terms of these 7 parameters is

F = FJ + FD + FA + FH (3.25) 1 q F = (q2 + q2)(1 − (1 − m2)(1 − m2)) J 2 x z 1 2

FD = −m1m2 cos β(qz cos θ2 + qx cos θ3 sin θ2)

−m1m2 sin βqx sin θ2 sin θ3 A F = cos2 θ (2 − m2 − m2) A 2 2 1 2 A + sin2 θ ((m cos θ )2 + (m sin θ )2) 2 2 1 1 2 1 Z 2π H cos θ2 p 2 2 FH = − 1 − (m1 cos u) − (m2 sin u) . 2π 0

R π/2 p 2 FH can be expressed in terms of the complete elliptic integral E(x) = 0 1 − x sin θdθ, i.e., q H 2 (m1 − m2)(m1 + m2) FH = − 1 − m2 cos θ2E − 2 . π/2 1 − m1 The free energy has been integrated analytically but the variational parameter minimization involves transcendental equations. Numerically solving these equations pro- duces the phase diagram in Figure 3.5 in the Dk = 0 limit.

50 4 D||=0 polarized tilted 3 FM FM 2 2

HJ/D elliptic cone 1

spiral 0 0 1 2 3 AJ/D2

Figure 3.5: Phase diagram obtained from variational Ansatz (3.23) for Dk = 0. Thick lines denote continuous transitions and thin lines denote ﬁrst order transitions. The variational state does not allow for skyrmion phases so we do not expect them in this phase diagram. The main result is that the phase boundaries at H = 2A and AJ/D2 = 2 agree to arbitrary precision with the numerical results, i.e., the polarized FM-elliptic cone and tilted FM-elliptic cone phase boundaries are identical to the boundaries in the Dk = 0 limit of Fig. 1 of the main text.

Important features of this phase diagram are the polarized FM-elliptic cone and tilted FM-elliptic cone phase boundaries. These boundaries agree to arbitrary precision with the phase boundaries found using numerics, i.e., H = 2A for the polarized

FM-elliptic cone transition and A = 2 for the tilted FM-elliptic cone transition. The spiral phase boundary is only qualitatively correct due to the simple Fourier structure of the variational state.

3.8.5 Phase Transitions

In this Appendix we discuss the nature of the various phase transitions shown in Fig. 1 of the main text, where thick lines indicate continuous transitions and

51 thin lines indicate first order transitions. All phase transitions are determined by comparing energy curves near the phase boundary (see Fig. 3.6). Energy curves that have the same slope on both sides of the phase boundary correspond to continuous transitions, discontinuous slopes correspond to first order transitions. In the D⊥ = 0 limit of the phase diagram we find only one continuous phase transition, between the polarized FM and the cone (vertical). In the Dk = 0 limit we find four continuous phase transitions, three of which meet at a point called a Lifshitz point [61]. Below we give reasoning for the continuous or first-order nature of the various transitions.

Dresselhaus limit: The D⊥ = 0 limit of the phase diagram has been previously studied [31]. The phase transitions are all ﬁrst order except the polarized FM to cone transition, which is continuous. At this transition the cone radius goes continuously to zero.

The spiral-cone, spiral-skyrmion crystal and cone-skyrmion crystal transitions are all first order. In each of these transitions the phases on either side of the phase boundary have distinct broken symmetries. Generically we expect to find first order transitions between two phases with different broken symmetries.

For the ﬁrst order polarized FM-spiral phase transition the polarized FM phase has no broken symmetry while the spiral phase breaks translational symmetry. Fur- thermore, the wavelength of the spiral remains ﬁnite at the phase boundary.

Our numerical results for the polarized FM-skyrmion crystal show that the transition is ﬁrst order; however, it is an unusual ﬁrst order transition with a diverging length scale associated with the optimal SkX unit cell size [35].

Rashba limit: We discuss here only the phase transitions that are present in the

Dk = 0 phase diagram and are not present in the D⊥ = 0 phase diagram.

52 The elliptic cone transitions continuously to the polarized FM and tilted FM phases. Near the phase boundary the radius of the cone going continuously to zero.

There is a point, A = 2 and H = 4, where these three phases meet. This is a Lifshitz point [61] at which a “symmetric” phase, in our case the polarized FM, meets a broken symmetry phase, the tilted FM, and a spatially modulated phase, the elliptic cone; see Sec. 4.6 of ref. [61].

The elliptic cone-spiral phase transition is also continuous, in contrast with the vertical cone-spiral phase boundary. Near the elliptic cone-spiral phase boundary the height of the cone goes continuously to zero.

The elliptic cone-hexagonal skyrmion crystal, elliptic-cone square skyrmion crystal and hexagonal-square skyrmion crystal phase boundaries are all ﬁrst order with distinct broken symmetries on each side of the phase boundaries.

3.8.6 Rashba Limit Phase Diagram

In appendix 3.8.6 we comment in detail on how the Rashba limit phase diagram shown in the right-most panel of Fig. 1 of the main paper, goes beyond all previous works, as mentioned in the text.

The H = 0 results for A > 0 can be compared by taking the T = 0 limit of the

ﬁnite temperature analysis of ref. [50]. For H = 0 and A > 0 they ﬁnd three phases: a spiral, a cone, and an in-plane FM. These are same as the H = 0 limit of our results with our tilted elliptic cone simplifying to a cone whose axis is horizontal. Our phase boundaries are more accurate, however, because, as emphasized in the text, we do not restrict attention to a single-Q spiral as in ref. [50]. Further, these authors did not analyze H 6= 0.

53 HJ/D2=0.8 0.00

2 -0.01 Hex SkX

)J/D -0.02 Square SkX FM -0.03 Elliptic Cone

(F- Tilted FM -0.04 1.2 1.4 1.6 1.8 2.0 AJ/D2 HJ/D2=0.8 0.35 0.30 0.25 0.20 0.15 dF/dA 0.10 0.05 0.00 1.2 1.4 1.6 1.8 2.0 AJ/D2

Figure 3.6: Free energy relative to the tilted FM (top) as a function of AJ/D2 at fixed HJ/D2 = 0.8 and the derivative dF/dA of the free energy (bottom) for the hexagonal skyrmion crystal (red), square skyrmion crystal (blue), elliptic cone (black) and tilted FM (green) phases. Phase transitions are marked by cyan lines. In the plot of dF/dA it is easy to see which phase transitions are continuous (elliptic cone-tilted FM) and which are first order (hexagonal-square skyrmion crystal and square skyrmion crystal-elliptic cone) by examining the jump discontinuities in the derivative of the free energy. The stable phase is indicated by a darker line in the bottom figure.

54 For H 6= 0, our own earlier work [35] presented clear evidence for the development

of nontrivial spatial structure in the topological charge density, but in that paper we

worked with a SkX variational ansatz that forced the spins on the unit cell boundary

to be pointing along the ﬁeld direction. Thus we did not have the variational freedom

to see skyrmions where we must use a Chern number to understand the quantization

of topological charge. We see here the larger hexagonal SkX region than in ref. [35]

and also the square SkX.

Recently, ref. [60] predicted a small region of stability for the square SkX in the

A > 0 regime, and our results are consistent with theirs as far as this feature of the phase diagram is concerned. However, both refs. [35, 60] missed the elliptic cone phase.

3.8.7 Magnetic Anisotropy

In appendix 3.8.7 we discuss how Rashba spin-orbit coupling gives rise to an easy-plane magnetic anisotropy. In the paper we have a magnetic anisotropy term

2 FA = Amz in the free energy functional, where A is treated as a phenomenological parameter that can be either A < 0 (easy-axis) or A > 0 (easy-plane). As noted

there, many mechanisms contribute to A, including atomic SOC which gives rise to

single-ion anisotropy and and dipolar interactions that lead to shape anisotropy.

Here we comment on the anisotropy contribution of the SOC that leads to the

DM terms in the free energy functional, following refs. [49, 35]. Rashba SOC, which

gives rise to a DM term D⊥ ∼ λsoc linear in the SOC coupling constant λsoc, also

gives rise to a compass-Kitaev anisotropy of the form

x x y y −A⊥(Si Si+y + Si Si+x). (3.26)

55 2 This term is often ignored in the literature because A⊥ ∼ λsoc is small. However, this argument is ﬂawed because the energetic contribution of the DM term is of order

2 D⊥/J and hence of the same order as that of A⊥. In fact, one can show that for a large

2 class of exchange mechanisms A⊥J/D⊥ ' 1/2 in the limit of weak SOC; see ref. [35] and references therein. In the continuum limit, and ignoring higher order derivative

2 terms, eq. (3.26) leads to +|A⊥|mz. This is an important easy-plane contribution to the total anisotropy.

What about the analogous term arising from Dresselhaus SOC? This is of the form

x x y y z z −Ak(Si Si+x + Si Si+y + Si Si+z) (3.27)

2 2 with AkJ/Dk = 1/2. If we take the continuum limit, retain just the order m terms

2 2 2 and ignore higher order derivative terms, we get Ak(mx + my + mz) which is just an additive constant of no consequence, since m2 = 1. So it is only the Rashba SOC that gives rise to the interesting easy-plane anisotropy, as one can see from symmetry.

56 Chapter 4: Chiral bobbers

One of the fascinating predictions for thin films of non-centrosymmetric materials is the presence of new spin textures forming at the surfaces and interfaces including the “chiral bobber” and “stacked spiral” phases [36, 37]. In Figure 4.1, we illustrate the predicted chiral bobber phase (Figure 4.1a) and compare with the established bulk skyrmion crystal phase (Figure 4.1b). The skyrmion phase consists of a hexagonal array of skyrmion tubes that extend throughout the crystal and are aligned with the external magnetic field. Chiral bobbers (Figure 4.1) are like skyrmions tubes with a finite length. At the end of the tube there is a singular Bloch point. The Bloch point is a monopole in the magnetization field. In addition, chiral bobbers in B20

ﬁlms have a characteristic thickness scale LD/2, where LD is the helical pitch length.

Chiral bobbers are localized near the ﬁlm surface. The interior of the ﬁlm consists of the topologically trivial conical phase.

Thus far, it has been unclear whether the chiral bobber phase could be realized because previous calculations could only establish that it is a metastable state, not a true ground state [36]. The metastability arises from having surface twists that

This work is based in part on the publication Chiral bobbers and skyrmions in epitaxial FeGe/Si (111) ﬁlms, A. S. Ahmed, J. Rowland, B. D. Esser, S. R. Dunsiger, D. W. McComb, M. Randeria, R. K. Kawakami. Phys. Rev. M 2 (4), 041401 (2018).

57 a) b)

Figure 4.1: Illustration of (a) a skyrmion lattice and (b) a chiral bobber lattice. The insets show a cutout of (a) a single skyrmion and (b) a single chiral bobber. Color and opacity indicates the direction of spin as shown in the colorwheel.

arise near a free boundary [64, 36]. On the other hand, it is natural to expect surface Rashba DMI due to the broken mirror symmetry at the FeGe/Si interface.

To understand its impact, we simulate B20 ﬁlms with additional interfacial DMI near one ﬁlm surface and calculate the zero temperature phase diagram shown in Figure

4.2. Indeed, we ﬁnd that surface DMI stabilizes the chiral bobber at the interface while the bulk remains conical (which we denote as the “chiral bobber (surface) / cone (bulk)” phase). The explanation for stabilization of this phase in our model is simple. Surface DMI does not lower the energy of the cone no matter what magnitude or penetration of surface DMI is chosen, but surface DMI does lower the energy of the chiral bobber phase, so with increasing surface DMI eventually the chiral bobber becomes stable.

58 4.1 Model

To understand the eﬀect of interfacial DMI in B20 systems we perform micromag- netic simulations with two kinds of DMI

Z 3 Ebulk = d rDm · (∇ × m)

Z 3 Eint = d rDint(z)m · ([zˆ × ∇] × m).

This is similar to the model in the previous chapter, but the Rashba-type DMI is now z-dependent. This is meant to represent a thin B20 ﬁlm with interfacial DMI. We use material parameters for FeGe estimated in [67]: saturation magnetization Msat =

−1 −1 384kAm , ferromagnetic exchange 2J = Aex = 8.78pJm , and Dzyaloshinskii-

Moriya (DM) exchange D = 1.58mJm−2. Important scales that arise are the helical

2 pitch LD = J/D, the characteristic energy density AD = D /J, and the characteristic

ﬁeld scale HD = AD/µ0Msat. In addition to known material parameters for FeGe we include spatially varying interfacial DM exchange with strength Dint(z) for which we use a modiﬁed version of mumax3 [68].

The film axis is identified as the z-axis in our simulations. Interfacial DMI is allowed by broken mirror inversion symmetry at the interface of a film; however, previous calculations for thin films have only included bulk DMI, and have ignored possible sources of interfacial DMI. The value of interfacial DMI is determined by the microscopic physics of the interface and can be very different at a vacuum interface compared with a FeGe/Si(111) interface. The additional surface energy coming from Dint is crucial to stabilize the chiral bobber phase on that surface. Note that our simulations do include surface twist effects that arise due to interactions of the

59 skyrmion tube with the ﬁlm surface, and these eﬀects work along with interfacial

DMI to enhance the stability of surface spin textures.

To obtain the phase diagrams in Figure 4.2 we use a variational procedure where a set of initial configurations are evolved to a local minimum using conjugate gradient methods. The initial configurations we consider are: conical, helical, stacked spiral, skyrmion tube, and chiral bobber. We also vary system size to find the optimal size for each texture which has interesting implications for the susceptibility discussed below. After all local minima are found we accept the configuration with the lowest energy as the ground state. Our minimization procedure is repeated for different values of system thickness and applied magnetic field to create the phase diagrams shown in Figure 4.2. In contrast to previous result in the absence of interfacial

DMI [36, 37], we ﬁnd a wide region of stability for the chiral bobber phase in Figure

4.2 where we have included interfacial DM exchange near one surface of the simulated

film. The interfacial DM exchange is confined to a 30nm layer near one film surface

−2 with a magnitude equal to Dint = 1.19mJm . The increased stability of chiral bobbers in the presence of interfacial DM exchange has an intuitive explanation as follows. Surface spin textures are stabilized by two effects that arise from broken mirror symmetry at the surface, and these effects work together. Previous studies have considered only the effect of free boundary conditions at the interface and find that surface twists increase the stability of stacked spiral, skyrmion, and helical phases [37].

The broken mirror symmetry at a material interface allows for interfacial DMI Dint in addition to the surface twist effect. Broken mirror symmetry allows for both surface twist effects and interfacial DMI. By including both surface twist effects and interfacial DM exchange surface phases are enhanced relative to conical and field

60 ChB SkX Spiral SS Cone

Figure 4.2: Phase diagram for a thin film B20 magnet with thickness L, anisotropy A (A < 0 is easy-axis anisotropy), and external field H. There is interfacial DMI that stabilizes a chiral bobber phase and a stacked spiral phase near one film surface. The phases are the chiral bobber lattice (ChB), skyrmion crystal (SkX), a helical spiral phase, the stacked spiral phase (SS), and the cone phase. The fraction of the system that is cone phase is indicated by blending colors.

polarized phases, and the eﬀects are compounded since both are compatible with broken mirror symmetry. With the combined eﬀects of surface twist and interfacial

DMI the chiral bobber phase becomes stable shown in Figure 4.2. This is in contrast with previous results where only the surface twist is considered and the cone phase is always more stable than chiral bobbers [64, 36, 37].

4.2 Estimate for Anisotropy of FeGe Thin-Films

The anisotropy we use includes both intrinsic easy-axis anisotropy, in addition to easy-plane anisotropy arising from dipolar fields. We have used our experimental magnetization curves to estimate A = 0.25AD as follows. From our simulations we have obtained the upper critical field (Hc2) for FeGe thin films with anisotropy and

61 film thickness as variables as shown in Figure 4.3. To make fitting easier we first use our calculations to find a heuristic form of Hc2 that captures the effects of anisotropy and surface twists:  0 field polarized  ∗ ∗ Hc2(A, L) = HD + αAMsat + H exp(−L/L ) skyrmion  HD + 2AMsat cone

The parameter HD is the critical ﬁeld in a bulk system in the absence of anisotropy.

The remaining ﬁtting parameters (α, H∗, and L∗) account for surface twist eﬀects

∗ in the in the skyrmion crystal phase that change Hc2 in thin ﬁlms L L so we

∗ ∗ expect L ∼ LD. By ﬁtting to simulations we obtain α = 1.0, H = 0.29HD, and

∗ L = 0.43LD. In Figure 4.3 we plot Hc2 as a function of A and L. There are phase transitions when the critical ﬁelds are equal, i.e., A(2 − α) = H∗ exp(−L/L∗) and

∗ ∗ αAMsat = HD + H exp(−L/L ) shown as thick black lines in Figure 4.3. Using this heuristic form for Hc2 we can extract anisotropy values from experimental data.

4.3 Experimental Evidence for Chiral Bobbers

We report evidence for an interfacial spin texture through magnetization measurements on a series of epitaxial FeGe thin ﬁlms grown by molecular beam epitaxy

(MBE), and theoretical analysis shows that the spin texture is likely to be the proposed chiral bobber phase [36]. We investigate the magnetic phase diagram through magnetization measurements (M vs. H) and analysis of the susceptibility curves

(χ vs. H) for diﬀerent temperatures and ﬁlm thicknesses. The key signature of the interfacial spin texture is the observation of a negative slope of the susceptibility

(dχ/dH) that scales inversely with the ﬁlm thickness (L). dχ/dH probes the spin texture through its magnetic response, and the 1/L thickness dependence conﬁrms that the spin texture is localized to the surface/interface. By including a surface Rashba

62 1.0

0.5 Hc2/HD Cone → FM

2 3.0 0.0 2.5

AJ / D 2.0 -0.5 1.5 1.0 SkX → FM 0.5 -1.0 Easy-axis FM 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0

L/LD

Figure 4.3: Saturation ﬁeld Hc2 as a function of ﬁlm thickness L and anisotropy A. This can be used to determine A if Hc2 and L are known.

DMI, arising from the broken surface inversion in a thin film, we find that the equilibrium ground state consists of chiral bobbers at the interface with a cone phase in the interior. Additionally, we include the effects of uniaxial magnetic anisotropy into our calculations, consistent with the experimentally observed anisotropy, which explains the large magnetic field region where the 1/L surface effect is dominant. This combination of experimental results and theoretical analysis show that the observed interfacial spin texture is likely to be the chiral bobber phase.

To search for novel magnetic phases in the FeGe films, we follow the methodology established by Bauer et al. [69, 70], and perform magnetization measurements as a function of magnetic field, temperature, and film thickness. For all FeGe/Si(111) thin

ﬁlms, we employ a zero-ﬁeld cool protocol to the desired temperature and measure

M vs. H ramping from H = 0 Oe to 20,000 Oe in a superconducting quantum

63 interference device (SQUID) magnetometer. We then take a numerical derivative to

obtain the susceptibility χ = dM/dH as a function of H. For a 35 nm ﬁlm of FeGe,

the χ vs. H curve shown in Figure 4.4 exhibits a variety of features. The inﬂection points of chi correspond to phase transitions into diﬀerent magnetic states (indicated by solid lines), while local maxima correspond to magnetic structures in a state of coexistence (indicated by dashed lines) [71]. Exploring the thickness dependence of

χ vs. H identifies exciting new behavior. Figure 4.4a shows chi vs. H for FeGe films with thickness ranging from 14nm to 1000nm. In the field range from 2.5 kOe to

5.0 kOe (blue regions) away from phase boundaries, the slope of chi clearly shows a thickness dependence. As the thickness approaches 1000 nm, the slope approaches zero. The data are summarized in Figure 4.4b, which plots the magnitude of the slope of susceptibility, |dχ/dH|, as a function of 1/L. We see that there are two

distinct thickness regimes with a transition near LD/2, where LD ∼ 70 nm. For

small thicknesses (L < LD/2), we ﬁnd that |dχ/dH| is independent of ﬁlm thickness

L, while for thicker ﬁlms (L > LD/2) the slope magnitude scales like 1/L. We next argue that the 1/L thickness dependence is an unambiguous signature for an interfacial spin texture, and the crossover at LD/2 is consistent with the chiral bobber phase.

As seen from Figure 4.1b, the chiral bobber is akin to a skyrmion at the surface, but which pinches oﬀ at a Bloch point at a depth LD/2 from the interface, as it merges with the conical texture in the interior. We note that the conical phase has a

ﬁeld-independent χ, so it makes no contribution to dχ/dH. Because χ is an intensive quantity (i.e., per unit volume), the contribution of chiral bobbers would scale like

1/L, as seen in Figure 4.4. On the other hand, for small thickness (L < LD/2), the

64 skyrmion tubes extend throughout the entire ﬁlm like in Figure 4.1a, and there is no

Bloch point. In this case, the contribution of the topological spin texture to chi is

independent of ﬁlm thickness. Thus, the scaling observed in Figure 4.4 implies that

there is a chiral bobber that occupies a region 40nm, close to LD/2, as estimated from the crossover between constant and 1/L behavior in |dχ/dH|. We note that if chiral bobbers were present at both interfaces, the crossover would have occurred at

LD instead of LD/2. The microscopic origin of the negative dχ/dH for skyrmions and chiral bobbers provides the ﬁnal element of the scaling argument.

In a large portion of the phase diagram in Figure 4.2, a constant thickness line cut between L = 2LD to 7LD would yield both skyrmions and chiral bobbers over a large magnetic field range. However, we find a large region of constant negative slope in χ below the field polarized state that extends for several thousand Oersteds. Our

MBE grown films have easy-plane anisotropy. We include the effects of easy plane anisotropy and we show that the chiral bobber/cone phase is further stabilized over the skyrmion phase down to smaller thicknesses. By including the effects of easy- plane anisotropy in our calculations, this explains the sharp transition at 40 nm, and is attributed to the transition from skyrmion to chiral bobber/cone.

4.4 Conclusions

The impact of interfacial DMI on surface spin textures is dramatic. Previous calculations have shown that surface twists induced by boundary conditions can lead to metastable surface spin textures [64, 36, 37]. Here we have shown that interfacial DMI further increases the stability of surface spin textures, and with suﬃciently large interfacial DMI the cone phase is completely removed from the phase diagram

65 a) b)

LD/L

Figure 4.4: (a) Magnetic susceptibility curves χ(H) for FeGe films with varying thickness, and (b) a scatter plot of |dχ/dH| along with a model fit assuming chiral bobbers (red line). The dashed red line is a fit to the 1/L behavior for thick films. The plot clearly shows intrinsic scaling (no L dependence) for films with thickness less than 40nm and extrinsic scaling for films with thickness greater than 40nm, indicative of a surface magnetic texture with thickness 40nm.

and a chiral bobber lattice phase emerges. We have observed this phenomenon experimentally by studying the magnetization of FeGe samples as a function of film thickness. In the thin film regime L LD the slope of magnetic susceptibility scales extrinsically (proportional to L). In the thick film regime L LD the slope of χ scales intrinsically (independent of L), which indicates the existence of a surface spin texture. The effect of surface spin textures is difficult to observe in measurements of bulk samples with thickness L LD since surface spin textures contribute LD/L to any thermodynamic quantity; however, surface modulations are expected even in large samples with L LD and can be observed with surface sensitive experiments like MFM or spin-polarized STM.

66 Chapter 5: Skyrmion Lifetimes

In the previous three chapters we focused on the ground state properties of mag-

netic skyrmions. In this chapter we explore properties of metastable skyrmions, and

the various mechanisms for skyrmion collapse. We study the collapse of skyrmions

in 2D and in 3D thin-ﬁlms without disorder. We focus on properties of the skyrmion

lifetime that do not depend on details of the microscopic exchange parameters. Our

key results are shown in Figure 5.1, where we plot skyrmion lifetime as a function of

anisotropy AJ/D2, and external ﬁeld HJ/D2 (the system size is 64 × 64 sites, and the lattice spacing is a = J/D/32). We note that in systems with disorder skyrmion lifetimes can be longer when skyrmions are pinned, or shorter if there is a site with weaker ferromagnetic exchange [72, 73, 74]. In a racetrack geometry the lifetime of skyrmions can be greatly reduced [75].

To calculate skyrmion lifetimes we use the Arrhenius law,

τ = τ0 exp(β∆F ), (5.1)

with a free energy barrier ∆F = Fsp −Fsk, where Fsk is the free energy of a metastable skyrmion, and Fsp is the free energy of a saddle point state that separates the skyrmion state from the ferromagnetic state. The time constant τ0 is determined by the equations of motion for magnetic spins as discussed in Section 5.2.3. At low temperature

67 T = 0.1J T = 0.5J 1025 2 Polarized FM Polarized FM 0.8 20

10 [s]

1015 τ HJ/D 0.6 1010 5 0.4 Skyrmion Skyrmion 10 crystal crystal 1 0.2 10−5

External Field Spiral Spiral −10

10 Skyrmion lifetime 0.0 −15 1.5 1.0 0.5 0.0 1.5 1.0 0.5 0.0 10 Anisotropy AJ/D2 Anisotropy AJ/D2

Figure 5.1: Numerical results for log skyrmion lifetime as a function of anisotropy (AJ/D2, A < 0 is easy-axis), and applied ﬁeld (HJ/D2), for two diﬀerent values of temperature. At low temperatures the lifetime is dominated by ∆E (shown in Figure 5.3) which is large near the phase boundary between skyrmion crystal and ferromagnet (polarized), and also near the phase boundary between the spiral and ferromagnet. At higher temperature the lifetime is dominated by ∆S (shown in Figure 5.7) which diverges near the phase boundary between spiral and ferromagnet.

the energy barrier (∆E = Esp − Esk) is the dominant contribution ∆F . In Section

5.1.1 we use a novel numerical method to calculate ∆E. We ﬁnd that skyrmion collapse occurs through the formation of a spacetime Bloch point. We show that skyrmions are stable close to the phase boundary between skyrmion crystal and ferromagnet. We also provide variational calculations of the skyrmion radius, and we show it is monotonically increasing with ∆E (which determines the lifetime). This interdependence of skyrmion radius and lifetimes restricts the allowed values of skyrmion radius, and generally leads to the result Rsk ∼ J/D.

The entropy barrier ∆S = Ssp − Ssk is often ignored, but recently ∆S was found to be an important contribution to the lifetime of skyrmions in some systems [76, 77].

In Section 5.2.2 we calculate ∆S and τ0 explicitly. Our calculations show that the

68 entropy barrier is large near the zero ﬁeld phase transition from spiral to ferromagnet.

We show that the enhancement of entropy is due to low energy modes that appear

when Rsk J/D. The energy of these modes vanishes in the limit Rsk → ∞, and

the entropy barrier becomes anomalously large. In contrast to ∆S, we ﬁnd that

2 2 τ0 is essentially constant as a function of AJ/D and HJ/D . Replacing τ0 with

the phenomenological value ~/J (J is the ferromagnetic exchange constant) leads to errors in τ of a few tens of percent, which is acceptable given that τ is exponentially

−14 sensitive to ∆E and ∆S, and only linear in τ0. For reference τ0 ≈ 10 s for materials

with TC close to 1000K.

Finally, in section 5.3 we discuss skyrmion lifetimes for 3D systems with a thin-ﬁlm

geometry. We ﬁnd that skyrmion collapse occurs through the formation of a Bloch

point in real space at one surface of the ﬁlm. This collapse mechanism was recently

observed experimentally [77]. We also discuss a variational ansatz that captures

the qualitative behavior of the crossover from 2D to 3D, and brieﬂy discuss our

understanding of entropy for the 3D collapse mechanism.

5.1 Energy Barrier for Skyrmion Collapse

The dominant contribution to skyrmion lifetimes is ∆E = Esp −Esk. To calculate

energy we choose a phenomenological energy functional with ferromagnetic exchange

J, Rashba type Dzyaloshinskii-Moriya interaction D, anisotropy A, and an external

ﬁeld with strength H, given by the integral

Z 1 E = d2r J(∇m)2 + Dm · ([zˆ × ∇] × m) − A(1 − m2) + H(1 − m ) . (5.2) 2 z z

The natural length scale for skyrmions is J/D, and the natural scale for energy density is D2/J. It is also useful to introduce dimensionless parameters h = HJ/D2

69 and k = −AJ/D2 (the negative sign is useful for subsequent derivations). In dimen-

sionless variables the energy is

Z 1 E = d2r (∇m)2 + m · ([zˆ × ∇] × m) + k(1 − m2) + h(1 − m ) . (5.3) 2 z z

For numerical calculations we convert the integral into a discrete Riemann sum as

discussed in Chapter 3; then m is a collection of spins on a square lattice. In the

discrete case we use periodic boundary conditions in the xy-plane.

5.1.1 Numerical Results

To calculate Esk we follow the same procedure described in Chapter 3, e.g., we discretize (5.3) and perform conjugate gradient calculations to ﬁnd a local minimum.

To avoid ﬁnite size eﬀects, we choose the atomic lattice spacing a small enough so

that Esk is independent of a. In Figure 5.4 we show how the energy of a skyrmion

converges rapidly as a function of a−1.

Finding a saddle point is in general more diﬃcult than ﬁnding a minimum. This

is because a saddle point is unstable along some directions, and stable along others,

so simple gradient based methods fail. To calculate Esp we developed a novel method

that uses interpolation to ﬁnd a minimum energy path between the skyrmion state

and the ferromagnetic state (see Section 5.5.1 for a detailed discussion of the method).

The coordinate along the path is commonly referred to as the “reaction coordinate”.

Given a minimum energy path between two states, the saddle point is found by taking

the maximum along the reaction coordinate as illustrated in Figure 5.2.

In Figure 5.3 we show numerical results for ∆E in the (k, h) plane. Skyrmion

lifetime depends exponentially on ∆E, so this phase diagram determines the domi-

nant behavior of ln τ. The key feature is that skyrmion lifetimes are long near the

70 (a) Energy landscape (b)

Saddle point Energy Excited Ground state state

“Reaction Coordinate”

Figure 5.2: Illustration of the minimum energy path in a two-dimensional energy landscape. (a) Contour plot of the energy with the minimum energy path overlayed. Color indicates the value of the “reaction coordinate”. The black cross is a saddle point. (b) Line plot of energy along the minimum energy path. The endpoints of the path are local minima, and the maximum along the path is a saddle point of the energy.

phase boundary between the skyrmion crystal phase and the ferromagnetic phase.

We note that there is an alternative method for skyrmions collapse in systems with

open boundary conditions, e.g., in a racetrack geometry. Recently it was shown

that skyrmions can be removed by leaving the boundary without forming a singular

state [75]. This can signiﬁcantly reduce the energy barrier for skyrmion collapse.

We ﬁnd that skyrmion collapse occurs through the formation of a spacetime Bloch

point (a monopole in the magnetization), with the reaction coordinate serving as a

timelike variable. In this case the saddle point is a singular skyrmion, i.e., a skyrmion

with vanishing radius. In physical systems the singularity is regularized by the atomic

lattice, so the radius of the singular skyrmion is set by a. Such a conﬁguration depends sensitively on microscopic details of the exchange interactions [72, 73, 74]. We ﬁnd

71 that the saddle point energy converges slowly as a function a−1, as shown in Figure

−1 5.4. We discuss the dependence on a in Section 5.1.2. Given the sensitivity of Esp to microscopic parameters, we cannot calculate Esp accurately using Equation (5.3).

However, it is safe to treat Esp as a phenomenological parameter. In Section 5.1.2 we show that the Bloch point energy is not sensitive to anisotropy and external field, which justifies the treatment of Esp as a phenomenological parameter. In Figure 5.3 we use the numerical value of Esp = 8.76 that we calculated for a system with 64 × 64 sites. If the value of Esp is determined experimentally or using ab-initio calculations, then Figure 5.3 can be corrected by adding a constant that is the difference in our numerical value from the true value.

5.1.2 Variational Results

In this section we calculate ∆E using variational methods to understand some of the features we found in Section 5.1.1. First we calculate Esk using a variational ansatz. The ansatz, shown in Figure 5.5, is motivated by the form of a domain wall [78]. Near the zero ﬁeld phase transition from spiral to ferromagnet the skyrmion radius diverges (Rsk J/D); however, the skyrmion always has a ﬁnite wall width w. To describe this phenomenon let us take a variational ansatz of the form

m(r, ϕ) = (sin θ(r) sin φ(ϕ), sin θ(r) cos φ(ϕ), cos θ(r)). (5.4)

The functions θ and φ are spherical coordinates in spin-space, and the parameters r and ϕ are polar coordinates in real space. With Rashba DMI the optimum value of φ is φ(ϕ) = ϕ. To describe a circular domain wall we take the polar angle of the skyrmion to be sinh R/w θ(r) = 2 tan−1 . (5.5) sinh r/w

72 Energy barrier 7.2 Polarized FM 0.8 6.4

2 5.6 0.6 4.8 HJ/D 4.0 E

0.4 3.2 ∆ Skyrmion crystal 2.4

Applied ﬁeld 0.2 1.6 Spiral 0.8 0.0 0.0 1.50 1.25 1.00 0.75 0.50 0.25 0.00 Anisotropy AJ/D2

Figure 5.3: Numerical results for the energy barrier (∆E = Esp − Esk) for skyrmion collapse as a function of anisotropy (AJ/D2, A < 0 is easy-axis), and applied ﬁeld (HJ/D2). The system is two-dimensional and the lattice spacing is a = J/D/32. The energy barrier is largest near the transition to the polarized phase, and decays to zero deep in the polarized phase. The lifetime of the skyrmion is exponentially sensitive to the energy barrier (∆E ∝ ln τ).

73 10 Esp

Energy 5 Esk

0 0.025 0.050 0.100 Lattice spacing a × D/J

Figure 5.4: Numerical results for the energy of a skyrmion (blue) and the saddle point (cyan) as a function of dimensionless lattice spacing a × D/J. The skyrmion state converges exponentially fast to the continuum value. The saddle point is a singular skyrmion state, so the energy decays to the continuum value with a power law form as discussed in Section 5.1.2. The dashed lines represent the a → 0 limit.

(a) (b) 1.0

0.5 )

= 0) w R x, y y

0.0 ( z x, y ( m z

m -0.5

x x -1.0

Figure 5.5: Illustration of domain wall skyrmion with radius R and wall width w. For the skyrmion in this ﬁgure α = R/w = 3. (a) shows a line cut through the center of the skyrmion. (b) shows a 2D contour plot of the skyrmion.

74 The parameter

α = R/w (5.6)

is central to our discussion. Using this ansatz the energy is

Z 1 sin2 θ sin 2θ E(α, w) = 2π rdr θ02 + − θ0 + + k sin2 θ + h(1 − cos θ). 2 r2 2r (5.7)

The parameter w can be scaled out of the integration leaving a polynomial in w

1 E = I (α) − I (α)w + (kI (α) + hI (α))w2 (5.8) J D 2 K H

where the IX (X = J, D, K, H) are dimensionless integrals discussed in detail in

Section 5.5.2. We note that in dimensionless variables the energy is measured in units

of J. The stationary equation for w can be solved analytically

I w∗(α) = D . (5.9) kIK + hIH

Using the optimum value of w the energy becomes a function of α only

2 ∗ 1 ID E (α) = IJ − . (5.10) 2 kIK + hIH

To ﬁnd the ground state we optimize E∗(α) numerically to ﬁnd α∗. In Section 5.5.2

we discuss how we evaluate the integrals IX for 0 ≤ α < ∞. The variational results

for energy agree with numerical results to within a few percent [78].

In the limit α → ∞ the integrals IX have explicit forms that we can use to study

the divergence of α analytically near the zero ﬁeld spiral to ferromagnetic phase

transition. In this region the optimum skyrmion radius is (in dimensionful units) s J πD 1 R = αw ≈ , (5.11) sk D 4(−A) −AJ/D2 − π2/8

75 5 Anisotropy k

sk 4 0.5 1.3 R 1.0 1.5 3 1.2

1 Skyrmion radius

0 0.0 2.5 5.0 7.5 10.0 12.5 Energy barrier ∆E

Figure 5.6: Parametric plot of skyrmion radius vs skyrmion energy barrier as a function of applied ﬁeld for several values of anisotropy (AJ/D2). The energy of the saddle point is 4π. The curves are bounded above and below. The upper bound for Rsk corresponds to zero applied ﬁeld. The lower bound for Rsk corresponds to zero anisotorpy. The dashed line at ∆E = 4π indicates the point where the ferromagnetic phase becomes unstable.

ﬁrst derived by Wang et al. [78]. It is surprising that the skyrmion radius is propor-

tional to D. This is counter-intuitive based on our understanding of the skyrmion

crystal phase where we show the typical skyrmion size is set by J/D [35, 79]. In the

rest of this section we will show that the energy barrier puts limits on the allowed

values of skyrmion radius if long skyrmion lifetimes are required at room temperature.

To change the radius without changing the lifetime we must keep J and k = AJ/D2

ﬁxed, so the expression s πJ 1 R ≈ (5.12) sk 4kD k − π2/8

is more transparent for understanding the behavior of Rsk.

Next we discuss the saddle point energy Esp. The topological character of skyrmions

precludes them from being removed by continuous deformations; strictly speaking,

76 the saddle point that we are searching for is not well deﬁned in model (5.3). How-

ever, if we allow m to develop singularities, i.e., points where m is undeﬁned, then

a skrymion can be removed by squeezing it through a singularity. In discrete sys-

tems the lattice oﬀers a natural mechanism for the formation of singularities in m.

This explains how we obtained a singular skyrmion configuration when we calculated the saddle point numerically in Section 5.1.1. As we squeeze the skyrmion down to atomic scales, the skyrmion energy becomes independent of DMI, anisotropy, and the external field. We can understand this using simple dimensional analysis. The energy density for DMI is smaller than the energy density for ferromagnetic exchange by an amount aD/J 1, and the energy for anisotropy and the external field is smaller again by aD/J. Thus, we can calculate the saddle point energy using only ferromagnetic exchange. This justifies our claim in Section 5.1.1 that the saddle point energy is insensitive to anisotropy and the external magnetic field, so we treat the saddle point energy as a phenomenological parameter. This also justifies our claim that the saddle point energy converges slowly, in fact, it converges proportional to a−1.

We have just shown that only ferromagnetic exchange is relevant to determine Esp.

If we consider only ferromagnetic exchange the minimum energy cost for removing a

skyrmion was ﬁrst calculated by Polyakov and Belavin [80]. The energy bound hinges

on the inequality 2 1 ∂mα ∂mγ 1 2 0 ≤ − αβγµνmβ = (∇m) − m · (∂xm × ∂ym) . (5.13) 2 ∂xµ ∂xν 2 The ﬁrst term is the integrand for ferromagnetic exchange, and the second term is the

integrand used to calculate the skyrmion number. Integrating this innequality we ﬁnd

that the exchange energy is bounded from below by 4πJNsk. A similar expression

holds for −Nsk, so the bound is Esp ≥ 4πJ|Nsk|. The lower bound is saturated using

77 our variational ansatz by taking α → 0 and w → 0. Thus, we take Esp = 4πJ as the

saddle point energy.

Combining the results for Esk and Esp the energy barrier for skyrmion collapse is

2 1 ID ∆E = 4π − IJ + . (5.14) 2 kIK + hIH

In Figure 5.6 we show a parameteric plot of Rsk as a function of ∆E for several values

of k and as a function of h. This shows that there is a lower bound for Rsk for any

ﬁxed value of ∆E. There is a tradeoﬀ between skyrmion size and skyrmion stability.

This has an intuitive explanation as follows: when Rsk approaches the atomic lattice

spacing, then the skyrmion conﬁguration is similar to the saddle point conﬁguration,

so ∆E vanishes. Near the zero ﬁeld spiral to ferromagnetic transition we can use the

limit α 1 to calculate the energy barrier analytically. In dimensionful units r ! k − π2/8 ∆E ≈ 4πJ 1 − . (5.15) k

The energy barrier depends on D only through the combination k = −AJ/D2. Thus,

we can change skyrmion size without aﬀecting lifetime if we keep J and k ﬁxed; so,

Equation (5.12) is more natural than Equation (5.11).

5.2 Entropic Corrections to Skyrmion Lifetimes

In Section 5.1 we showed that the energy barrier for skyrmion collapse is large near

the phase boundary between the skyrmion crystal phase and the ferromagnetic phase.

This determines the dominant behavior of ln τ. However, recent experimental and theoretical results suggest that ∆S can be an important contribution to the skyrmion lifetime in some systems [76, 77]. In this section we calculate τ0 and ∆S using

78 harmonic transition state theory (HTST) for spin systems. We provide a derivation

of HTST to setup some of the terminology that we will use throughout this section.

5.2.1 Harmonic Transition State Theory for Spin Systems

In this section we derive the expression for the rate of skyrmion collapse using

harmonic transition state theory. For a review of transition state theory and harmonic

transition state theory see [81]. For HTST applied to spin systems see [82]. Harmonic

transition state theory (HTST) is an approximate form of transition state theory that

includes only gaussian ﬂuctuations of the energy around saddle points. To derive the

HTST expression we start with the more general transition state theory expression

R ∂B D[m(r)]δ(g(m))v⊥Θ(v⊥) exp(−βJE[m]) kTST = R . (5.16) B D[m(r)]δ(g(m)) exp(−βJE[m])

The region B is the basin of attraction of the metastable skyrmion state, and ∂B

is the boundary of this region. The δ-function of a vector should be interpreted as Q δ(g) = i δ(gi), and we use Θ to denote the unit step function. The function g(m) implements constraints. For spin systems we can write g(m) explicitly as

2 gi(m) = 1 − |mi| . (5.17)

Lastly, the velocity v⊥ is determined by the dynamics of the system in question. For

spin systems we use the Landau-Lifshitz equation of motion

m˙ = −γm × Beﬀ. (5.18)

Then v⊥ is the projection of m˙ normal to ∂B. Now proceed by using spherical

coordinates for m = (sin θ cos φ, sin θ sin φ, cos θ). We choose to deﬁne x = (θ, φ) to

simplify notation. These coordinates implement the constraint explicitly at the cost

79 of introducing a factor of det J in the integral measure (J is the Jacobian of the

coordinate transformation, not to be confused with the exchange energy J). Then R ∂B D[x(r)] det Jv⊥Θ(v⊥) exp(−βJE[x]) kTST = R (5.19) B D[x(r)] det J exp(−βJE[x]) The dominant contribution to the numerator occurs for x near a saddle point, and the

dominant contribution to the denominator occurs for x near the metastable skyrmion

state. Near these points we can deﬁne x = xsp + η and expand the energy to second order in η, and similar for xsk. This introduces the Hessian matrices Hsp and Hsk

(the Hessian is the second derivative of E with respect to x). We can also replace

J with the value at the saddle point and expand v⊥ to linear order in η. These

approximations are all valid to zeroth order in βJ.

We need to evaluate v⊥ to linear order in η. Fist we deﬁne the dynamical matrix

Asp implicitly in terms of the equation x˙ = Aspη. To calculate Asp explicitly we

expand the Landau-Lifshitz equation of motion to linear order in η. Asp has the

dimension of a rate. The order of magnitude of Asp is set by γ and J in the Landau-

Lifshitz equation. We can obtain an estimate for the order of magnitude of Asp by

the value ~/J. Let e0 denote the vector normal to ∂R near the saddle point. Then

T v⊥,sp ≈ e0 Aspη. It is easy to show that e0 must be the lowest energy eigenvector of

Hsp. Using these approximations leads to the HTST integral:

det0 J R D[η(r)]v Θ(v ) exp(−βJηT Hη) k = exp(−βJ∆E) sp ∂B ⊥,sp ⊥,sp . (5.20) HTST R T det Jsk B D[η(r)] exp(−βJη Hη) This integral can be evaluated explicitly. To proceed we perform a change of variables

(an aﬃne transformation) which leads to s det0 J det βJH R D[η(r)]v Θ(v ) exp(−ηT η) k = exp(−βJ∆E) sp sk ∂B ⊥,sp ⊥,sp . HTST 0 R T det Jsk det βJHsp B D[η(r)] exp(−η η) (5.21)

80 0 The function det βHsp represents the determinant with the unstable e0 mode re-

0 moved, and similarly for det Jsp; this mode is explicitly ignored when we integrate

over the region ∂B. The Hessian appears in the expression

T −1/2 v˜⊥,sp = e0 Asp(βJHsp) x. (5.22)

It is useful to deﬁne the vector

T −1/2 T asp = (e0 Asp(Hsp) ) (5.23)

The vector asp has dimensions of a rate since Asp has dimensions of a rate and Hsp

is dimensionless. We perform a rotation so that asp is along the ﬁrst axis. Then the

transition rate is s det0 J det βH R D[η(r)]x Θ(x ) exp(−ηT η) k = |a | exp(−βJ∆E) sp sk ∂B 1 1 , HTST sp 0 R T det Jsk βJ det βHsp B D[η(r)] exp(−η η) (5.24)

which can be evaluated readily as

0 s |asp| det Jsp det Hsk kHTST = exp(−βJ∆E) 0 . (5.25) 2π det Jsk J det Hsp

In the large β limit the entropy of the skyrmion and the saddle point are

det0 J exp(S = sp (5.26) sp p 0 βJ det βHsp det Jsk exp(Ssk) = √ , (5.27) det βHsk and we deﬁne 2π τ0 = . (5.28) |asp| With this we arrive at the desired result

τ = τ0 exp(−∆S) exp(β∆E) = τ0 exp(β∆F ) (5.29)

81 where we have made E dimensionful by absorbing the factor of J. This derivation

assumed there is a single saddle point. If there are additional saddle points on the

dividing surface, then the total rate is the sum over rates for each saddle point.

5.2.2 Numerical Results

Using the skyrmion and saddle point that we found in Section 5.1.1, we evaluate

τ0, Ssk, and Ssp numerically using Equations (5.26) and (5.28). We ﬁnd τ0 is almost constant in the (k, h) phase diagram. Moreover, replacing τ0 with ~/J is accurate to within a few tens of percent. This error is acceptable since τ is exponentially sensitive to ∆E and ∆S, and it varies only linearly with τ0.

When the skyrmion size is much less than J/D we ﬁnd that ∆S vanishes. This is not surprising since the skyrmion state and the saddle point are similar when

Rsk J/D. Even when Rsk ∼ J/D the entropy barrier is not too large. Thus, for skyrmions with Rsk . J/D, the skyrmion lifetime is essentially determined by

τ ≈ ~ exp(β∆E). (5.30) J

In contrast, near the zero-ﬁeld spiral to ferromagnet phase transition we ﬁnd a surprising result that the entropy barrier diverges. This divergence coincides with the divergence of Rsk near that phase transition. We can only calculate the entropy barrier for skyrmion sizes up to 2J/D in a simulation with 64 × 64 lattice sites; nevertheless, even for skyrmions with a modest size of Rsk = 2J/D, the entropy barrier is much

larger than the entropy barrier for Rsk = J/D. To understand the divergence we

study the eigenmodes of the skyrmion state. In Figure 5.8 we show three low energy

2 eigenmodes for a skyrmion with Rsk = 2J/D at the point AJ/D = −0.53 and

HJ/D2 = 0.37. The eigenmodes are string-like excitations that describe variations

82 Entropy barrier Polarized FM 0.8 25 2 20 0.6 HJ/D

15 S ∆

0.4 − Skyrmion 10 crystal

Applied ﬁeld 0.2 5 Spiral 0.0 0 1.50 1.25 1.00 0.75 0.50 0.25 0.00 Anisotropy AJ/D2

Figure 5.7: Numerical results for the entropy barrier (−∆S = Ssk − Ssp) for skyrmion collapse as a function of anisotropy (AJ/D2, A < 0 is easy-axis), and applied ﬁeld (h = HJ/D2). The system is two-dimensional and the lattice spacing is a = (J/D)/32. The entropy barrier diverges near the phase transition at AJ/D2 = −π2/8 and H = 0. Near the phase transition the skyrmion radius diverges and there are low energy bound states of the skyrmion. The entropy barrier becomes the dominant contribution to τ at high temperature when −∆S β∆E.

83 iegs hl h alwidth wall the while diverges, states. scattering essentially are modes the point, saddle singular the the of Besides core 5.8. the Figure near in behavior shown also point saddle the angle of polar eigenmodes the the of with function a as radius skyrmion the of point. saddle the of mode oe fti betaeflcutoso h krinrdu ihteform the with radius skyrmion the of fluctuations R are object this of modes spiral to field due zero square the a mode near of goldstone vanish symmetry The additional states have effects. bound modes size the The finite of transition. energy ferromagnet the to and states, bound utain is fluctuations state bound the for except mode spin-waves, stone of states scattering local are with with the skyrmion onto skyrmion a projected singular eigenmodes of (a eigenmodes point energy saddle Low responding 5.8: Figure sk ertezr edsia ofroantpaetasto h krinradius skyrmion the transition phase ferromagnet to spiral field zero the Near +

δR Saddle point Skyrmion

n modes modes cos( n oen=3 mode n=2 mode ntson.Telresymo atrsasre flwenergy low of series a captures skyrmion large The shown). (not 2 = nϕ ≈ .I eto .. eso htteeeg fteestring-like these of energy the that show we 5.2.3 Section In ). ∆ sk + γ sk n 2 w with n=4 n=3 ean osat so constant, remains θ ˆ n ieto.Teegnoe ftesdl point saddle the of eigenmodes The direction. n 0 = 84 snticue si acl the cancels it as included not is 1 = , 1 ... , h a ∆ gap The . n=5 n=4 R sk ϕ ≈ w nra pc.Cnrs this Contrast space. real in R a sk .Teiae hwthe show images The ). sk 2 = R n n h stiﬀness the and sk ,adteGold- the and 1, = J/D h o energy low The . n=6 n=5 n h cor- the and , n R 2 = R γ = sk sk both vanish when the skyrmion radius diverges. This leads to a divergence in the entropy barrier which leads to anomalously large skyrmion lifetimes.

5.2.3 Variational Results

Our numerical results show that the entropy barrier diverges when the skyrmion radius is large. To understand this phenomenon we use a simple ansatz that allows us to evaluate the skyrmion spectrum explicitly in the limit α 1. In this region we will show the entropy diverges with the functional form r 1 HJ AJ −∆S ∼ 2R − (5.31) sk 2 D2 D2 where Rsk is the skyrmion radius. We have used Stirling’s approximation to derive this result; hence, it is only valid in the limit α → ∞.

To derive ∆S we ﬁrst calculate the entropy of a skyrmion in Section 5.2.3. Then we calculate the entropy of the saddle point in Section 5.2.3. Finally, in Section

5.2.3 we combine these results and derive a simple expression for the entropy barrier for skyrmion collapse. The result is summarized in Figure 5.9 where we plot or anlaytic result for ∆S. The result we derive reproduces the qualitative behavior of our numerical results shown in Figure 5.7.

Entropy of a Metastable Skyrmion

We start our derivation by considering the low energy modes of a skyrmion. We use the ansatz sinh(α + ψ(ϕ)) θ(r, ϕ) = 2 tan−1 (5.32) sinh(r/w)

85 Entropy barrier Polarized FM 0.8 25 2 20 0.6 HJ/D

15 S ∆

0.4 − Skyrmion 10 crystal

Applied ﬁeld 0.2 5 Spiral 0.0 0 1.50 1.25 1.00 0.75 0.50 0.25 0.00 Anisotropy AJ/D2

Figure 5.9: Analytic estimate (5.55) of the entropy barrier for skyrmion collapse as a function of anisotropy AJ/D2 (A < 0 is easy-axis), and applied ﬁeld HJ/D2. The entropy barrier diverges near the zero ﬁeld spiral to ferromagnetic transition. The analytic results are only valid near the phase transition where Rsk → ∞. The entropy of the skyrmion is dominated by vibrational modes that become soft when Rsk → ∞. In contrast, the entropy of the saddle point is gapped.

86 where we assume ψ 1 to study small ﬂuctuations. We keep terms only up to order

ψ2 in the energy

Z 2π 2 2 ! 1 dϕ ∂ψ ∂ E0 2 0 E = E0 + I11 + 2 ψ . (5.33) 2 0 2π ∂ϕ ∂α

The parameter E0 is the energy of a skyrmion in the absence of ﬂuctuations. The dimensionless integral I110 depends only on α and they are discussed in detail in

2 Section 5.5.2. The functions I110 , E0, and ∂αE0 should be evaluated at the equilibrium value for w and α. The energy has translation symmetry in ϕ, so the eigenmodes are waves, i.e., ψ(ϕ) = ψn cos(nϕ) and ψ(ϕ) = ψn sin(nϕ) where n is an integer. The energy spectrum for these modes is

2 ( 1 ∂ E0 2 2 d ∂α2 ψ0 n = 0 E = 2 (5.34) n 1 2 2πn 2 ∂ E0 2 4 d w I11 d + ∂α2 ψn otherwise

We have just calculated the Hessian of E with respect to the ψn modes; but, to calculate the entropy, we actually need the Hessian of E with respect to ﬂuctuations of m, not ψn. In other words, we have ignored the Jacobian for the ψn modes. To proceed we will treat ∂m/∂ψn as a variational state. We need to normalize the state, which amounts to dividing by √ s 2 ( 2 ∂m Z ∂m(r) dw I11 n = 0 3 q = d r = 1 2 (5.35) ∂ψn ∂ψn 2 dw I11 otherwise Then we use the relation ∂m(r) ∂m(r0) ∂2E ∂m 2 Z H(r, r0) = d3rd3r0 ∂ψn ∂ψn . (5.36) 2 ∂ψn∂ψn ∂ψn ∂m ∂ψn were H(r, r0) is the Hessian matrix for variations of m(r), which we need to calculate

Ssp. The integral is a projection of H into the subspace of the ψn modes which allows us to read oﬀ the eigenvalues

2 1 1 ∂ E0 I110 1 2 2 n = 2 2 + 2 n = ∆sk + γskn (5.37) w I11 ∂α I11 w 87 Near the zero ﬁeld spiral to ferromagnet phase transition α diverges and the integrals

can be evaluated explicitly to ﬁnd

1 1 γsk ≈ 2 2 = 2 (5.38) α w Rsk 1 w2 w2 w3 ∆sk ≈ 2 4 + h = 4 + h . (5.39) w α α Rsk Rsk

We ﬁnd that the gap vanishes when Rsk diverges. In zero ﬁeld the gap vanishes

−4 −1 rapidly with the power law Rsk . This divergence is reduced to Rsk when an external ﬁeld is applied. The string-like modes of the skyrmion become soft with the power

−2 law Rsk .

Entropy of a Singular Skyrmion

In Section 5.1.1 we showed that the saddle point state is a singular skyrmion. Any skyrmion can be represented as

p 2 msp(r) = mzzˆ + 1 − mzrˆ (5.40)

parameterized by the function mz(r). To describe a singular skyrmion we require that m(r) = −1 in a small region near r = 0, and mz(r) = 1 otherwise. Given that mz(r) = 1 almost everywhere, we might naively expect the saddle point has the same spectrum of excitations as the ferromagnetic state, e.g., ferromagnetic spin waves with a spectrum gapped by

∆sw = k + h/2 (5.41)

. However, we know that the saddle point has a single negative energy mode.

This negative energy mode is a bound state of spin waves captured by the singular skyrmion. To calculate the spectrum explicitly consider the saddle point in the

88 presence of ﬂuctuations described by

m(r, ϕ) = msp(r, ϕ) + ψϕ(r, ϕ)ϕˆ + ψχ(r, ϕ)χˆ. (5.42)

The ﬁelds ψϕ 1 and ψχ 1 describe ﬂuctuations around the saddle point. We

p 2 deﬁne χˆ = (mzrˆ− 1 − mzzˆ) so that msp, ϕˆ, and χˆ form an orthonomal coordinate

system. These coordinates are chosen so that ﬂuctuations are transverse to msp.

Expanding the energy to second order in ψϕ, ψχ we ﬁnd

Z 1 1 1 E = E + d2r (∇ψ ϕˆ)2 + (∇ψ rˆ)2 + ∆ (ψ2 + ψ2 ) + u(r)ψ2 . (5.43) sp 2 ϕ 2 χ sw ϕ χ 2 χ

We have lumped all of the mz dependence into a function u(r), and we have ignored

contributions to E that vanish in the singular limit. In the absence of u(r) this

expression describes spin-wave ﬂuctuations in a ferromagnetic background. Then the

1 2 eigenmodes have have the gapped spectrum ∆sw + 2 q . With u(r) included the ψχ

ﬂuctuations interact with the skyrmion, while the ψϕ ﬂuctuations remain free spin- waves. In the singular limit only the exchange energy contributes to u(r) so

u(r) = (∇χˆ)2 − (∇rˆ)2. (5.44)

It is easy to compute (∇rˆ)2 = (∇(x/r))2 + (∇(y/r))2 = 1/r2. Next we organize the

terms in (∇χˆ)2 as 2 0 ! 2 0 mz (∇χˆ) = m [rˆrˆ]ij + mz[∇rˆ]ij + [rˆzˆ]ij . (5.45) z p 2 1 − mz This is easy to evaluate using (rˆ · ∇)rˆ = 0, so there are no cross terms. Then

02 2 mz 1 − mz u(r) = 2 − 2 . (5.46) 1 − mz r To ﬁnd the eigenmodes we solve the stationary equation

2 ∇ ψχ − uψχ = λψχ. (5.47)

89 The solution has the form ψχ(r, ϕ) = fn(r) cos(nϕ). Only the n = 0 sector can host bound states. Modes with n > 0 have energy above the gap. The n = 0 sector satisﬁes the equation 1 ∂ (r∂ f ) − uf = λf . (5.48) r r r 0 0

We solved this equation numerically to obtain the low energy excitations. The ansatz

η mz(r) = 1−2 exp((r/l) ) leads to the formation of a single bound state for η < 1.179.

The saddle point should have one negative energy mode, so we expect to find a bound state, and the bound state energy must be greater in magnitude than k + h/2 to overcome the ferromagnetic gap. The presence of absence of bound modes depends on the form of mz(r). In the limit singular limit l → 0 we find the bound state is below the ferromagnetic gap, so it is an unstable mode. The unstable mode is an expansion mode that tends to inflate the skyrmion. The higher energy states are all scattering state with energy above the spin wave gap. We have studied the scattering spectrum as a function of η, and we find that the spectrum is nearly identical to the ferromagnetic spectrum 1 (q) ≈ ∆ + q2 (5.49) sw 2 regardless of the value of η. This suggests that the spectrum of a singular skyrmion does not depend on the specific form of mz(r); hence, Ssp is not sensitive to microscopic details of the ferromagnetic exchange.

Entropy Barrier for Skyrmion Collapse

In Section 5.2.3 we found a variational estimate for the low energy modes of a skyrmion sk,i, and in Section 5.2.3 we argued that sp,i, the energy spectrum of the saddle point, follows the spectrum of ferromagnetic spin waves. Using these estimates

90 and the deﬁnition of entropy in Section 5.2.1 we can calculate the entropy barrier

N 1 X −∆S = − ln( / ). (5.50) 2 sk,i sp,i i We keep only the smallest N eigenvalues. We assume that high energy modes of the skyrmion and the saddle point are scattering states with eigenvalues approximately given by the spin-wave spectrum (it also suﬃcient to have ln(sk,i/sp,i) randomly distributed with zero mean for i > N). In the thermodynamic limit the spacing

2 2 between qi vanishes so for low energy modes qi ∆sw i.e., the singular skyrmion spectrum is dominated by the gap. This allows us to make the approximation sp,i ≈

∆sw. The skyrmion modes are doubly degenerate for n = 1, 2, ..., and there is a single mode with n = 0 so the entropy barrier is

2 1 ∆sk X ∆sk + γskn −∆S ≈ − ln − ln (5.51) 2 ∆ ∆ sp n sp Near the zero ﬁeld spiral to ferromagnet phase transition the skyrmion radius diverges

with α → ∞. Then ∆sk and γsk vanish, so the entropy is anomalously large. We

2 2 consider the limit ∆sk γsk to make the approximation ln(∆sk + γskn ) ≈ ln γskn .

This approximation is valid when h 1. Then the entropy barrier is

1 ∆ γ −∆S ≈ − ln sk − N ln sk − 2 ln N!. (5.52) 2 ∆sw ∆sw

We choose the value of N to be the largest integer that satisﬁes the condition ∆sk +

2 N γsk < ∆sw. We have ∆sw ∆sk so we can safely ignore ∆sk. Skyrmion excitations

with i larger that N are scattering states which we assume have a spin-wave spectrum

similar to the saddle point. When α → ∞ then N is large, so we can use Stirling’s

approximation

1 ∆ γ −∆S ∼ − ln sk − N ln sk − 2N ln N + 2N. (5.53) 2 ∆sw ∆sw 91 p Substituting N = ∆sw/γsk leads to a cancellation and s ∆ 1 ∆ −∆S ∼ 2 sw + ln sw . (5.54) γsk 2 ∆sk

When α → ∞ we can derive an explicit formula for ∆sk and γsk (see the end of

Section 5.2.3) to ﬁnd

4 p 1 ∆swRsk −∆S ∼ 2Rsk ∆sw + ln 2 3 3 (5.55) 2 w + hw Rsk

The skyrmion radius and w are dimensionless. To restore dimensions recall that the

natural length scale is J/D. In Figure 5.9 we plot the entropy barrier given by the

estimate in Equation (5.55). We relied on several approximations to derive (5.55):

(1) we assume the low energy excitations of a skyrmion are described by (5.32), (2)

we assume ∆ γn valid only when h 1, (3) we cutoﬀ the summation over modes

at the spin-wave gap, (4) we evaluate the dimensionless integral IX (α) in the limit

α 1, (5) we use Stirling’s approximation when N is large. The most restrictive

approximation is (5) which is only valid in the limit N → ∞. We use ∼ to denote that −∆S diverges with the form given in (5.55). It is remarkable that with these approximations the entropy barrier Figure 5.9 is qualitatively similar to the numerical result Figure 5.7.

5.3 Skyrmion Collapse in 3D

In Section 5.1 and Section 5.2.3 we discussed the energy barrier and entropy barrier for skyrmion collapse in a two-dimensional system. In this section we discuss skyrmion collapse in a 3D system with a thin-ﬁlm geometry. A naive estimate for the energy of skyrmion collapse in 3D with ﬁlm thickness d is simply ∆E3D = (d/a)∆E2D.

This relationship holds when a skyrmion tube collapses uniformly along its length.

92 Figure 5.10: Illustration of the collapse path for a 3D. The reaction coordinate is increasing from left to right. The surface represents the skyrmion radius defined by the condition mz = 0. In thick films, a Bloch point forms on one film surface; then, the Bloch point propagates through the film and leaves the other film surface.

We call this “uniform collapse” of the skyrmion, and it represent an upper bound for

∆E in 3D systems. Notice that this estimate for the energy barrier increases with

thickness without bound. In this section we will show that the energy barrier actually

saturates as a function of ﬁlm thickness. We also argue that the entropy barrier is

negligible in the bulk limit d → ∞.

5.3.1 Numerical Results

To simulate a thin-film geometry we use periodic boundary conditions in the xy- plane, and open boundary conditions along the z-axis with thickness d. We use the same procedure described in Section 5.1.1. We find that the minimum energy path for skyrmion collapse occurs through the formation of a Bloch point at one surface of the film. The Bloch point propagates through the film and is removed by passing through the other film surface. The saddle point state occurs when the Bloch point is

93 40

Energy barrier 20 ∆E = Ets − Esk Energy barrier

0 0.0 0.2 0.4 0.6 0.8 1.0 Film thickness dD/J

Figure 5.11: Numerical results for energy barrier for skyrmion collapse in a film geometry as a function of film thickness for AJ/D2 = −0.53, HJ/D2 = 0.37, and J/D = 64a. For thin films with d J/D the barrier increases proportional to d. In contrast, in thicker films the barrier saturates to the bulk value. The dashed gray lines represent the bulk and thin-film limits.

close to the surface where it ﬁrst appeared. In Figure 5.10 we show an illustration of skyrmion collapse in 3D This mechanism was recently observed experimentally [77].

In Figure 5.11 we plot numerical results for the energy barrier for skyrmion collapse as a function of thickness for AJ/D2 = −0.53 and HJ/D2 = 0.37. This ﬁgure show

that for d J/D the energy of the saddle point grows nearly proportional to d; this

limit can be obtained by making the uniform collapse assumption. When d J/D

the energy of the saddle point saturates. The important qualitative features of our

result are: (1) linear in d behavior when d J/D, (2) and saturation of Esp when

d J/D.

94 5.3.2 Variational Results

To better understand the qualitative features of skyrmion collapse in 3D we study the saddle point using a simple variational ansatz

sinh α(z) θ(r, z) = 2 tan−1 , (5.56) sinh r/w(z) where α = R/w and w functions of z. The energy is

Z d 0T 0 ! 1 wα I11 I12 wα E = dz 0 0 + E0(α(z), w(z)) (5.57) 0 2 w I12 I22 w where I11, I12, and I22 are dimensionless integrals arising from the exchange energy introduced by variations of α and w along the z-axis. We discuss these integrals in detail in Section 5.5.2. The function E0 is the energy of a 2D skyrmion in layer z.

The Euler-Lagrange (EL) equations are not analytically tractable in general. To proceed we will make the simplifying assumption α0(z) = 0. We have solved the EL equations numerically, and we ﬁnd that the assumption α0(z) = 0 introduces errors in ∆E of a few tens of percent. This is signiﬁcant given that the lifetime depends exponentially on ∆E. However, the solution to the EL for w is useful for the insight it provides into the qualitative features of Esp. Moreover, it is far more accurate than the uniform collapse assumption. The EL equation for w(z) is simple:

00 0 = I22w − ID + (kIK + hIH )w. (5.58)

The solution is w∗ − w∗e−q(z−z0) z > z w(z) = 0 (5.59) 0 z < z0

p ∗ where q = (kIK + hIH )/I22, and w = ID/(kIK +hIH ) is the equilibrium value. For typical values of k and h the dimensionless integrals are ∼ 1 so q ∼ 1 (in dimensionful

95 units q ∼ D/J). In Figure 5.10 we show an illustration of a solution with z0 → −∞,

z0 = 0, and z0 ≈ d/3. For z0 < 0 the system contains a skyrmion tube that extends

throughout the entire thickness of the ﬁlm. For 0 < z0 < d the system contains a

skyrmion tube that ends in a Bloch point at position z = z0. The system is fully

polarized when z0 > d. The energy for these three cases is evaluated separately;

2 −2qd  ID −2q|z0| 1−e Esk + e z0 < 0  kIK +hIH 2q 2 −2q(d−z ) d−z0 ID 1−e 0 E(z0) = Esk + 0 < z0 < d (5.60)  d kIK +hIH 2q  0 z0 > d

The energy attains a maximum for z0 = 0. This is the energy of the saddle point within our variational ansatz

2 −2qd ID 1 − e Esp = Esk + . (5.61) kIK + hIH 2q In the 3D limit the saddle point energy is

2 s 4 ID 1 1 IDI22 lim Esp = Esk + = Esk + 3 . (5.62) d→∞ kIK + hIH 2q 2 (kIK + hIH ) The energy barrier is determined by the core energy of the Bloch point and the energy

of the surrounding region where the Bloch point relaxes to a skyrmion conﬁguration.

Using our ansatz we estimate this core energy to be

s 4 1 IDI22 ∆E = 3 . (5.63) 2 (kIK + hIH ) This is an upper bound on the energy barrier for skyrmion collapse in bulk systems.

5.3.3 Entropy of a Skyrmion Tube

In this section we will ﬁnd the low energy modes of a skyrmion tube. We will

proceed in a similar fashion to Section 5.2.3. We assume the ﬂuctuations are described

by the ansatz sinh(α + ψ(ϕ, z)) θ(r, ϕ, z) = 2 tan−1 . (5.64) sinh(r/w)

96 with ψ 1 and keep terms only up to order ψ2 in the energy

Z d Z 2π 2 2 2 ! 1 dϕ 2 ∂ψ ∂ψ ∂ E0 2 0 E = E0 + dz I11w + I11 + 2 ψ . (5.65) 2 0 0 2π ∂z ∂ϕ ∂α

See Section 5.2.3 and Section 5.5.2 for a discussion of the dimensionless integrals I11 and I110 . The energy has translation symmetry in z and ϕ, so the eigenmodes are waves. Using the same procedure as in Section 5.2.3 we can read oﬀ the eigenvalues

2 1 1 ∂ E0 I110 1 2 2 2 2 2 2 n,kz = 2 2 + 2 n + 4π qz + = ∆ + γnn + 4π qz (5.66) w I11 ∂α I11 w

Similar to the 2D case, the gap ∆ and the stiﬀness γn both vanish in the α → ∞

limit. The vibration modes along the length of the tube satisfy qz = nzπ/d where nz

is in an integer; these vibration modes only become soft in thick ﬁlms when d a.

5.3.4 Entropy Barrier in 3D

It is diﬃcult to calculate the entropy of the saddle point in 3D, but we can make

estimates based on our understanding of the entropy in 2D. For thin ﬁlms with d

J/D the problem remains essentially two-dimensional. As we showed in Section

5.3.3 the low energy skyrmion ﬂuctuations of a skyrmion tube are essentially two-

dimensional when d J/D. In Section 5.3.1 we showed that skyrmion tubes collapse

uniformly in the d J/D limit. In this limit the saddle point is a string of singular

skyrmions skyrmions with an atomic scale radius. For such a conﬁguration the saddle

point spectrum should be similar to the spin wave spectrum except for the formation

of a single unstable mode bound to the string. Thus, in the thin-ﬁlm limit (d J/D)

∆S is given by the 2D result that we calculated in Section 5.2.3.

For thick ﬁlms with d J/D the saddle point diﬀers from the equilibrium

skyrmion tube only in a ﬁnite region or radius ∼ J/D surrounding a Bloch point. The

97 saddle point skyrmion tube has length ∼ d − J/D. We will consider a tube of length

d − z0 to study the entropy barrier as the Bloch point passes through the thickness of

the ﬁlm. Like the equilibrium tube, the saddle point tube has low energy vibrational

modes. However, the z-axis ﬂuctuations for the saddle point have a stiﬀer dispersion

2 2 2 2 2 2 2 2 qz ∼ nzπ /(d − z0) , compared with the dispersion qz = nzπ /d for the equilibrium skyrmion state. Using this we can estimate

Nsp 2 2 Nsk 2 2 X nzπ X nzπ − ∆S ≈ ln 2 − ln 2 (d − z0) d nz nz π π = 2 ln Nsp + 2Nsp ln − 2 ln Nsk − 2Nsk ln (5.67) d − z0 d

2 2 2 with cutoﬀs Nsp and Nsk determined implicitly by Nspπ /(d − z0) < ∆sw and

2 2 2 Nskπ /d < ∆sw. By choosing these values for the cutoﬀ we are ignoring states

above the spin wave gap. Nsp and Nsk are large in the limit d 1 so we can use

Stirling’s approximation and to ﬁnd

2 p p −∆S ∼ ln ∆ − 1 z ∆ (5.68) π sw 0 sw

At the saddle point z0 ≈ 1, and typically ∆sw ∼ 1 in dimensionless units, so ∆S ∼ 1.

This suggests that the entropy barrier is negligible in the bulk limit d J/D.

5.4 Conclusions

In this Chapter we studied skyrmion lifetimes in 2D and 3D systems as a function

of anisotropy AJ/D2, and external ﬁeld HJ/D2, and we derived several key results.

We showed that the prefactor τ0 in the Arrhenius law (5.29) is almost constant as a

2 2 function of AJ/D and HJ/D . Moreover, τ0 can be replaced with the phenomenological value ~/J with errors of only a few tens of percent. We also showed that the behavior of skyrmion lifetimes across the phase diagram is typically dominated by

98 ∆E, the energy barrier for skyrmion collapse. Thus, in most situations the skyrmion

lifetime can be approximately as

τ = ~ exp(β(E − E )). (5.69) J sp sk

In Section 5.1.2 we show that, although Esp is sensitive to microscopic parameters

and cannot be calculated accurately in a phenomenological model, it is safe to treat

Esp as a single phenomenological parameter. Therefore, the skyrmion energy Esk accounts for most of the variations of τ. In Section 5.1.2 we showed that J/D sets

the scale for the skyrmion radius when skyrmion lifetime is taken into account.

In Section 5.2.3 we showed that there exists an important exception to Equation

5.69; namely, we found that the entropy barrier diverges when the skyrmion radius

is much larger than J/D. This occurs in a small region near the zero ﬁeld spiral

to ferromagnet phase transition. In Section 5.2.3 we used this divergence to our

advantage to calculate the leading order divergence to the entropy barrier explicitly

and we ﬁnd p −∆S ∼ 2Rsk ∆sw. (5.70)

Finally, in Section 5.3 we studied skyrmion lifetimes in a ﬁlm geometry with thickness d. We found an interesting crossover from thin ﬁlms with d J/D, to

thick ﬁlms with d J/D. In the thin ﬁlm limit skyrmion collapse follows the

uniform collapse mechanism, and ∆E ≈ dD/J × ∆E2D, where E2D is the energy barrier for skyrmion collapse in 2D. In the thicker films the collapse mechanism is more interesting; a singular magnetic Bloch point forms at one surface of the film, as illustrated in Figure 5.10, and the Bloch point propagates through the film unzipping

99 the skyrmion tube. In the limit d J/D the energy barrier saturates to a value determined by the core energy of the Bloch point.

5.5 Technical Details of Calculations

5.5.1 Novel Method for Finding Saddle Points

To calculate the skyrmion lifetime we only need access to the metastable skyrmion state and the saddle point state (in contrast to quantum systems where the whole collapse path determines the lifetime). However, finding a saddle point is usually achieved by first constructing a minimum energy path from the metastable state to the ground state. For generality, consider a function f(x) (for magnetic systems x is the collection of spins, and f is the energy). A minimum energy path is a function x(λ) with derivative v = ∂λx that satisfies the necessary conditions

0 = ∇f(x) − (vˆ · ∇f(x))vˆ, (5.71)

along with the boundary conditions x(0) = x0 and x(1) = x1. In the skyrmion problem x0 corresponds to the metastable skyrmion state, and x1 corresponds to the ferromagnetic ground state. If x0 and x1 are local minima then ∇f(x) = 0 at λ = 0 and λ = 1. By the mean value theorem f(x) must attain a maximum f ∗ = f(x∗) in the interval λ ∈ (0, 1). We also demand that f ∗ is the lowest energy maximum taken over all paths. Under these conditions x∗ is a ﬁrst order saddle point, since f has negative curvature along vˆ, and ∇f(x) has positive curvature along other directions, otherwise, we could ﬁnd a smaller value for f ∗. The point x∗ is the saddle point.

Minimum energy paths are found numerically using “string methods”. To implement string methods the path x(λ) is discretized into x(λi) with i = 0, ..., N − 1 called a string. One common string method is the nudged elastic band method. This

100 method includes elastic couplings between neighboring points on the string, and gra-

dient optimization is used to relax the string to a local minimum. The forces are

modiﬁed during the gradient optimization, called nudging, so that the string is not

aﬀected by sharp bends in the true minimum energy path [83]. We use an alterna-

tive approach ﬁrst introduced in [84]. Interpolation is used to enforce the spacing

of points along the string, as opposed to elastic coupling in the nudged elastic band

method.

The interpolation method is divided into two steps. First, optimization of f(x) is

performed independently for each x(λi). Second, interpolation is used to restore even spacing of x(λi) along the string. These steps are alternated until convergence of x(λi). The number and type of optimization steps are hyperparameters that should be tune for good performance. The number and type of optimization steps can vary throughout the string optimization.

During the interpolation step the spacing between points on the string is measured using a distance function. For spins we use the Euclidean distance between points on the sphere and we ﬁnd it has good performance. Geodesic interpolation can also be used to improve performance on the sphere.

5.5.2 Dimensionless Integrals IX(α)

Our analytic results depend on the dimensionless integrals IX (α) with X = J, D, etc. In this section we discuss how we construct numerical functions for IX deﬁned on the range 0 ≤ α < ∞. Consider the energy functional

Z 1 E[m] = d2r (∇m)2 + m · ([zˆ × ∇] × m) + k(1 − m2) + h(1 − m ) (5.72) 2 z z

101 We introduce cylindrical coordinates r → (r, ϕ) for real-space, and spherical coordinates m → (m, θ, φ) for the magnetization, with m = 1. If we assume a skyrmion with polar angle θ(r) and azimuthal angle φ = ϕ, then the energy is

Z 1 sin2 θ sin 2θ E[θ] = 2π rdr θ02 + + θ0 + + k sin2 θ + h(1 − cos θ) . 2 r2 2r (5.73)

Next we introduce a two parameter ansatz for the skyrmion

sinh R/w θ(R, w; r) = 2 tan−1 . (5.74) sinh r/w

The dimensionless variable α = R/w is a key parameter that distinguishes bubble

skyrmions with α 1, from compact skyrmions with α . 1. Using this ansatz then energy is 1 E (α, w) = I − I w + (kI + hI )w2. (5.75) 0 J D 2 K H

where we have deﬁned the dimensionless integrals

Z 2 1 02 sin θ IJ (α) = 2π rdr θ + −−−→ 2πα + 2π/α (5.76) 2 r2 α→∞ Z −1 0 sin 2θ 2 ID(α) = w 2π rdr −θ − −−−→ 2π α (5.77) 2r α→∞ Z −2 2 IK (α) = w 2π rdr(sin θ) × 2 −−−→ 8πα (5.78) α→∞ Z −2 2 3 IH (α) = w 2π rdr(1 − cos θ) × 2 −−−→ 4πα + π /3. (5.79) α→∞

The α → ∞ limit converges exponentially fast. Skyrmions in 3D decay through the

formation of a Bloch point. To study Bloch point formation we consider an ansatz

θ(r, z) = θ(α(z), w(z); r). The energy of a skyrmion with this ansatz is

Z d 0 0 1 wα I11 I12 wα 1 2 E = dz 0 0 + IJ − IDw + (kIK + hIH )w (5.80) 0 2 w I12 I22 w 2

102 with dimensionless integrals

Z 2 −2 ∂θ I11(α) = w 2π rdr −−−→ 4πα (5.81) ∂α α→∞ Z −1 ∂θ ∂θ 2 3 I12(α) = w 2π rdr −−−→ 4πα + π /3 (5.82) ∂α ∂w α→∞ Z 2 ∂θ 3 3 I22(α) = 2π rdr −−−→ 4πα + π α. (5.83) ∂w α→∞

The Euler-Lagrange equations for α and w are

00 0 0 0 00 0 0 0 02 0 = −I22w − I22α w − I12(wα + w α ) − I12wα − ID + (kIK + hIH )w (5.84)

2 00 0 0 0 2 02 00 02 0 0 0 0 = −I11(w α + 2ww α ) − I11w α − I12(ww + w ) − I12ww α 0 0 0 0 1 wα I11 I12 wα 0 0 1 0 0 2 + 0 0 0 0 + IJ − IDw + (kIK + hIH )w α. (5.85) 2 w I12 I22 w 2

To study the entropy of skyrmions we need to ﬁnd low energy excitations. Consider excitations of a skyrmion of the form α → α + ψ(z, θ) with ψ 1. The energy of a skyrmion with this ﬂuctuations is (to order ψ2)

Z d Z 2π 2 2 2 ! dϕ 1 ∂ψ ∂ψ ∂ E0 2 0 E = E0 + dz I11 + I11 + 2 ψ (5.86) 0 0 2π 2 ∂z ∂ϕ ∂α with dimensionless integral

Z 1 ∂θ 2 I 0 (α) = 2π rdr → 4π/α. (5.87) 11 r2 ∂α

2 with E0 and ∂αE0 evaluated at ψ = 0. The explicit formulas we derived are valid only in the limit α → ∞. When α ≈ 1

−4 we calculate the integrals numerically. However, when α & 20 or when α . 10 the numerical integration becomes unstable. Thus, we use numerical integration in the region 10−4 < α ≥ 20. We use a table of values in this region to build an

103 interpolating function. For α > 20 we use the explicit formulas we derived. For

−4 α ≤ 10 most of the integrals vanish, except for IJ , which decays to 4π, I11, which diverges logarithmically, and I110 , which diverges with a power law. In this region we

ﬁt the integrals to a low order polynomial to capture the decay. For I11 and I110 we

ﬁnd the the power law form is (within 0.1% error)

I11 → −3ϕ − 8π ln(α) (5.88)

2 I110 → 4π/α (5.89) where ϕ is the golden ratio.

5.5.3 Viariational Methods for Entropy Calculations

To study the entropy of skyrmions and the singular skyrmion state we have ap- pealed to a variational theorem for the entropy. Let i denote the eigenvalues of the

Hessian matrix (sorted); this is the exact spectrum of the Hessian. Let µi denote eigenvalues of the Hessian matrix projected into an N-dimensional subspace. Then

i < µi for all i, so N N ˜ X X ˜ S = − ln βi ≥ − ln βµi = Sµ (5.90) i i where S˜ denotes the entropy with only the ﬁrst N eigenvalues included in the sum.

We showed for the singular skyrmion state that the eigenmodes are scattering states with energy above the gap energy ∆ (except for the bound mode that is explicitly ignored in the deﬁnition of S). We can use this to set an upper bound

S˜ ≤ −N ln β∆. So we have the lower bound

N ˜ X ∆S ≥ N ln β∆ − ln βµi. (5.91) i

104 ˜ Suppose we have found Sµ and ∆. If the modes with i > N are high energy

scattering states, then the energies will be similar to the ferromagnetic spectrum.

The modes can be ignored in calculating ∆S, so we can put an approximate lower

bound on ∆S. N N X X ∆S & N ln β∆ − ln βµi = − ln(µi/∆). (5.92) i i

5.5.4 Bose Statistics

In Harmonic Transition State Theory we use the result

X S = − ln (βi) (5.93) i

to calculate skyrmion lifetimes. This formula relies on a saddle point approximation

and is valid when βi 1. What happens if we treat each mode of the Hessian as a

quantum harmonic oscillator? Then we would obtain the entropy

e−βi −βi Si = − ln 1 − e + βi . (5.94) 1 − e−βi

In the limit βi 1 we recover − ln(βi) with corrections linear in βi. The classical limit is only valid when the number of Bosons in each state is large. This is at odds with the low temperature limit needed to justify the saddle point approximation in

HTST. We use HTST in the intermediate regime βi & 1.

5.5.5 Zero Field Spiral to Ferromagnet Phase Transition

Take the ansatz m(x, y, z) = (cos θ(x), 0, sin θ(x)). In a 1D chiral magnet with

Rashba SOC in zero ﬁeld the energy is

Z 1 E = dx θ02 − θ0 + k(1 − sin2 θ) . (5.95) 2

105 The Euler-Lagrange equation for θ is the Sine-Gordon equation

θ00 = −k sin 2θ (5.96)

The solution is the Jacobi amplitude θ(x) = ϕ(qx, m), with m = 2k/q2. The deriva- p tive satisﬁes θ0 = q 1 − m sin2 θ. The function sin(θ(x)) is periodic with period

Lϕ = 4K(m)/q, where K is the complete elliptic integral of the ﬁrst kind. Using this

the energy in one period is

Z Lϕ 1 2 2 E = q Lϕ − 2π + kLϕ − 2k dx sin θ. (5.97) 2 0

With the change of variables dθ/θ0 = dx we can evaluate the integral

Z Lϕ 1 Z 2π sin2 θ 4 K(m) − E(m) dx sin2 θ = dθ = (5.98) p 2 0 q 0 1 − m sin θ q m

where E(m) is the complete elliptic integral of the second kind. For a periodic texture

we should optimize energy density to ﬁnd q∗. The energy density is

E E 1 πq = q2 − − + k. (5.99) Lϕ K 2 2K

The optimum wavevector satisﬁes the equation

∂(E/L ) q2E(2qE − π) 0 = ϕ = . (5.100) ∂q 2(q2 − 2k)K2

Numerical root ﬁnding shows that the lowest energy solution is a root of (2qE −π). We can solve the equation exactly for a boundary case. The elliptic integral of the second kind is only deﬁned for arguments m < 1 and it is monotonically decreasing with m (monotonically increasing with q). The function π/2q is monotonically decreasing

with q. Therefore, the solution occurs when m = 2k/q2 = 1. Then q = π/2 and

2 k = π /8. The energy density for this solution is E/Lϕ = 0. This point is the

106 transition from the spiral phase to the polarized phase. For k < π2/8 the energy

density is negative. For k > π2/8 the energy density for the spiral phase is positive.

The general solution very nearly satisﬁes

π/2 − 1 q∗ ≈ 1 + k ≈ 1 + 0.46k. (5.101) π2/8

We can deﬁne the width of the domain wall wDW as the length needed for sin θ to

transition from −1/2 to +1/2. The resulting expression is

2 π 2k w = F , (5.102) DW q∗ 6 (q∗)2

where F is the incomplete elliptic integral of the ﬁrst kind.

5.5.6 Lattice Spin Models in TensorFlow and PyTorch

We used TensorFlow to ﬁnd the minimum energy path for skyrmion collapse. Ten-

sorFlow, and similar frameworks like PyToch, Keras, etc., are ideal for implementing

lattice spin models. To understand why this is true, consider a general lattice spin

model 1 X E = mT H m . (5.103) 2 i i,j j

The tensor Hi,j (the Hessian) captures the eﬀects of on-site interactions, arbitrary range exchange interactions, long-range dipole-dipole interactions, and disorder. If mi is a saddle point then it must satisfy the equation

∂E X 0 = = J T H m (5.104) xi i,j j ∂xi

where xi = (θi, φi), and Jxi = ∂xi m is part of the Jacobian matrix. The solution

Jφi = 0 occurs when there is a coordinate singularity (θi = 0 or π). This can

107 be removed by redeﬁning θi and φi with respect to new coordinate axes. With no coordinate singularities the solution is Hi,jmj = 0. Then the Hessian is

2 ∂ E T H 0 = = J H J 0 . (5.105) xi,xj 0 xi i,j xj ∂xi∂xj

For models with translation symmetry Hi,j depends only on rj − ri. In that case the energy can be calculated using convolution. We use this fact to deﬁne E in

TensorFlow using a convolution operation that is highly optimized to run on modern

GPUs. Other packages (PyTorch, Keras, etc.) can also be used for convolution. On- site terms also be added trivially in TensorFlow. Once we have defined an energy functional in TensorFlow, then we can reap the benefits of automatic differentiation of E with respect to xi. This allows us to perform gradient optimization and calculate the Hessian without writing additional code.

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