Skyrmions in Chiral Magnets with Rashba and Dresselhaus Spin-Orbit Coupling
SSkyrmions in Chiral Magnets with Rashba and Dresselhaus Spin-Orbit Coupling
James Rowland, Sumilan Banerjee, and Mohit Randeria Department of Physics, The Ohio State University, Columbus, Ohio, 43210 Department of Condensed Matter Physics, Weizmann Institute of Science, Israel, 7610001 (Dated: November 5, 2018)Skyrmions are topological spin textures of interest for fundamental science and applications.Previous theoretical studies have focused on systems with broken bulk inversion symmetry, whereskyrmions are stabilized by easy-axis anisotropy. We investigate here systems that break surface-inversion symmetry, in addition to possible broken bulk inversion. This leads to two distinctDzyaloshinskii-Moriya (DM) terms with strengths D ⊥ , arising from Rashba spin-orbit coupling(SOC), and D (cid:107) from Dresselhaus SOC. We show that skyrmions become progressively more sta-ble with increasing D ⊥ /D (cid:107) , extending into the regime of easy-plane anisotropy. We find that thespin texture and topological charge density of skyrmions develops nontrivial spatial structure, withquantized topological charge in a unit cell given by a Chern number. Our results give a design prin-ciple for tuning Rashba SOC and magnetic anisotropy to stabilize skyrmions in thin films, surfaces,interfaces and bulk magnetic materials that break mirror symmetry. Recently there has been a surge of interest in skyrmionsin chiral magnetic materials , ranging from fundamen-tal science to potential device applications. A skyrmionis a spin texture characterized by a topological invariantthat, in metallic magnets, gives rise to the topologicalHall effect and may also have implications for non-Fermi liquid behavior . The ability to write and erase in-dividual skyrmions , along with their topological stabil-ity, small size, and low depinning current density , pavesthe way for potential information storage and processingapplications.Experiments have focussed primarily on skyrmions innon-centrosymmetric crystals with broken bulk inversionsymmetry, e.g., metals like MnSi, FeGe and insulatorslike Cu OSeO . In these bulk materials, the skyrmioncrystal (SkX) phase is stable only in a very limited re-gion of the magnetic field ( H ), temperature ( T ) phasediagram . On the other hand, the skyrmion phase isfound to be stable over a much wider region of ( T, H ) inthin films of the same materials , even extendingdown to T = 0 in some cases . (A class of two di-mensional (2D) systems shows atomic-scale skyrmions arising from competing local interactions, distinct fromthe spin-orbit induced chiral interactions that we focuson here.)A key question that we address is this paper is: Howcan we enhance the domain of stability of skyrmion spintextures? We are motivated in part by the thin film ex-periments, and also by the possibility of chiral magnetismin new 2D systems like oxide interfaces . We showhow the SkX become progressively more stable over everlarger regions in parameter space of field H and mag-netic anisotropy A , as the effects of broken surface inver-sion dominate over those of broken bulk inversion. Thekey parameter responsible for this behavior is the ratio D ⊥ /D (cid:107) of the strength of the chiral magnetic interactionarising from broken bulk inversion ( D (cid:107) ) to that arisingfrom broken surface inversion ( D ⊥ ); see Fig. 1.We begin by summarizing our main results, which re-quires us to introduce some terminology. We focus on magnets in which spin textures arise from the inter-play between ferromagnetic exchange J and the chiralDzyaloshinskii-Moriya (DM) interaction D ij · ( S i × S j ).Spin-orbit coupling (SOC) determines the magnitude ofthe D vector, while symmetry dictates its direction. Bro-ken bulk inversion symmetry ( r → − r ) leads to the Dres-selhaus DM term with D ij = D (cid:107) (cid:98) r ij , where (cid:98) r ij = r ij / | r ij | with r ij = ( r i − r j ). On the other hand, broken sur-face inversion or mirror symmetry ( z → − z ) leads to theRashba DM term with D ij = D ⊥ ( (cid:98) z × (cid:98) r ij ). In the limitof weak SOC, D/J (cid:28)
1, where D = ( D (cid:107) + D ⊥ ) / , thelength scale of spin textures is ( J/D ) a (cid:29) a (the micro-scopic lattice spacing) and we can work with a continuum“Ginzburg-Landau” field theory.We show in Fig. 1 the evolution of the T = 0 phase di-agram going from the pure Dresselhaus limit to the pureRashba limit. Each phase diagram is plotted as a func-tion of the (dimensionless) field HJ/D and anisotropy AJ/D . Here A >
A <
0) corresponds to easy-plane(easy-axis) anisotropy. Our main results are:(1) As the Rashba D ⊥ is increased relative to the Dres-selhaus D (cid:107) , the spiral and skyrmion phases become in-creasingly more stable relative to the vertical cone phase,and penetrate into the easy-plane anisotropy side of thephase diagram.(2) With increasing D ⊥ /D (cid:107) , the textures change con-tinuously from a Bloch-like spiral to a Neel-like spiral.Correspondingly, the skyrmion helicity evolves with avortex-like structure in the Dresselhaus limit to a hedge-hog in the Rashba limit, which is shown to impact theferrotoroidic moment.(3) In the pure Rashba limit, we find the largest do-main of stability for the hexagonal skyrmion crystal. Inaddition we also find a small sliver of stability for a squareskyrmion lattice, together with an elliptic cone phase,distinct from the well-known vertical cone phase in theDresselhaus limit.(4) We see in Fig. 2 that in the Rashba limit the spintexture of the skyrmion and their topological charge den- a r X i v : . [ c ond - m a t . s t r- e l ] O c t - / D H J / D D ⟂ = - / D D ⟂ = ∥ Hex SkX - / D D ⟂ = ∥ Hex SkXSpiral C one Polarized FM - / D D ∥ = H J / D m ⟂ Hex SkXSpiralPolarized FMSquare SkX E lli p t i cc one T il t ed F M m FM, z 〈 m EC, z 〉 FIG. 1. Phase diagrams as a function of
AJ/D and HJ/D for four values of D ⊥ /D (cid:107) . Easy-axis anisotropy corresponds to A <
A >
0. The cone, elliptic cone, and tilted FM phases are shown schematically, with the Q-vectorshown in red and the texture traced out by spins shown in black. The color bar on the right indicates m z for the elliptic coneand tilted FM phases in the D (cid:107) = 0 panel. Insets: Unit cell in the hexagonal (Hex) skyrmion crystal (SkX) phase with whitearrows indicating the projection of magnetization on the x - y plane. The colors indicates the magnitude and direction of thespin projection following the convention of ref. 3 indicated in the color wheel. Thick lines denote continuous transitions, whilethin lines indicate first-order phase transitions. sity χ ( r ) begins to show non-trivial spatial variations asone changes anisotropy, but the total topological charge N sk = (cid:82) d r χ ( r ) in each unit cell remains quantized,even when χ ( r ) seems to “fractionalize” with positiveand negative contributions within a unit cell.(5) For H > A , one can have isolated skyrmions ina ferromagnetic (FM) background, and their topologicalcharge N sk is quantized, as usual, by the homotopy group π ( S ) = Z . For H < A , we find that skyrmions cannotexist as isolated objects, and N sk must now be defined bythe Z Chern number classifying maps from the SkX unitcell, a two-torus T to S , the unit sphere in spin-space,a definition that works for all values of H/ A . Free energy:
We consider a continuum (free) energyfunctional F [ m ] = (cid:82) d r F ( m ) with F = F J + F DM + F A − Hm z (1)whose form is dictated by symmetry. The isotropic ex-change term F J = ( J/ (cid:80) α ( ∇ m α ) ( α = x, y, z ) con-trols the gradient energy through stiffness J . The DMcontribution in the continuum F DM = D cos β m · ( ∇× m )+ D sin β m · [(ˆ z ×∇ ) × m ] (2)is the sum of the two terms discussed above. The D (cid:107) = D cos β term arises from Dresselhaus SOC in theabsence of bulk inversion and D ⊥ = D sin β from RashbaSOC with broken surface inversion. The anisotropy term F A = Am z can be either easy-axis ( A <
0) or easy-plane(
A > A ,including single-ion and dipolar shape anisotropies. Inaddition, Rashba SOC naturally leads to an easy-plane,compass anisotropy A ⊥ (cid:39) D ⊥ / J , which is energetically comparable to the DM term . We treat A as a free,phenomenological parameter.We focus on T = 0 where the local magnetization isconstrained to have a fixed length m ( r ) = 1, and itshould be hardest to stabilize skyrmions; once | m ( r ) | canbecome smaller due to thermal fluctuations, skyrmionsshould be easier to stabilize. It is convenient to scale alldistances by the natural length scale J/D (setting the mi-croscopic a = 1) and scale the energy F by D /J . All ourresults will be presented in terms the three dimension-less parameters that describe F , namely field HJ/D ,anisotropy AJ/D , and tan β = D ⊥ /D (cid:107) . Phase Diagram:
In Fig. 1, we show the evolutionof the (
A, H ) phase diagram as a function of tan β = D ⊥ /D (cid:107) , increasing from left to right. These results wereobtained by minimizing the energy functional F DM sub-ject to m ( r ) = 1. The energies of the fully polarizedferromagnet (FM), the tilted FM, and the vertical conestates can be determined analytically, while the energiesfor the spiral, the skyrmion crystals and the elliptic conestate were found by a numerical, conjugate gradient min-imization approach. In all cases, the results were checkedby semi-analytical variational calculations. Details of themethodology are described in the Supplementary Mate-rials; here we focus on the results.We begin with well known results in the Dressel-haus limit (left panel of Fig. 1), where the hexagonalSkX and spiral phases are stable only in a small regionwith easy-axis anisotropy ( A ≤ A > m cone ( z ) =(cos ϕ ( z ) sin θ , sin ϕ ( z ) sin θ , cos θ ) with ϕ ( z ) = D (cid:107) z/J and cos θ = H/ [2 A + D (cid:107) /J ]. The phase boundary be- m ⟂ AJ / D = / D = / D = / D = χ - - - FIG. 2. Evolution of the spin texture m (top row) and the topological charge density χ (bottom row) for four values of AJ/D at fixed HJ/D = 0 . D (cid:107) = 0). White arrows indicate the projection of m into the x - y plane. The colorsalso indicates the magnitude and direction of the spin projection following the convention of ref. 3 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 overa single unit cell (cid:82) d r χ ( r ) = − tween the vertical cone and polarized FM is given by H = 2 A + D (cid:107) /J .We note a change of variables that greatly simplifiesthe analysis of skyrmion crystal and spiral phases. Thistransformation is useful when m = m ( x, y ) has no z -dependence (along the field). Using a rotation R z ( − β )by an angle − β about the z -axis, we define n ( x, y ) = R z ( − β ) m ( x, y ). It is then easy to show that the twoterms in (2) combine to give a pure Dresselhaus form F DM = D n · ( ∇× n ). The other terms in (1) are invariantunder this transformation, and thus F greatly simplifiesusing the transformation m → n .We choose, without loss of generality, ˆ x as the prop-agation direction for the spiral of period L , so that n sp ( x ) = (0 , sin θ ( x ) , cos θ ( x )) with n sp ( x + L ) = n sp ( x ).(Note that this is not in general a single- q spiral). Weminimize F to find the optimal L and optimal func-tion θ ( x ), which is a 1D minimization problem. Forthe SkX, we first pick a unit cell, hexagonal or square.We then find its optimal size and optimal texture n =(cos ϕ sin θ, sin ϕ sin θ, cos θ ), by solving a 2D minimiza-tion problem to determine ϕ ( x, y ) and θ ( x, y ) within aunit cell, subject to periodic boundary conditions. Wecalculate the optimal n and transform it back to the mag-netization m at the end.With increasing D ⊥ /D (cid:107) , we see that the SkX and spi-ral phases become more stable relative to the verticalcone, and their region of stability extends into the easy-plane regime. To understand this, consider increasing theRashba D ⊥ keeping D (cid:107) fixed. The energy of cone m ( z )depends only on D (cid:107) , and is unchanged as D ⊥ increases.In contrast, the SkX and spiral, with m = m ( x, y ), uti- lize the full D = ( D (cid:107) + D ⊥ ) / to lower their energy. Helicity and Ferrotoroidic Moment:
We findthat the spin textures smoothly evolve as a function of D ⊥ /D (cid:107) . The spiral continuously changes from a Bloch-like (helical) spiral in the Dresselhaus limit to a Neel-like(cycloid) spiral in the Rashba limit. In between, the spinstumble around an axis at an angle β = tan − ( D ⊥ /D (cid:107) )to the q -vector of the spiral. Similarly the skyrmionssmoothly evolve from vortex-like textures in the Dressel-haus limit to hedgehogs in the Rashba limit, as seen inthe insets of Fig. 1. In fact, γ = π/ − β is the “helicity” of the skyrmions.Our results imply that D ⊥ /D (cid:107) controls the helicity γ ,where Rashba D ⊥ could be tunable by electric field at aninterface or by strain in a thin film. The ability to tune γ could be important in several ways. There is a recent pro-posal to use helicity to manipulate the Josephson effectin a superconductor/magnetic-skyrmion/superconductorjunctions . Another interesting phenomenon that de-pends on the helicity of skyrmions is the “ferrotoroidicmoment” t = (1 / (cid:82) d r [ r × m ( r )] . We will showelsewhere that t = t sin γ (cid:98) z for the SkX. Rashba limit:
Next we turn to the D (cid:107) = 0 resultsin the right panel of Fig. 1, where one has the maximumregime of the stability for the spiral and the hexagonalSkX, in addition to a small region with a square latticeSkX (first predicted in ref. 25), an elliptic cone phaseand a tilted FM. This phase diagram improves upon allprevious works as explained in detail in the Sup-plementary Material.The tilted FM, which spontaneously breaks the U (1)symmetry of F (in a field), has m z = H/ A and exists inthe regime 2 A > H and
AJ/D ⊥ > D (cid:107) = 0. We alsosee a new phase where the spins trace out a cone withan elliptic cross-section. The elliptic cone axis makes anangle θ = cos − ( H/ A ) with ˆ z , and the spatial variationof m is along a q -vector in the x - y plane.The nature of various phase transitions is discussed inthe Supplementary Materials. In Fig. 1, thick lines de-note continuous while thin lines denote first-order transi-tions. ( A = 2 , H = 4) J/D is a Lifshitz point at whicha state without broken symmetry (polarized FM) meetsa broken symmetry (tilted FM) and a modulated (ellipticcone) phase.Let us next consider deviations from pure Rashba limitto see how the extreme right panel of Fig. 1 evolves intothe D (cid:107) (cid:54) = 0 phase diagrams. As soon as one breaks bulkinversion, an infinitesimal D (cid:107) leads to the tilted FM beingoverwhelmed by the vertical cone, which gains Dressel-haus DM energy. On the other hand, the elliptic andvertical cone states compete for D (cid:107) (cid:54) = 0 and for somesmall, but finite, D (cid:107) the vertical cone wins. Spin textures and topological charge:
There areinteresting differences between the skyrmions for
H < A and H > A . ( H = 2 A is marked as a dashed line in thephase diagrams of Fig. 1). First, let us look at the spintextures. For SkX with H > A , which have been thefocus of all the past work, the spins at the boundary ofthe unit cell (u.c.) are all up, parallel to the field. Henceone can think of isolated skyrmions in a fully polarizedFM background; see Fig. 2 left-most panels. It is theidentification of the point at infinity in real space for anisolated skyrmion that lets us define a map from S → S and use the homotopy group π ( S ) = Z to characterizethe topological charge or skyrmion number N sk .In contrast, when H < A , we find that the spins at theboundary are not all pointing up and the only constraintis periodic boundary conditions on the u.c.; see Fig. 2.There is no way to isolate this spin texture in a FMbackground. We must now consider the map r → m ( r )from the u.c., which is a 2-torus T to S in spin space.(Such maps are well known when T represents a Bril-louin zone in k -space, but the mathematics is identical.)This map is characterized by an integer Chern number N sk = (cid:82) u . c . d r χ ( r ), where χ ( r ) = m · ( ∂ x m × ∂ y m ) / π is the topological charge density. In fact, one can use thisdefinition of N sk for all values of H/ A .From the A = 0 panel on the left side of Fig. 2, wesee 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 > A .With increasing A , once H < A , we see that χ ( r ) beginsto spread out and even changes sign within the u.c. Inthe square SkX phase χ is again concentrated, but thistime 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 < ∼ A , the spin textures in the SkX phases are es-sentially composed of vortices and anti-vortices. Never-theless, the Chern number argument shows that the totaltopological charge in each u.c. is an integer; N sk = − Discussion:
Previous theories on understandingthe increased stability on skyrmions in thin filmsof non-centrosymmetric materials focussed pri-marily on the changes in uniaxial magnetocrystallineanisotropy with thickness, or on finite-size ef-fects . In fact, the latter can give rise to spin-texturesmore complicated than skyrmion crystals, with variationsin all three directions. However, none of these theoriestake into account the role of broken surface inversion andRashba SOC. As we have shown here, a non-zero Rashba D ⊥ leads to a greatly enhanced stability of the SkXphase, particularly for easy-plane anisotropy, while at thesame time giving a handle on the helicity of skyrmionswith interesting internal structure.We note that the phase diagrams in Fig. 1 apply to allsystems with broken mirror symmetry, with or withoutbulk inversion. Mirror symmetry can be broken by cer-tain crystal structures in bulk materials, by strain inthin films, or by electric fields at interfaces. For systemswith D (cid:107) = 0 the vertical cone phase, which dominatesmuch of the phase diagram for D (cid:107) (cid:54) = 0, simply does notexist. After our paper was written, we became aware ofthe very recent observation of hedgehog-like skyrmionsin the magnetic semiconductor GaV S , a polar mate-rial with broken mirror symmetry that is dominated byRashba SOC. Skyrmions are, however, stabilized in thismaterial only at finite temperature due to the large easy-axis anisotropy.In conclusion, we have made a comprehensive study ofthe T = 0 phases, with a focus on skyrmion crystals inchiral magnets with two distinct DM terms. D ⊥ , arisesfrom Rashba SOC and broken surface inversion, while D (cid:107) comes from Dresselhaus SOC and broken bulk inver-sion symmetry. We predict that increasing the RashbaSOC, via strain or electric field, and tuning magneticanisotropy towards the easy-plane side will greatly helpstabilize skyrmion phases in thin films, surfaces, and in-terface magnetism. Our results are very general, basedon a continuum “Ginzburg-Landau” energy functionalwhose form is dictated by symmetry. We hope that itwill motivate ab-initio density functional theory calcu-lations of the relevant phenomenological parameters en-tering our theory and an experimental investigations ofskyrmions in Rashba systems. Acknowledgments : We thank C. Batista, S. Lin andN. Nagaosa for useful discussions. MR acknowledges sup-port from NSF DMR-1410364. JR was supported by anNSF Graduate Research Fellowship, and by the CEM, anNSF MRSEC, under grant DMR-1420451.
APPENDICES
CONTENTS
A. Continuum Free Energy 5B. Ferromagnetic and Cone Phases 6C. Numerical Methods 7D. Variational Solution 8E. Phase Transitions 9F. Rashba Limit Phase Diagram 10G. Magnetic Anisotropy 11References 11
Appendix A: Continuum Free Energy
In this appendix we first introduce the continuum freeenergy functional that we use to model the spin texturesin a chiral magnet. The free energy for a magnetic sys-tem with broken bulk inversion and mirror symmetries is F [ m ( r )] = (cid:82) d r F ( m ( r )) where F ( m ( r )) = ( J/ ∇ m ) (A1)+ D (cid:107) m · ( ∇ × m )+ D ⊥ m · ((ˆ z × ∇ ) × m )+ Am z − Hm z . and ( ∇ m ) is shorthand for (cid:80) i,α ( ∂ i m α ) . The z -axisis the axis of broken mirror symmetry. Here J is theexchange stiffness, D ij = D (cid:107) ˆ r ij + D ⊥ ˆ z × ˆ r ij is the DMvector, A is the magnetic anisotropy, and H is the field.We measure all the lengths in units of microscopic latticespacing a , which we set to unity, so that J , D (cid:107) , D ⊥ , A and H all have units of energy. We are interested in lowtemperature behavior so we ignore fluctuations in themagnitude of the local magnetization m and impose theconstraint that | m ( r ) | = 1.In our free energy functional we take the normal to theplane in which mirror symmetry is broken and the easy-axis direction to be the same, namely ˆ z . For simplicity,we also choose the external field to be along the samedirection. One can, of course, imagine more general situ-ations in which these directions are not all the same, butthe “simple” case treated here has sufficient complexitythat it must be thoroughly investigated first.Next we define parameters D and β such that D (cid:107) = D cos β and D ⊥ = D sin β. (A2)It is convenient to rewrite (A1) using the natural energyand length scales in the problem. We measure energies in units of D /J and lengths in units of J/D . In scaledvariables the free energy density is given by F ( m ( r )) = (1 / ∇ m ) (A3)+ m · ([cos β ∇ + sin β ˆ z × ∇ ] × m )+ Am z − Hm z , which depends on three dimensionless variables AJ/D , HJ/D and β = tan − ( D ⊥ /D (cid:107) ). In the main paper,we explicitly write the anisotropy and field as AJ/D and HJ/D , but in the Appendices we simplify notationand denote them as just A and H . (The total energy F depends on an inconsequential overall factor of ( J/D ) coming from the integration over the volume.)Next we briefly discuss the phases that we find as afunction of A , H and β . The two ferromagnetic (FM)phases – fully polarized and tilted – are states with nospatial variations. The (vertical) cone phase is a non-coplanar state which has only z -variations, along themagnetic field direction, so that m = m ( z ). These statescan be treated analytically, as discussed in Appendix B.The spiral and skyrmion phases have a local magne-tization of the form m = m ( x, y ), as does the ellipticcone phase. Specifically, the spiral has spatial variationalong a single direction, say x , in the plane perpendicu-lar to the field. The SkX phases have magnetic texturevarying in both x and y . We use numerical methods forthe analysis of phases with m = m ( x, y ) as described inAppendix C. Note that we do not consider states where m has non-trivial variations in all three coordinates; see,e.g., ref. 30. Rotation:
We next give the details of a transforma-tion (introduced in the main text) that greatly simplifiesthe analysis for states where m = m ( x, y ). We make therotation n = cos β sin β − sin β cos β
00 0 1 m ≡ R z ( − β ) m . (A4)where β = tan − ( D ⊥ /D (cid:107) ). Expressed in terms of m thefree energy density (A3) simplifies to F = (1 / ∇ n ) + n · ( ∇ × n ) + An z − Hn z , (A5) provided that m , and thus n , depends only on x and y ,but not on z . This result (A5) has the same form as thefree energy density in the pure Dresselhaus limit. Aftersolving the problem in the n representation, we musttransform back to m to find the actual spin texture.It is easy to see that the the exchange term is invari-ant under m → n , i.e., ( ∇ m ) → ( ∇ n ) , as are theanisotropy term and Zeeman term coupling to H . Theonly term that has a nontrivial transformation is DMterm in (A3). We symbolically write the DM term as as m · ( D × m ) where D ≡ cos β ∇ + sin β (ˆ z × ∇ )= cos β∂ x − sin β∂ y sin β∂ x + cos β∂ y . (A6)Here we set D z = cos β∂ z →
0, because we focus on tex-tures that have no z -variation, as already stated above.A straightforward calculation, using m = R z ( β ) n , thenallows us to derive m · ( D × m ) = n · ( ∇ × n ) (A7)which in turn leads to eq. (A5).There is a slick way to obtain this same result by notingthat the Free energy is invariant under a combined rota-tion in spin-space and in real space about the z -axis. (Acombined spin and spatial rotation is needed because ofSOC. and the z -axis is singled out by broken mirror sym-metry). In fact, it is simple to proceed with the generalcase where we retain D z = cos β∂ z . The transformation R z ( − β ) of eq. (A4) acts only in spin-space. Thus to writethe 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 ) ∂ z ˆ z ] . (A8)Thus, for any magnetic texture m ( x, y, z ), the DM termcan we written in general as+ n · ( ∇ × n ) − (1 − cos β ) n · ( ∂ z (ˆ z × n )) (A9)For a magnetic texture m ( x, y ) that does not vary alongthe z -axis the z -derivative terms vanish and this resultsimplifies to (A5) derived above. Appendix B: Ferromagnetic and Cone Phases
In this appendix (B) we consider phases which canbe treated analytically: the polarized ferromagnet (FM),tilted FM and the vertical cone phase.
Ferromagnets:
The free energy density for a FMstate is F = Am z − Hm z . (B1)Minimizing the free energy is easy in this case; we needto solve 0 = ∂ F ∂m z = 2 Am z − H (B2)along with the constraint | m | = 1. The solution is m ∗ z = (cid:26) H ≥ A H A H < A ⇒ F ∗ = (cid:26) A − H H ≥ A − H A H < A . (B3)We call the solution with m ∗ z = 1 a polarized FM sincethe magnetization is aligned with the magnetic field. The solution with m ∗ z < U (1) symmetry of spin rotation around z axis in B1. Cone:
Another simple class of magnetic states aretextures m ( z ) that do not break translational symme-try in the x - y plane. We will find that the optimumconfiguration is a cone texture. This texture is called acone because the magnetic moments trace out a cone asa function of z ; see the illustration in figure B.1.For textures m ( z ) with translation symmetry in the x - y plane the Rashba term in (A3), with strength sin β ,does not contribute to the free energy. To implement theconstraint | 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 ) . (B4)In terms of θ ( z ) and φ ( z ) the free energy density is F ( θ ( z ) , φ ( z )) = 12 ( θ (cid:48) ) + 12 ( φ (cid:48) ) sin θ − φ (cid:48) cos β sin θ + A cos θ − H cos θ. (B5)The Euler-Lagrange equation for φ ( z ),0 = ∂∂z ∂ F ∂φ (cid:48) = ∂∂z ( φ (cid:48) sin θ − cos β sin θ ) , (B6)can be integrated with the result φ (cid:48) sin θ − cos β sin θ = C. (B7)Thus the free energy density can be written F ( θ ( z )) = 12 ( θ (cid:48) ) + f ( θ ) (B8)where f does not depend on θ (cid:48) . 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 (B5).Given that a constant θ ( z ) = θ ∗ minimizes the free en-ergy it is easy to find the optimum value θ ∗ and theextremal function φ which is φ ∗ ( z ) = z cos β. (B9)There are two solutions for θ ∗ . The first solution is θ ∗ = 0which is a ferromagnetic solution with energy F = A − H .The second solution, with θ ∗ = cos − ( H/ (2 A + cos β )) , (B10)is called a cone texture. To make contact with themain text, we must recall that lengths are measuredin units of J/D , so that eq. (B9) becomes φ ∗ ( z ) → z ( D/J )( D (cid:107) /D ) = zD (cid:107) /J , and energies A and H in unitsof D /J , so that θ ∗ → cos − [ H/ (2 A + D (cid:107) /J )].We will occasionally refer to the cone as a vertical coneto distinguish it from the elliptic cone found in Appen-dices C and D. The elliptic cone varies in the x - y plane FIG. B.1. Illustration of the vertical cone phase, with q -vector (red arrow) along the z -axis, and the elliptic conephase. In the elliptic cone phase the magnetization traces outan elliptic cone, i.e., the cross section is an ellipse rather thana circle. The elliptic cone phase shown here is for the D (cid:107) = 0limit. If the cone height is decreased so that the spins lie inthe x - z plane the configuration becomes a cycloid (Neel-likespiral). while the vertical cone varies along the z -axis. An illus-tration of the vertical and elliptic cone textures is givenin figure B.1. In the Rashba limit, D (cid:107) = 0, the conephase is not stable for any values of H and A . For finite D (cid:107) the tilted FM phase is not stable anywhere and thevertical cone takes its place in the phase diagram. Appendix C: Numerical Methods
In appendix C we consider phases that cannot betreated analytically: the spiral, elliptic cone, squareskyrmion crystal and hexagonal skyrmion crystal phases.For these phases the free energy density needs to be inte-grated and minimized numerically; we use conjugate gra-dient minimization to achieve this. All of these phasesare of the form m ( x, y ) so we use the transformed freeenergy density (A5) and we will refer to n ( x, y ) as the tex-ture in the spin rotated frame. To implement numericalintegration of the free energy some boundary conditionsneed to be specified. We use three boundary conditions:periodic along the x -axis, periodic with square symmetryand periodic with hexagonal symmetry.For each boundary condition the minimization proce-dure is very similar: convert the function n ( x, y ) to a vector s ( i, j ), write the integral as a sum, then minimizeto find the optimum vector. We discuss this procedurein detail only for the simplest case (periodic along the x -axis) and we discuss the important differences for theother two cases. Spiral and elliptic cone:
First we consider textureswhich are periodic along a single axis; here we choose the x axis without loss of generality. The periodic boundarycondition means that the texture n ( x ) satisfies n ( x ) = n ( x + L ). Since the free energy density is uniform alongthe y and z axes the free energy is just a one-dimensionalintegral along the x axis.To evaluate this integral numerically we can convert itto a sum. This involves discretizing the function n ( x );let s ( i ) = n (( i − x ) where ∆ x = L/ ( N − s ( i + N ) = s ( i ) is the periodic boundary condition. Nextwe replace all derivatives in the free energy density withfinite differences, i.e., ∂ x n ( x ) → ( s ( i + 1) − s ( i )) / ∆ x forthe point x = ( i − x . Lastly we replace the integralwith a sum ( (cid:82) L dx → ∆ x (cid:80) Ni =1 ). At this point the freeenergy can be computed given a configuration s ( i ) andthis will be a good approximation to the true free energywhen N is large, i.e., F ( s ) ≈ F [ n ] is a good approxima-tion when N is large.We also have the constraint | n ( x ) | = 1. In termsof s the constraint is | s ( i ) | = 1. We impose the con-straint 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 2 N compo-nent vector { θ (1) , ..., θ ( N ) , φ (1) , ..., φ ( N ) } = { θ , φ } = p .We can minimize the vector function F ( p ) using con-jugate gradient minimization. Conjugate gradient mini-mization accepts as its input a function, in this case F ,the gradient of that function ∇ p F , and an initial vector,call it p ; the output of conjugate gradient minimiza-tion is a local minimum p ∗ . We choose single- q spirals m ( x ) = (cos qx, sin qx,