1. Topological phases driven by skyrmions crystals
Joaquín Fernández Rossier 1,2
INL, Braga, Portugal
Univ. Alicante, Spain
SPIN ORBIT COUPLING AND TOPOLOGY
IN LOW DIMENSIONS
Spetses, June

2. 5 minute summary

3. Topological phases: the Quantum Hall trioW
Topological phases: the Quantum Hall trio
S. Oh, Science 340, 153 (2013)
Gapped bulk
Chiral edge states
Persistent non-dissipative currents
Quantized, material independent Hall response given by winding number

4. W
A skyrmion lattice
Alejandro Roldán-Molina, A. S. Núñez, J. Fernández-Rossier, New J. Phys. 18, (2016)

5. 1. Quantum Anomalous Hall effect in graphene with B=0 and “no SOC !!”
Two topological phases
Graphene
Magnetic material
With skyrmion lattice
1. Quantum Anomalous Hall effect in graphene with B=0 and “no SOC !!”
J. L. Lado, and J. Fernández-Rossier, Phys. Rev. B 92, (2015)

6. 1. Quantum Anomalous Hall effect in graphene with B=0 and “no SOC !!”
1) AHE for graphene on skyrmions
Graphene
Magnetic material
With skyrmion lattice
1. Quantum Anomalous Hall effect in graphene with B=0 and “no SOC !!”
J. L. Lado, and J. Fernández-Rossier, Phys. Rev. B 92, (2015)

7. 2) Skyrmion crystal magnons are topologicalW
2) Skyrmion crystal magnons are topological

8. People Alejandro Roldán-Molina (Chile) Álvaro S. Núñez (U. Chile)
José Luis Lado (INL, Portugal)
Francesca Finnochiaro (IMDEA-NANO, Madrid)

9. What are skyrmions?

10. W Skyrmions Non-coplanar spin texture
Skyrmion lattice = ground state in some materials (MnSi)
Competition of exchange and DM interaction
Topological spin texture

11. W
Magnetic skyrmions
N=+1
N=-1

12. W Skyrmions: different sizes STM Images, spontaneous skyrmion lattice
Fe(ML)/Ir(111)
Ta(5nm)/Co20Fe60B20(CoFeB)(1.1nm)/TaOx(3nm
STM Images, spontaneous skyrmion lattice
(B=0)
W. Jiang et al,
Science DOI: /science.aaa1442
S. Heinze et al., Nat. Phys. 7, 713 (2011)

13. W Graphene + Skyrmions: experiments !!
Jens Brede et al, Nature Nano (2014)

14. Electrons surfing non-collinear magnetic landscapes

15. Electrons surfing a non collinear magnetic landscapeW
Electrons surfing a non collinear magnetic landscape
Bruno, P., V. K. Dugaev, and M. Taillefumier. "Topological Hall effect and Berry phase in magnetic nanostructures." Physical Review Letters (2004)

16. Electrons surfing a non collinear magnetic landscapeW
Electrons surfing a non collinear magnetic landscape

17. Strong coupling : spinless fermions in a “magnetic” fieldW
Strong coupling : spinless fermions in a “magnetic” field X X X X

18. W
Total flux= skyrmion number
Pseudo field flux = Skyrmion number

19. W Total flux= skyrmion number Pseudo field flux = Skyrmion number
CONSEQUENCES
Non-quantized anomalous Hall effect (aka topological Hall effect) expected for electrons interacting with skyrmions (in the strong coupling ) Yi, Onoda, Nagaosa, Han arXiv: (2009)
Quantized Anomalous Hall effect predicted for Kagome Lattice with non-coplanar magnetization K. Ohgushi, S. Murakami, and N. Nagaosa, Phys. Rev. B 62, R6065 (2000).

20. Graphene electrons surfing a skyrmion lattice

21. W The Quantum Hall trio in graphene No reports so far
Haldane (88): weird B
Qiao (‘10):
M + Rashba SOC
Kane-Mele (‘05): SOC
Abanin-Lee-Levitov (‘06):
B + FM order
(Routinely) observed experimentally
B>1T
No reports so far

22. W The Quantum Hall trio in graphene Haldane (88): weird B Qiao (‘10):
M + Rashba SOC
Kane-Mele (‘05): SOC
Abanin-Lee-Levitov (‘06):
B + FM order
(Routinely) observed experimentally
B>1T
Quantum Hall trio in graphene: B and/or Spin-orbit needed

23. Quantum Anomalous Hall effect in graphene with B=0 and no SOC !!
Graphene coupled to Skyrmions
Graphene
Magnetic material
With skyrmion lattice
J. L. Lado, and J. Fernández-Rossier , Phys. Rev. B 92, (2015)
Quantum Anomalous Hall effect in graphene with B=0 and no SOC !!

24. W Hall conductivity Hall conductivity Berry curvature Berry connection
Quantized Hall Conductance in a Two-Dimensional Periodic Potential TKNN PRL 49, 405 (1982)

25. Quantum Hall conductivity as a topological invariantW
Quantum Hall conductivity as a topological invariant
Chern number =
Integer number
Berry curvature
Berry connection
Quantized Hall Conductance in a Two-Dimensional Periodic Potential TKNN PRL 49, 405 (1982)

26. W Haldane model Bulk Spinless fermions in honeycomb lattice
Local magnetic flux, zero total B
Broken time reversal symmetry
Gapped bulk
Chern number = 1
Quantized Hall conductance
1 Edge states (index theorem)
Quantum Anomalous Hall effect
Strip
(2 edges)
1 edge
F. D. M. Haldane
Phys. Rev. Lett. 61, 2015 (1988)

27. Quantized Anomalous Hall effectW
Quantized Anomalous Hall effect
Wanted:
2 Dimensional system
Broken time reversal symmetry
Insulating

28. Turning graphene magnetic: “theory”W
Turning graphene magnetic: “theory”
Graphene
FM material

29. Opening a gap in magnetic grapheneW
Opening a gap in magnetic graphene
Spin mixing term needed:
Spin orbit coupling
Non-collinear magnetism

30. Qiao et al model for QAH in grapheneW
Qiao et al model for QAH in graphene
Bulk
Graphene + Zeeman off-plane + Rashba
Gapped bulk
Chern number = 2
Quantized Hall conductance
2 Edge states (index theorem)
Quantum Anomalous Hall effect
Strip
(2 edges)
1 edge
Z. Qiao, S. A. Yang, W. Feng,
W. Tse, J. Ding, Y. Yao, J. Wang, Q. Niu Phys. Rev. B82, R (2010)

31. QAHE in graphene + SkyrmionsW
QAHE in graphene + Skyrmions
J. L. Lado, and J. Fernández-Rossier, Phys. Rev. B 92, (2015)

32. 1st neighbor tight-bindingW
Models and methods
Classical spins
1st neighbor tight-binding

33. 1st neighbor tight-bindingW
Models and methods
Classical spins
1st neighbor tight-binding
Dirac cones

34. W Open source code: Quantum Honeycomp (Contact José Luis Lado)
Models and methods
Classical spins
1st neighbor tight-binding
J. L. Lado, and JFR, PRB 92, (15)
Weak coupling:
Two Geometries:
2D crystal: standard band calculation enlarged unit cell
Edge states (semi-infinite 2D crystal): recursive Green function
Open source code: Quantum Honeycomp
(Contact José Luis Lado)

35. Graphene + triangular lattice SkyrmionsW
Graphene + triangular lattice Skyrmions
Gap opens

36. Graphene + triangular lattice SkyrmionsW
Graphene + triangular lattice Skyrmions
Gap opens
Finite Berry curvature
Chern number= 2 N
Topological imprinting
Quantized Anomalous Hall phase

37. W Edge states, 1 per spin channel 2 co-propagating edge states

38. Topological wires: 2 and 4 lanes
J. L. Lado, and J. Fernández-Rossier, Phys. Rev. B92, (2015)

39. Non-quantized Anomalous Hall effectW
Non-quantized Anomalous Hall effect
Triangular lattice

40. Dirac electrons interacting with 1 skyrmion
Trying to understand:
Dirac electrons interacting with 1 skyrmion

41. Dirac electrons surfing 1 skyrmion
valley
Dirac Hamiltonian + exchange
spin
sublattice

42. Dirac electrons surfing 1 skyrmion
Dirac Hamiltonian + exchange
Skyrmion field:

43. Hamiltonian in the skyrmion frame
Rotated Dirac Hamiltonian + exchange

44. Hamiltonian in the skyrmion frame
Rotated Dirac Hamiltonian + exchange
Spin dependent magnetic field

45. Hamiltonian in the skyrmion frame
Rotated Dirac Hamiltonian + exchange
Effective Rashba-type spin-orbit coupling

46. A hand waving argument

47. Does it work without skyrmions?

48. W Do we need skyrmions? “in-plane skyrmion” opens a gap
Zero Berry curvature
Trivial insulator

49. W
How big is the gap?

50. W How big is J? Skyrmion J Gap
[1]
[2]
[3]
J (meV) 37 70 50
t (eV)
2.8
2.6
2.7
Material
EuO
BiFeO3
YIG
Skyrmion Gap (mev)
0.1
0.4
0.2 J
Skyrmion
Gap
[1] H. X. Yang et al, PRL. 110, (‘13)
[2] Z. Quai et al, PRL. 112, (‘14)
[3] Z. Wang, PRL 114, (‘15)

51. Graphene on skyrmion lattice material (Fe ML/Ir(111)) [1]
Wish list
Graphene Hall bar on top of insulating magnetic material with skymion lattice
Graphene on skyrmion lattice material (Fe ML/Ir(111)) [1]
AHE in graphene/YIG: ferromagnetic proximity [2]
Skyrmions in insulating material Cu2OSeO3 [3]
[1] Jens Brede et al, Nature Nano (2014)
[2 ]Z. Wang, Phys. Rev. Lett. 114, (2015)
[3] S. Sekil et al., Science 336, 198 (2012)

52. Turning graphene magnetic: experimentsW
Turning graphene magnetic: experiments
AHE
Z. Wang, Phys. Rev. Lett. 114, (2015)

53. A new way to have a Chern insulator:
Conclusions part 1
A new way to have a Chern insulator:
Graphene interacting with a skyrmion lattice
No B and no graphene-SOC needed
Weak exchange coupling is ok
Topological imprinting:

54. Topological spin waves in the atomic-scale magnetic skyrmion crystal
PART II
Topological spin waves in the atomic-scale magnetic skyrmion crystal
A. Roldán-Molina, A. S. Núñez, J. Fernández-Rossier, New J. Phys. 18, (2016)

55. Skyrmion lattice: the modelW
Skyrmion lattice: the model

56. Skyrmion lattice: the modelW
Skyrmion lattice: the model
(1)
(2)
(3)
(4)
Heisenberg exchange (FM) (J=1 meV)
DM coupling (D=1 meV) (breaks inversion symmetry, requires SOC)
Zeeman
Uniaxial anisotropy K=0.5 meV

57. Skyrmion lattice: the modelW
Skyrmion lattice: the model
(1)
(2)
(3)
(4)
J Promotes (FM) (J=1 meV)
K defines easy axis
D promotes helical phase
Off plane Zeeman promotes non-coplanarity

58. Skyrmion lattice: classical approximationW
Skyrmion lattice: classical approximation
A. Roldán-Molina, A. S. Núñez, J. Fernández-Rossier, New J. Phys. 18, (2016)

59. Spin waves: reminder of Holstein-Primakoff
Representation of spin operators in terms of HP Bosons
Spin quantization axis is
Density of HP bosons =
= deviation from the maximal spin projection along the quantization axis
= metric of quantum fluctuations
T. Holstein and H. Primakoff, Phys. Rev. 58, 1098 (1940).

60. Spin waves: reminder of Holstein-Primakoff
Representation of spin operators in terms of HP Bosons
Linearization: small density of HP bosons
T. Holstein and H. Primakoff, Phys. Rev. 58, 1098 (1940).

61. Spin waves: reminder of Holstein-Primakoff
Representation of spin operators in terms of HP Bosons
Linearization: small density of HP bosons
Quadratic Hamiltonian for bosons (Bogoliubov de Gennes type)

62. Spin waves for magnetic crystals
Bloch quadratic Hamiltonian for bosons (Bogoliubov de Gennes type)
After Bogoliubov transformation, the true excitations are found

63. For classical ground states with crystal symmetry
After Bogoliubov transformation, the true excitations are foundc
And their wave functions are also found:
That permit to compute the density of HP bosons
A. Roldán-Molina , MJ Santander, AS Núñez and J. Fernández-Rossier Phys. Rev. B (2015)

64. A simple case: a FM 1D chainW
A simple case: a FM 1D chain
No anomalous terms:
HP bosons = magnon excitation

66. Spin waves of a single skyrmion
A. Roldán-Molina , MJ Santander, AS Núñez and J. Fernández-Rossier Phys. Rev. B (2015)

67. Lowest energy bands magnons skyrmion lattice
A. Roldán-Molina, A. S. Núñez, J. Fernández-Rossier, New J. Phys. 18, (2016)

68. W
Density of HP bosons

69. Berry curvature of magnons in skyrmion crystalW
Berry curvature of magnons in skyrmion crystal
Berry curvature for a given band:

70. Chern number for magnons in skyrmion crystalW
Chern number for magnons in skyrmion crystal
Berry curvature for a given band:
Chern number of a given band:
Winding numbers

71. What are the consequences of finite winding numbers?

72. W
Edge states 2 3 2 1 1

73. W
Edge states

74. W
Edge states

75. Other “topological magnon systems”W
Other “topological magnon systems”
Zhang L, Ren J, Wang J S and Li B Phys. Rev. B (2013 )
Mook A, Henk J and Mertig I Phys. Rev. B (2014 )
Mook A, Henk J and Mertig I Phys. Rev. B (2015 )

76. Other “topological magnon systems”W
Other “topological magnon systems”

77. Other “topological magnon systems”W
Other “topological magnon systems”
Self-assembled nanostructure
Artificial
nanotructure
Atomic crystal

78. W Quarks, electrons Atoms, electrons Crystals, Magnets
Emergence: symmetry breaking and topology
Quarks, electrons
Atoms, electrons
Crystals, Magnets
Skyrmion crystal
Topological spin waves

79. Thanks for your attention
Conclusions part 2
Magnons in skyrmion crystal have finite Chern number
Chiral edge states for magnons
First example in a self-assembled mesoscale structure
Thanks for your attention