1. Ferromagnetism and Quantum Anomalous Hall Effect in Saturated Germanene
Shin-Ming Huang (黃信銘)
Department of Physics, National Tsing Hua University, Hsinchu, Taiwan
In collaboration with Prof. Chung-Yu Mou (牟中瑜) and Shi-Ting Lee (李詩婷)
Reference: arXiv: (to be published in Phys. Rev. B)
May 22th ,

2. Outline
Introduction to quantum Hall effects and topology: quantum Hall, quantum spin Hall, quantum anomalous Hall
Theoretical and experimental examples for QAH insulators
Silicene/Germanene, a QSH insulator
Our study: QAHE in germanene via saturation
Conclusion

3. Quantum Hall Trio
Oh, Science 340, 153 (2013)

4. Quantum Hall Effect Hall effect (Hall, 1879)
IQHE (Klitzing, Dorda, and Pepper, 1980)
classical theory
Common: GaAs heterojunction
𝑅 xy = 1 𝜈 h 𝑒 2 , 𝜈: filling factor
𝑅 xy = V H I = B z e𝜌
Lorentz force
Landau levels

5. Connection Hall Conductivity to Chern Number
PRL 49, 406 (1982)
𝜎 𝑥𝑦 =𝑖 𝑒 2 ℏ 𝐸 𝑎 < 𝐸 𝐹 < 𝐸 𝑏 𝑎 𝑣 𝑥 𝑏 𝑏 𝑣 𝑦 𝑎 − 𝑎 𝑣 𝑦 𝑏 𝑏 𝑣 𝑥 𝑎 𝐸 𝑎 − 𝐸 𝑏 2
Kubo formula:
First Chern number, TKNN invariant (an integer)
𝜎 𝑥𝑦 =− 𝑒 2 ℎ 𝛼∈𝑜𝑐𝑐 𝐶 𝛼
“flux” number within the BZ
𝐶 𝛼 = 𝐵𝑍 𝑑 𝑘 2 2𝜋 𝜕 𝐴 𝑦 𝛼 𝐤 𝜕 𝑘 𝑥 − 𝜕 𝐴 𝑥 𝛼 𝐤 𝜕 𝑘 𝑦 ≡ 𝐵𝑍 𝑑 𝑘 2 2𝜋 𝐹 𝛼 𝐤
𝐴 𝑖 𝛼 𝐤 =−𝑖 𝑢 𝛼 𝐤 𝜕 𝜕 𝑘 𝑖 𝑢 𝛼 𝐤 , the Berry's connection 𝐹 𝛼 𝐤 = ϵ 𝑖𝑗 𝐴 𝑗 𝛼 𝐤 , the Berry's curvature

6. Topology in Quantum Hall Effect
cont. mapping
Vector space of Hamiltonian, 𝕄
𝕄 is isomorphic to a mathematical space
D-dim, BZ is a TD torus
( 𝑇 𝐷 =𝑆 1 × 𝑆 1 ×…× 𝑆 1 )
𝐻 𝑘 : 𝑘 →𝕄
If 𝐻 𝑘 →∞ →𝑐𝑜𝑛𝑠𝑡, BZ can be regarded as a SD.
(SD: D-dim sphere)
nth homotopy group, 𝜋 𝑛 𝕄 : maps from Sn into a given space 𝕄
A group has different homotopy classes. Each class is labeled by an integer.
Two mappings are homotopic if one can be continuously deformed into the other ⟹ same class
Without tearing/cutting, deformation will change the class.

7. Example 𝜋 2 𝑆 2 =ℤ 𝑘 = 𝑘 𝑥 , 𝑘 𝑦 𝐻 𝑘 = 𝑑 𝑘 ∙ 𝜎
𝑘 = 𝑘 𝑥 , 𝑘 𝑦
𝐻 𝑘 = 𝑑 𝑘 ∙ 𝜎
𝑑 𝑘 = sin 𝜃 𝑘 cos 𝜙 𝑘 , sin 𝜃 𝑘 sin 𝜙 𝑘 , cos 𝜃 𝑘
𝐸 𝑘 =± 𝑑 𝑘 (assume 𝑑 𝑘 >0)
Cont. mapping
𝑆 2
𝜋 2 𝑆 2 =ℤ
Nontrivial insulators: 𝑑 𝑘 sweeps over 𝑆 2 ; 𝜃 𝑘 :0→𝜋 and 𝜙 𝑘 :0→2𝜋

8. Bulk-Edge Correspondence
𝑛=−1
𝑛=−2
𝑛=2
𝑛=1
2 branches for 2 sides
Chiral edge states are topologically protected.
No back-scattering

9. Topological Phase Transition
It is meaningful when the band is isolated and completely filled.
When two bands touch, the good definition is the whole.
Sum of topological numbers is conserved:
𝑛 1 + 𝑛 2 = 𝑛 1 ′ + 𝑛 2 ′ [Avron, Seiler, and Simon PRL (1983)]
n1=+1
n2=-1
n’2=0
n’1=0

10. Quantum Spin Hall Effect
𝐶=0 when time-reversal symmetric. However, nontrivial topological invariants are possible in TR symmetric systems (topological insulators).
QSH: “spin-up” and “spin-down” states (TR partners) possess opposite Chern numbers.
𝐶 ↑ + 𝐶 ↓ =0 and 𝐶 ↑ − 𝐶 ↓ ≠0
QSH insulator is like two TR QH insulators.
Edge states are helical: spin and momentum are correlated.

11. Quantum Anomous Hall: Haldane Model
Breaks time reversal symmetry as QH.
QAH: QH w/o Laudau levels. Need SOC.
First theoretical model: F.D.M. Haldane PRL 61, 2015 (1988)
Osci. magnetic flux; zero net flux
The Hamiltonian near Dirac points, α=±1
𝐻 𝛼 =−𝑐 𝑝 𝑚 𝛼 𝑝 𝑥 −𝑖𝛼 𝑝 𝑦 𝑝 𝑥 +𝑖𝛼 𝑝 𝑦 − 𝑚 𝛼
m α =M−α3 3 t 2 sinϕ,
t 2 : NNN hopping
M: on-site energy diff.
𝜈≠0⟺ m +1 × m −1 <0 (𝑏𝑎𝑛𝑑 𝑖𝑛𝑣𝑒𝑟𝑠𝑖𝑜𝑛)

12. Graphene: Rashba SOC & Exchange Field
Enlarge SOC via adatoms
Qiao, Niu et al., PRB (2010)

13. Magnetically Doped TI
Science 329, 61 (2010)
3QL
4QL
5QL

14. Bi2Se3 Surface states couple + exchange field +Cr/Fe
Zhang et al., Nature Phys. 6, 584 (2010)
Gap~40meV in a 5 QL film

15. Experimental Realization of QAH
Cr0.15(Bi0.1Sb0.9)1.85Te3 film on SrTiO3(111), 5QL
TCurie~30K
Temperature for QAHE is limited by TCurie and the gap of surface states.
C.-Z. Chang, Qi-Kun Xue et al., Science 340, 167 (2013)

16. To achieve High-temperature QAHE
Large SOC
High Tcurie

17. Enhancement of SOC in Graphene
SOC in graphene was too weak. ( 𝜆 𝑆𝑂 ~1𝜇𝑒𝑉) Min et al., PRB 2006; Yao et al., PRB 2007
Adatom (thallium, indium)
hydrogenation
C.Weeks et al. RPX 1, (2011)
J.Balakrishnan et al. Nature Phys. 9, 284(2013)

18. Silicene and Germanene
First-principles calc. predict stable Si and Ge planar lattices.
Takeda and Shiraishi (1994); Cahangirov et al. (2009); …
Group IVA elements
Different to C, Si and Ge like to form sp3 bonds → buckling
Liu, Feng & Yao, PRL 107, (2011)

19. Silicene and Germanene
Different to C, Si and Ge like to form sp3 bonds
buckling
Planar silicene
buckled silicene
Planar graphene
ΔSO~1.5meV
buckled germanene
Planar germanene
sp2 →sp3
σ and π-bands mix
ΔSO~24meV
Liu, Feng & Yao, PRL 107, (2011)

20. Low-Energy Effective Hamiltonian
Fǁ contributes intrinsic SOC
F⊥ contributes Rashba SOC
QSHE can be observed
Liu, Jiang and Yao, PRB 84, (2011)

21. Silicene Sheet on Substrate
On Al(111) substrate
B.Lalmi et al., APL 97, (2010)
P. Vogt et al., RPL 108, (2012)

22. Proposal of QAH in Germanene
Germanene has large SOC. (essential)
Q1: Can it shows QAHE without magnetic doping or external Zeeman field?
A1: Magnetic transition due to interaction.
Q2: p orbitals used to weakly interact?
A2: Reduce bandwidth!
HOW?

23. Theoretical Model parameters for germanene: Site energy:
Liu, Jiang and Yao, PRB 84, (2011)
Site energy:
Hubbard on-site interaction :
Because of the buckled structure, we can saturate only sublattice-A
(narrow band) FM mean field:
𝑚 𝑖 = 𝑚 𝐴 , 𝑚 𝐵
eV will be the unit for energies

24. Saturate One Sublattice
Let sub-A has site potential VA (relative to sub-B)
𝜏 : sub-lattice space
𝜎 : spin space
Eigen-energies: Lo Up VA
SO (+ΔFM)
Most contributed from sub-A
Most contributed from sub-B
Each branch has two bands (up and down spins)
Each bandwidth is of the order SOC (~10meV)
If considering FM, The ground state is insulating when 2 Δ 𝐹𝑀 >8 𝑡 2 / 𝑉 𝐴 .

25. Ferromagnetism from Interaction
Turn on the Hubbard on-site interaction:
Nagaoka’s and flat-band ferromagnetism
One hole in the half-filling Hubbard model with infinite U
Flat band with tiny interaction
H. Tasaki, Prog. Theor. Phys. 99, 489 (1998).
Stability of Nagaoka’s ferromagnetism
No exact proof for finite U and finite doping.
Empirically, it works for nearly flat bands and small doping, especially for a triangular (frustrated) lattice.
Our saturated system is expected to have FM.
FM Mean field for the interaction:
𝑚 𝑖 = 𝑚 𝐴 , 𝑚 𝐵

26. Mean-Field Results for Finite VALo Up VA
SO (+ΔFM)
Most contributed from sub-A
Most contributed from sub-B
mA is small due to nA is small.
mB is quickly saturated.
U and VA play same rules (BW/U), nearly.
Note that even 𝑉 𝐴 →∞, BW is still finite limited by SOC.
Half-filled

27. Due to linear dispersion at low energy, 𝑈 𝑐 =𝑐 𝜆 𝑅2 .
Infinite VA
Later on, we will discuss the 𝑉 𝐴 →∞ case for simplicity.
𝑉 𝐴 →∞, 𝐵−𝑠𝑢𝑏𝑙𝑎𝑡𝑡𝑖𝑐𝑒 𝑜𝑛𝑙𝑦
FM self-consistent equation:
Critical U:
Due to linear dispersion at low energy, 𝑈 𝑐 =𝑐 𝜆 𝑅2 .

28. One-Side Semihydrogenation
Large gap opens
Flat band
Ferromagnetic insulator
Wang et al., Phys. Chem. Chem. Phys. 14, 3031 (2012)

29. Chern Number & Edge Statesky kx

30. Topological Phase Transition
Condition for the nontrivial phase:
U≈0.1eV

31. Fractional Saturation
The magnetization is too strong for full saturation. Bad for QAHE.
How about partial saturation?
We will call the positions with infinite 𝑉 𝐴 as vacancies.
Arrange vacancies periodically.
1A+2B/u.c.
2A+3B/u.c.
3A+4B/u.c.

32. intrinsic SOC & FM field
Narrow Bands
A unit cell with 𝑁 𝑐 (=NA+NB) lattice points (excluding vacancies)
2 𝑁 𝑐 bands (two spins)
intrinsic SOC & FM field
hopping
Rashba SOC
Chiral symmetry for 𝐾 k :
𝑐 𝐴 → 𝑐 𝐴 and 𝑐 𝐵 →− 𝑐 𝐵 , ⟹ 𝐾 k →− 𝐾 k
𝑛 and −𝑛 have opposite energies of 𝐾 k ( 𝐾 k ± 𝑛 =± 𝜖 k ± 𝑛 )
If 𝑁 𝑐 odd (odd vacancies), two zero-energy bands, 0,𝜎 .
0,𝜎 are composed of B-el only (NA<NB).
We will call 0,𝜎 the middle bands (two spins).
With SOC, has bandwidth~SOC.
If 𝜆 𝑆𝑂 , 𝜆 𝑅2 ≪𝑡 , are still B-el only.
Eff. Hamiltonian for the middle bands:

33. Bands for Fractional Saturation
1 𝑞 −𝑣𝑎𝑐𝑎𝑛𝑐𝑦:
A unit cell: 1 vacancies, 𝑞−1 A-sites and 𝑞 B-sites,
𝑁 𝑐 = 𝑁 𝐴 + 𝑁 𝐵 =2𝑞−1. (odd ⟹middle bands)
Neglect spins here. (# of bands doubles w/ spins.)
1 B-only bands + 𝑁 𝑐 −1 A+B bands
Each of these (A+B) bands can fill with 𝑁 𝐴 = 𝑞−1 𝑁 𝑐 −1 and 𝑁 𝐵 = 𝑞−1 𝑁 𝑐 −1 , that is, 𝑛 𝐴 = 1 𝑁 𝑐 −1 and 𝑛 𝐵 = 1 𝑞 𝑞−1 𝑁 𝑐 −1 .
The middle band can fill with 𝑁 𝐵 =1, that is, 𝑛 𝐵 = 1 𝑞 .
B, ↑↓ (nB=2/3)
A+B, ↑↓ (nA=1/2,nB=1/3)
𝑛 𝐴 ( 𝑛 𝐵 ): particle number per A (B) site.
Example: 1/3-vacancy.
𝑞=3, 𝑁 𝑐 =5,
1 𝑁 𝑐 −1 = 1 4 , 1 𝑞 𝑞−1 𝑁 𝑐 −1 = 1 6

34. Density of States (Nonmagnetic)
One fully-filled middle band can give nB=1/q (particle no. per B site)
½-vacancy
1/3-vacancy
↑+↓
¼ -vacancy

35. Mean-Field Results for Fractional Saturation
A fully-filled middle band contributes 𝑛 𝐵 = 1 𝑞
𝑚 𝐵 → 1 2𝑞 for U≫ 𝜆 𝑆𝑂 , 𝜆 𝑅2 at half-filling

36. Topology in ¼-vacancy Zigzag edges: Two transitions are at
Band touching → Topological phase transition

37. Topology in ¼-vacancy
Compare to U=0.1eV for the full-vacancy system, U increases to 0.34eV.

38. Critical U for Ferromagnetism with Vacancy Percentage
𝑈 to make 𝑚 𝐵 𝑈 reach 10% and 90% of 1 2𝑝 , and 𝑈 to make topological transition: Δ 𝐹𝑀 = 𝜆 𝑆𝑂

39. Conclusions
We investigate ferromagnetism induced by saturating one-side of germanene.
Due to buckled geometry, one-sublattice saturation is possible.
Saturation produces narrow bands. A flat-band ferromagnetism is induced.
The induced magnetization is directly related to the saturation fraction and is thus controllable in magnitude through the saturation fraction.
The narrow bands are proved to exist and are close to the chemical potential at half-filling.
The narrow bands may contain nontrivial topology. We find two cases with nonzero Chern numbers.
arXiv:

40. Discussion: Effect of Finite VA
T is the hopping between A and B sites.
Finite 𝑉 𝐴 will generate effective hoppings between B sites, 𝑡 𝑒𝑓𝑓 = 𝑡 2 𝑉 𝐴
The bandwidth from the effective hopping term is 8𝑡 2 𝑉 𝐴 .
To become an insulator instead of a half metal, 2 Δ 𝐹𝑀 > 8𝑡 2 𝑉 𝐴 .
SOC only
w/ eff. hopping
2 Δ 𝐹𝑀
w/ FM