quantum anomalous hall effect (qahe) and the quantum spin hall effect (qshe) shoucheng zhang,...
Post on 18-Dec-2015
229 views
TRANSCRIPT
![Page 1: Quantum anomalous Hall effect (QAHE) and the quantum spin Hall effect (QSHE) Shoucheng Zhang, Stanford University Les Houches, June 2006](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649d225503460f949f88e9/html5/thumbnails/1.jpg)
Quantum anomalous Hall effect (QAHE) and the quantum spin Hall effect (QSHE)
Shoucheng Zhang, Stanford University
Les Houches, June 2006
![Page 2: Quantum anomalous Hall effect (QAHE) and the quantum spin Hall effect (QSHE) Shoucheng Zhang, Stanford University Les Houches, June 2006](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649d225503460f949f88e9/html5/thumbnails/2.jpg)
References:
• Murakami, Nagaosa and Zhang, Science 301, 1348 (2003) • Murakami, Nagaosa, Zhang, PRL 93, 156804 (2004)• Bernevig and Zhang, PRL 95, 016801 (2005)• Bernevig and Zhang, PRL 96, 106802 (2006); • Qi, Wu, Zhang, condmat/0505308; • Wu, Bernevig and Zhang, PRL 96, 106401 (2006);
• (Haldane, PRL 61, 2015 (1988));• Kane and Mele, PRL95 226801 (2005); • Sheng et al, PRL 95, 136602 (2005); • Xu and Moore cond-mat/0508291……
![Page 3: Quantum anomalous Hall effect (QAHE) and the quantum spin Hall effect (QSHE) Shoucheng Zhang, Stanford University Les Houches, June 2006](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649d225503460f949f88e9/html5/thumbnails/3.jpg)
What about quantum spin Hall?
![Page 4: Quantum anomalous Hall effect (QAHE) and the quantum spin Hall effect (QSHE) Shoucheng Zhang, Stanford University Les Houches, June 2006](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649d225503460f949f88e9/html5/thumbnails/4.jpg)
Key ingredients of the quantum Hall effect:
• Time reversal symmetry breaking.• Bulk gap.• Gapless chiral edge states.
• External magnetic field is not necessary!
Quantized anomalous Hall effect:
• Time reversal symmetry breaking due to ferromagnetic moment.
• Topologically non-trivial bulk band gap.• Gapless chiral edge states ensured by the index theorem.
![Page 5: Quantum anomalous Hall effect (QAHE) and the quantum spin Hall effect (QSHE) Shoucheng Zhang, Stanford University Les Houches, June 2006](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649d225503460f949f88e9/html5/thumbnails/5.jpg)
Topological Quantization of the AHE (cond-mat/0505308)Magnetic semiconductor with SO coupling (no Landau levels):
General 2×2 Hamiltonian
Example
Rashbar Spin-orbital Coupling
![Page 6: Quantum anomalous Hall effect (QAHE) and the quantum spin Hall effect (QSHE) Shoucheng Zhang, Stanford University Les Houches, June 2006](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649d225503460f949f88e9/html5/thumbnails/6.jpg)
Topological Quantization of the AHE (cond-mat/0505308)Hall Conductivity
Insulator Condition
Quantization Rule
The Example
![Page 7: Quantum anomalous Hall effect (QAHE) and the quantum spin Hall effect (QSHE) Shoucheng Zhang, Stanford University Les Houches, June 2006](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649d225503460f949f88e9/html5/thumbnails/7.jpg)
Origin of Quantization: Skyrmion in momentum space
Skyrmion number=1
Skyrmion in lattice momentum space (torus)
Edge state due to monopole singularity
![Page 8: Quantum anomalous Hall effect (QAHE) and the quantum spin Hall effect (QSHE) Shoucheng Zhang, Stanford University Les Houches, June 2006](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649d225503460f949f88e9/html5/thumbnails/8.jpg)
Band structure on stripe geometry and topological edge state
![Page 9: Quantum anomalous Hall effect (QAHE) and the quantum spin Hall effect (QSHE) Shoucheng Zhang, Stanford University Les Houches, June 2006](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649d225503460f949f88e9/html5/thumbnails/9.jpg)
The intrinsic spin Hall effect
• Key advantage:• electric field manipulation, rather than
magnetic field.• dissipationless response, since both
spin current and the electric field are even under time reversal.
• Topological origin, due to Berry’s phase in momentum space similar to the QHE.
• Contrast between the spin current and the Ohm’s law:
lkh
ewhereEJorRVI Fjj
22
/
)(6
,2
LF
HFspinkijkspin
ij kk
eEJ
Bulk GaAs
Ene
rgy
(eV
)
![Page 10: Quantum anomalous Hall effect (QAHE) and the quantum spin Hall effect (QSHE) Shoucheng Zhang, Stanford University Les Houches, June 2006](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649d225503460f949f88e9/html5/thumbnails/10.jpg)
Spin-Hall insulator: dissipationless spin transport without charge transport (PRL 93, 156804, 2004)
• In zero-gap semiconductors, such as HgTe, PbTe and -Sn, the HH band is fully occupied while the LH band is completely empty.
• A bulk charge gap can be induced by quantum confinement in 2D or pressure. In this case, the spin Hall conductivity is maximal.
a
es 1.0
![Page 11: Quantum anomalous Hall effect (QAHE) and the quantum spin Hall effect (QSHE) Shoucheng Zhang, Stanford University Les Houches, June 2006](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649d225503460f949f88e9/html5/thumbnails/11.jpg)
Spin-Orbit Coupling – Spin 3/2 Systems
• Symplectic symmetry structure
Luttinger Hamiltonian
( : spin-3/2 matrix)
![Page 12: Quantum anomalous Hall effect (QAHE) and the quantum spin Hall effect (QSHE) Shoucheng Zhang, Stanford University Les Houches, June 2006](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649d225503460f949f88e9/html5/thumbnails/12.jpg)
• Natural structure5(4) (5)SU SO S
SO(5) VectorMatrices
SO(5) TensorMatrices
• Inversion symmetric terms: d- wave
• Inversion asymmetric terms: p-wave
Spin-Orbit Coupling – Spin 3/2 Systems
Strain:
Applied Rashba Field:
![Page 13: Quantum anomalous Hall effect (QAHE) and the quantum spin Hall effect (QSHE) Shoucheng Zhang, Stanford University Les Houches, June 2006](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649d225503460f949f88e9/html5/thumbnails/13.jpg)
Luttinger Model for spin Hall insulator
Bulk Materialzero gap
Symmetric Quantum Well, z-z mirror
symmetryDecoupled between (-1/2,
3/2) and (1/2, -3/2)
![Page 14: Quantum anomalous Hall effect (QAHE) and the quantum spin Hall effect (QSHE) Shoucheng Zhang, Stanford University Les Houches, June 2006](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649d225503460f949f88e9/html5/thumbnails/14.jpg)
Dirac Edge States
Edge 1
Edge 2
0L
xy
kx0
![Page 15: Quantum anomalous Hall effect (QAHE) and the quantum spin Hall effect (QSHE) Shoucheng Zhang, Stanford University Les Houches, June 2006](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649d225503460f949f88e9/html5/thumbnails/15.jpg)
From Dirac to RashbaDirac at Beta=0Rashba at Beta=1
0.0
0.2 1.0
0.02
![Page 16: Quantum anomalous Hall effect (QAHE) and the quantum spin Hall effect (QSHE) Shoucheng Zhang, Stanford University Les Houches, June 2006](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649d225503460f949f88e9/html5/thumbnails/16.jpg)
From Luttinger to Rashba
![Page 17: Quantum anomalous Hall effect (QAHE) and the quantum spin Hall effect (QSHE) Shoucheng Zhang, Stanford University Les Houches, June 2006](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649d225503460f949f88e9/html5/thumbnails/17.jpg)
Phase diagramRashba Coupling
10^5 m/s
0
1.1
2.2
-1.1
-2.2
![Page 18: Quantum anomalous Hall effect (QAHE) and the quantum spin Hall effect (QSHE) Shoucheng Zhang, Stanford University Les Houches, June 2006](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649d225503460f949f88e9/html5/thumbnails/18.jpg)
• Relate more general many-body Chern number to edge states: “Goldstone theorem” for topological order.
• Generalized Twist boundary condition: Connection between periodical system and open boundary system
Topology in QHE: U(1) Chern Number and Edge States
Niu, Thouless and Wu, PRB
Qi, Wu and Zhang, in progress
![Page 19: Quantum anomalous Hall effect (QAHE) and the quantum spin Hall effect (QSHE) Shoucheng Zhang, Stanford University Les Houches, June 2006](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649d225503460f949f88e9/html5/thumbnails/19.jpg)
Non-vanishing Chern number Monopole in enlarged parameter space
Topology in QHE: Chern Number and Edge States
Gapless Edge States in the twisted Hamiltonian
3d parameter space
MonopoleGapless point
boundary
![Page 20: Quantum anomalous Hall effect (QAHE) and the quantum spin Hall effect (QSHE) Shoucheng Zhang, Stanford University Les Houches, June 2006](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649d225503460f949f88e9/html5/thumbnails/20.jpg)
The Quantum Hall Effect with Landau Levels
Spin – Orbit Coupling in varying external potential?
for
![Page 21: Quantum anomalous Hall effect (QAHE) and the quantum spin Hall effect (QSHE) Shoucheng Zhang, Stanford University Les Houches, June 2006](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649d225503460f949f88e9/html5/thumbnails/21.jpg)
• 2D electron motion in increasing radial electric
raE charge
raE charge
GaAs
E• Inside a uniformly charged
cylinder
raE charge
• Electrons with large g-factor:
Quantum Spin Hall
![Page 22: Quantum anomalous Hall effect (QAHE) and the quantum spin Hall effect (QSHE) Shoucheng Zhang, Stanford University Les Houches, June 2006](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649d225503460f949f88e9/html5/thumbnails/22.jpg)
• Hamiltonian for electrons:
• Tune to R=2
Quantum Spin Hall
• Spin - effectiveB• Spin - effectiveB
• No inversion symm, shear strain ~ electric field (for SO coupling term)
![Page 23: Quantum anomalous Hall effect (QAHE) and the quantum spin Hall effect (QSHE) Shoucheng Zhang, Stanford University Les Houches, June 2006](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649d225503460f949f88e9/html5/thumbnails/23.jpg)
• Different strain configurations create the different “gauges” in the Landau level problem
• Landau Gap and Strain Gradient
Quantum Spin Hall
![Page 24: Quantum anomalous Hall effect (QAHE) and the quantum spin Hall effect (QSHE) Shoucheng Zhang, Stanford University Les Houches, June 2006](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649d225503460f949f88e9/html5/thumbnails/24.jpg)
• P,T-invariant system
• QSH characterized by number n of fermion PAIRS on ONE edge. Non-chiral edges => longitudinal charge conductance!
Helical Liquid at the Edge
• Double Chern-Simons
(Zhang, Hansson, Kivelson)(Michael Freedman, Chetan Nayak, Kirill Shtengel, Kevin Walker, Zhenghan Wang)
![Page 25: Quantum anomalous Hall effect (QAHE) and the quantum spin Hall effect (QSHE) Shoucheng Zhang, Stanford University Les Houches, June 2006](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649d225503460f949f88e9/html5/thumbnails/25.jpg)
Quantum Spin Hall In Graphene (Kane and Mele)
• Graphene is a semimetal. Spin-orbit coupling opens a gap and forms non-trivial topological insulator with n=1 per edge (for certain gap val)
• Based on the Haldane model (PRL 1988)
• Quantized longitudinal conductance in the gap
• Experiment sees universal conductivity, SO gap too small
• Haldane, PRL 61, 2015 (1988)• Kane and Mele, condmat/0411737• Bernevig and Zhang, condmat/0504147• Sheng et al, PRL 95, 136602 (2005)• Kane and Mele PRL 95, 146802 (2005)• Qi, Wu, Zhang, condmat/0505308• Wu, Bernevig and Zhang condmat/0508273• Xu and Moore cond-mat/0508291 …
![Page 26: Quantum anomalous Hall effect (QAHE) and the quantum spin Hall effect (QSHE) Shoucheng Zhang, Stanford University Les Houches, June 2006](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649d225503460f949f88e9/html5/thumbnails/26.jpg)
Stability at the edge• The edge states of the QSHE is
the 1D helical liquid. Opposite spins have the opposite chirality at the same edge.
• It is different from the 1D chiral liquid (T breaking), and the 1D spinless fermions.
yy SiSi eTeT 22 • T2=1 for spinless fermions and
T2=-1 for helical liquids.
)()( 1
11
RLLRRLLR
RLLR
TT
TTTT
• Single particle backscattering is not possible for helical liquids!
![Page 27: Quantum anomalous Hall effect (QAHE) and the quantum spin Hall effect (QSHE) Shoucheng Zhang, Stanford University Les Houches, June 2006](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649d225503460f949f88e9/html5/thumbnails/27.jpg)
• Quantum AHE in ferromagnetic insulators.
• Quantum SHE in “inverted band gap” insulators.
• Quantum SHE with Landau levels, caused by strain.
• New universality class of 1D liquid: helical liquid.
• QSHE is simpler to understand theoretically, well-classified by the global topology, easier to detect experimentally, purely intrinsic, can be engineered by band structure, enables spintronics without spin injection and spin detection.
Conclusions
![Page 28: Quantum anomalous Hall effect (QAHE) and the quantum spin Hall effect (QSHE) Shoucheng Zhang, Stanford University Les Houches, June 2006](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649d225503460f949f88e9/html5/thumbnails/28.jpg)
Topological Quantization of Spin Hall • Physical Understanding: Edge states
In a finite spin Hall insulator system, mid-gap edge states emerge and the spin transport is carried by edge states.
Energy spectrum on stripe geometry.
Laughlin’s Gauge Argument:
When turning on a flux threading a cylinder system, the edge states will transfer from one edge to another
![Page 29: Quantum anomalous Hall effect (QAHE) and the quantum spin Hall effect (QSHE) Shoucheng Zhang, Stanford University Les Houches, June 2006](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649d225503460f949f88e9/html5/thumbnails/29.jpg)
Topological Quantization of Spin Hall • Physical Understanding: Edge states
When an electric field is applied, n edge states with transfer from left (right) to right (left).
accumulation Spin accumulation
Conserved Non-conserved
+=
![Page 30: Quantum anomalous Hall effect (QAHE) and the quantum spin Hall effect (QSHE) Shoucheng Zhang, Stanford University Les Houches, June 2006](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649d225503460f949f88e9/html5/thumbnails/30.jpg)
Topological Quantization of SHE
LH
HH
SHE is topological quantized to be n/2
Luttinger Hamiltonian rewritten as
In the presence of mirror symmetry z->-z, <kz>=0d1=d2=0! In this case, the H becomes block-diagonal: