Skyrmions in quasi-2d chiral magnets

Transcription

1 MRSEC 1 Skyrmions in quasi-2d chiral magnets Mohit Randeria Ohio State University kitp ucsb August 2015

2 2 James Rowland Sumilan Banerjee (now at Weizmann) Onur Erten (now at Rutgers) * Banerjee, Erten & MR, Nature Physics 9, 626 (2013) MRSEC * Banerjee, Rowland, Erten & MR, Phys. Rev. X 4, (2014) * Rowland, Banerjee & MR (in preparation, 2015)

3 Outline: o Introduction * Skyrmions * Materials * Properties o How to stabilize skyrmion phases? * Role of Dresselhaus vs. Rashba SOC * Role of magnetic anisotropy o Conclusions 3

4 o Skyrmions topological solitons in field theory o Skyrmions in magnetism theory Stabilized by DM interac8on & H field o Skyrmions in magnetism experiments (~ 2006 onwards) C. Pfleiderer (Munich), Y. Tokura (Tokyo) 4

5 Chiral Magnets o Ferromagnetic Exchange J S i S j o Chiral DM interaction (Dzyaloshinskii-Moriya) D (S i S j ) Material constraints for DM: 1. Broken Inversion Symmetry à direction of D 2. Spin-orbit coupling (SOC) à magnitude of D

6 Chiral Magnets à Spin textures o Ferromagnetic Exchange J S i S j J cos J 2 o DM interaction (Dzyaloshinskii-Moriya) D (S i S j ) ' D sin ' D E( ) =J 2 /2 E = D J D Length scale E Spontaneous spin textures like Spirals or Skyrmions J D a Energy scale D 2 /J a

7 7 Skyrmion: Topological spin texture in magnetization M(r) =M bm(r) Skyrmion Crystal (SkX) Winding Number on unit sphere in spin-space Q = 1 4 Z d 2 r bm (@ x y bm) =0, ±1, ±2,... Topological Invariant

8 Chiral Magnetic Materials with broken bulk inversion symmetry o Non-centrosymmetric Crystals P B20 structure Metals: MnSi, Fe 1- x Co x Si, FeGe, Insulators: Cu 2 OSeO 3 Experiments: Pfleiderer; Tokura Theory: Bogdanov; Nagaosa; Rosch

9 Spin textures in chiral magnets 9 Skyrmion crystal (SkX) probed in q-space and in r-space FeCoSi Lorentz TEM MnSi MnSi SANS X. Z. Yu et al., Nature 465, 901 (2010) Fe 0.5 Co 0.5 Si Bauer et al, 200Å

10 Skyrmions are more stable in thin films 10 FeGe Tc = 278K Seki, et. al., Science 336, 198 (2012) Bulk Thin Film decreasing thickness à Mul8ferroic: Seki et. Al. PRBB 86, (2012) Z. Yu et. al., Nature Mat 10, 106 (2011)

11 Unusual Properties of Skyrmion phases Review: Nagaosa & Tokura, Nature Nano 8, 899 (2013) * Skyrmions in metallic magnets o Emergent electromagnetism o Topological Hall effect o Non-Fermi liquid phase Conduction electrons moving in Skyrmion texture acquire Berry phase à Effective E & B fields For a 10 nm Skyrmion Effective B ~ 100 T(!) 11 T 3/2 Ritz et al, Nature 497, 231 (2013)

12 Potential for novel applications in memory and logic à Low pinning compared to domain walls J C ~ 10 6 A / m 2 vs. J C ~ A / m 2 Skyrmions Domain walls Jonietz, et al., Science 330, 1648 (2010) Yu, et al., Nature Comm. 3, 988 (2012) à Reading & writing single Skyrmions Romming et al., Science 341, 636 (2013) * Proposals for Memories: Skyrmion slide cell magnetic tunnel junction (MTJ) memory Skyrmion race-track memory Rahman et al, JAP 111, 07C907 (2012) Reading & Writing Koshiabe et al (Tokura- Nagaosa), arxiv: (2015)

13 Chiral Magnetic Materials with broken surface inversion symmetry o Thin films of non-centrosymmetric crystals -- both bulk & surface inversion broken in general o 3d Magnetic monolayer on 5d metals with large SOC -- nano-skyrmions from competing exchange Experiments: Weisendanger Theory: Blugel o Bulk materials with broken z à -z mirror symmetry -- examples with magnetism? o Oxide interfaces: e.g. LAO/STO -- tantalizing hints of magnetism

14 A quick introduction to oxide interfaces 14 Emergent phenomena at the interface leads to properties totally different from either bulk material AlO 2 - LaO + AlO 2 - LaO + TiO 2 SrO TiO 2 SrO Ohtomo & Hwang, Nature (2004) LaAlO 3 (LAO) E g = 5.6 ev Interface SrTiO 3 (STO) E g = 3.2 ev Superconductivity Magnetism Caviglia et al, Nature (2008) T c (n) Li et al, Nat. Phys (2008) M(H)

15 Where do the electrons come from? AlO 2 - LaO + AlO 2 - LaO + TiO 2 SrO E V Polar Catastrophe à Electronic reconstruction 15

16 Where do the electrons come from? AlO - 2 LaO + AlO - 2 LaO + E V 0.5 e - TiO 2 SrO Polar Catastrophe à 0.5 e/ti à n 2D ~ 3.3 x cm -2 + additional carriers from Oxygen vacancies 16

17 LAO/STO Interface Polar catastrophe AlO 2-2DEG Thiel et al, Science (2006) Hall à n e - /Ti LaO + AlO 2 - LaO + TiO 2 SrO TiO 2 SrO Ohtomo & Hwang, Nature (2004) LaAlO 3 Interface SrTiO 3 Polar Catastrophe = n 2D ~ 3.3 x cm -2 + additional electrons from O-vacancies Hall à mobile carriers n 2D = 2x10 13 cm -2 ~ 10% of polar catastrophe Where are the missing electrons?

18 If the polar catastrophe does occur in LAO/STO, where are the electrons not seen in transport? Answers -- not certain -- seem to be sample & probe dependent However, many experiments indicate large density of Localized electrons that behave like local moments 18 Scanning SQUID M Bert et al. Nature Phys. (2011) * Inhomogeneous; M=0 * Isolated micron-size patches of in-plane FM * Susceptometry: local moments ~ 0.5 e/ti H 2T Torque Magnetometry Li et al, Nature Phys (2011) * M µ B /Ti * Exchange scale ~ 100K * No hysteresis Spectroscopy: Ti d 1 states XPS Sing et al, PRL (2009) XMCD ~ 0.1 e/ti Lee et al, Nature Mat. (2013) MFM: Room Temp FM Bi et al, Nature Comm. (2014)

19 LAO/STO summary 19 * 2D interface * Broken inversion z à -z * Rashba spin-orbit coupling tunable by gate voltage [Expt: Caviglia et al., PRL (2010)] *Many experiments see evidence for magnetism * Disorder & inhomogeneity H soc = soc bz (k ) * sample-to-sample variations & many open questions! If there is magnetism at oxide interfaces, then we have all the ingredients for Spirals & Skyrmions.

20 Outline: 20 o Introduction * Skyrmions * Materials * Properties o How to stabilize skyrmion phases? * Role of Dresselhaus vs. Rashba SOC * Role of magnetic anisotropy o Conclusions

21 Form of the DM Interaction Symmetry à direction of DM vector o broken bulk inversion à Dresselhaus SOC D ij (S i S j ) Form of DM terms in continuum Ginzburg-Landau Theory à bd D ij = br ij D D bm (r bm) o broken surface inversion à Rashba SOC à bd R ij = bz br ij D R bm [(bz r) bm] 21

22 Continuum field theory F [ bm(r)] = J 2 r bm 2 FM exchange D D D R bm (r bm) bm [(bz r) bm] DM -- Dresselhaus DM -- Rashba + Am 2 z Anisotropy Hm z Field Minimize F subject to the constraint bm(r) 2 = 1 à Optimal bm(r)

23 Magnetic Anisotropy + Am 2 z A > 0 Easy-plane A < 0 Easy-axis Sources of Anisotropy: Single-ion + Dipolar + + Rashba SOC Broken z-inversion à Rashba SOC * Symmetry allows an easy-plane Compass-Kitaev term X A c (S y i Sy i+ˆx + Sx i Si+ŷ) x i * Easy-plane anisotropy with same microscopic origin as DM Ignored in literature A c O 2 soc D O( soc ) However, both A c and D impact energy at the same order à Cannot ignore A c! A c D 2 /J E E( )

24 Microscopic Interlude I: Rashba SOC + Double exchange H 0 = t X c i c X j i.ˆd ij c i c j +h.c. H int = <ij>, J H 2 X c i c i S i i <ij>, with ˆd ij = ẑ ˆr ij 24 To derive effective H, consider two-site problem: H 0 = t X (c i [ei#. ˆd ij ] c j +h.c.) t = p t tan # = /t SU(2) Gauge Transformation ec i =[e i(#/2). ˆd ij ] c i etc. In the rotated basis H 0 = et X ec i ec j <ij>, H int = X J H es i es i i

25 Microscopic Interlude II: Rashba SOC + Double exchange H 0 = et X <ij>, ec i ec j H int = This is the standard Anderson-Hasegawa problem -- in the transformed variables H DE = J F X <ij> X J H es i es i i J H!1 Classical S 0 is h i 1/2 1+S i R(2#ˆd ij )S j With J F et Expand square root to get three terms: FM exchange Rashba-DM Compass Anisotropy J = J F cos 2# D R = J F sin 2# A c = J F (1 cos 2#) A c J/D 2 =1/2 t 25

26 Microscopic Interlude: summary 26 Rashba SOC à significant easy-plane anisotropy For a wide variety of exchange mechanisms in both metals & insulators with Rashba SOC A c J /D 2 =1/2 ( soc t) Exchange = J Rashba DM = D R compass anisotropy = A c for pair-wise exchange mechanisms Banerjee, Rowland Erten & MR, PRX 4, (2014) * AFM Superexchange + SOC Moriya, PR (1960); Shekhtman et al, PRL (1992) * RKKY + SOC Imamura et al, PRB (2004) * Double exchange + SOC Banerjee, Erten & MR, Nature Phys. (2013) * Bosons + SOC Cole, Zhang, Paramekan8 & Trivedi, PRL (2012)

27 T=0 Phase Diagram as function of *Anisotropy AJ/D 2 * Field HJ/D 2 * Rashba/Dresselhaus Minimize continuum Energy functional F * Variational calculations -- analytical * Numerical minimization -- conjugate gradient * Classical analysis

28 Broken Bulk Inversion: Dresselhaus SOC 28 Wilson, Butenko, Bogdanov & Monchesky, PRB 89, (2014).

29 Stability of Spiral & Skyrmion crystal enhanced with Rashba HJ/D 2 D Dresselhaus limit D R =0 Increasing Rashba à D R =0.3D D D R =0.5D D HJ/D D 2 HJ/D D 2 AJ/D 2 D Circular cone: Spiral: Skyrmion: m(z) m(x) m(x, y) AJ/D 2 D à gains energy only from à D = q D 2 D + D2 R AJ/D 2 D D D 29

30 Analysis of Spiral & Skyrmion Phases where Define: D D = D sin D R = D z 0 m(x, y) 30 Rewrite the sum of two DM terms in a simple form via change of variables m! n n = Rẑ ( /2 ) m D D m (r m) D R m [(bz r) m] = D n (r n) q With D = DD 2 + D2 R and = tan 1 DD D R = helicity

31 Evolution from Dresselhaus à Rashba = /2 Dresselhaus limit Vortex-like / Bloch Rashba limit =0 = /4 Rashba = Dresselhaus Hedgehog / Neel 31

32 Phase Diagram For Rashba Limit Elliptic Cone Tilted FM Elliptic Cone 32

33 Phase Diagram For Rashba Limit Elliptic Cone Tilted FM 33 H=0, A>0 results consistent with Li, Liu, & Balents [PRL 112, (2014)], who focus on T>0

34 Evolution of spin-texture in Rashba limit with A at fixed H A=0.0, H=0.7 circular skyrmions Spin texture Topological charge density χ = 1 4 bm (@ x y bm) Z d 2 r = 1 unit cell

35 Evolution of spin-texture in Rashba limit with A at fixed H A=0.0, H=0.7 circular skyrmions A=0.6, H=0.7 A=1.1. H=0.7 Topological charge density Spin texture χ

36 Evolution of spin-texture in Rashba limit with A at fixed H A=1.1, H=0.7 A=1.4, H=0.7 Topological charge density Spin texture χ

37 Evolution of spin-texture in Rashba limit with A at fixed H A=1.1 A=1.4 Cannot use Homotopy group 2 (S 2 )=Z Use Chern Number for maps from 2-Torus à 2-Sphere Spin texture χ à Each unit cell Q = -1 Topological charge density

38 Outline: o Introduction * What is a Skyrmion? * Skyrmion materials & properties o How to stabilize skyrmion phases? * Role of Dresselhaus vs. Rashba SOC * Role of magnetic anisotropy o Conclusions 38

39 Importance of * Broken surface inversion * Rashba SOC * easy-plane anisotropy à Predict enhanced stability of Skyrmions Rashba limit Summary 39 Dresselhaus limit Rashba = Dresselhaus * Nature Physics 9, 626 (2013) * Phys. Rev. X 4, (2014) * In preparation (2015)

40 Could not discuss this for lack of 8me Related Work in Cold Atoms: Bose Hubbard Model with Rashba SOC FM + Rashba DM + Compass Anisotropy Arbitrary D/J References: Cole, Zhang,Paramekan8, & Trivedi PRL 109, (2012) See also: Radic, Di Ciolo, Sun & Galitski, PRL (2012)