the hall states and geometric phase
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The Hall States and Geometric Phase. Jake Wisser and Rich Recklau. Outline. Ordinary and Anomalous Hall Effects The Aharonov - Bohm Effect and Berry Phase Topological Insulators and the Quantum Hall Trio The Quantum Anomalous Hall Effect Future Directions. - PowerPoint PPT PresentationTRANSCRIPT
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The Hall States and Geometric Phase
Jake Wisser and Rich Recklau
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Outline
I. Ordinary and Anomalous Hall EffectsII. The Aharonov-Bohm Effect and Berry PhaseIII. Topological Insulators and the Quantum Hall
TrioIV. The Quantum Anomalous Hall EffectV. Future Directions
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I. The Ordinary and Anomalous Hall Effects
Hall, E. H., 1879, Amer. J. Math. 2, 287
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The Ordinary Hall EffectVH
Charged particles moving through a magnetic field experience a force
Force causes a build up of charge on the sides of the material, and a potential across it
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The Anomalous Hall EffectVH
“Pressing effect” much greater in ferromagnetic materials
Additional term predicts Hall voltage in the absence of a magnetic field
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Anomalous Hall Data
Where ρxx is the longitudinal resistivity and β is 1 or 2
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II. The Aharonov-Bohm Effect and Berry Phase Curvature
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Vector Potentials
Maxwell’s Equations can also be written in terms of vector potentials A and φ
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Schrödinger’s Equation for an Electron travelling around a Solenoid
Solution:
Where
Key: A wave function in the presence of a vector potential picks up an additional phase relating to the integral around the potential
Where For a solenoid
ψ’ solves the Schrodinger’s equation in the absence of a vector potential
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Vector Potentials and Interference
If no magnetic field, phase difference is equal to the difference in path length
If we turn on the magnetic field:
There is an additional phase difference!
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Experimental Realization
Interference fringes due to biprism
Due to magnetic flux tapering in the whisker, we expect to see a tilt in the fringes
Critical condition:
Useful to measure extremely small magnetic fluxes
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Berry Phase CurvatureFor electrons in a periodic lattice potential:
The vector potential in k-space is:
Berry Curvature (Ω) defined as:
Phase difference of an electron moving in a closed path in k-space:
An electron moving in a potential with non-zero Berry curvature picks up a phase!
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A Classical Analog
Parallel transport of a vector on a curved surface ending at the starting point
results in a phase shift!
Zero Berry Curvature Non-Zero Berry Curvature
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Anomalous Velocity
Systems with a non-zero Berry Curvature acquire a velocity component perpendicular to the electric field!
How do we get a non-zero Berry Curvature?
By breaking time reversal symmetry
VH
E
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Time Reversal Symmetry (TRS)Time reversal (τ) reverses the arrow of time
A system is said to have time reversal symmetry if nothing changes when time is reversed
Even quantities with respect to TRS: Odd quantities with respect to TRS:
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III. The Quantum Trio and Topological Insulators
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The Quantum Hall Trio
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The Quantum Hall Effect• Nobel Prize Klaus
von Klitzing (1985)• At low T and large
B– Hall Voltage vs.
Magnetic Field nonlinear
– The RH=VH/I is quantized
– RH=Rk/n• Rk=h/e2
=25,813 ohms, n=1,2,3,…
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What changes in the Quantum Hall Effect?
• Radius r= m*v/qB• Increasing B, decreases r• As collisions increase, Hall resistance increases• Pauli Exclusion Principle• Orbital radii are quantized (by de Broglie
wavelengths)
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The Quantum Spin Hall Effect
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The Quantum Spin Hall EffectKönig et, al
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What is a Topological Insulator (TI)?
Bi2Se3
Insulating bulk, conducting surface
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V. The Quantum Anomalous Hall Effect
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Breaking TRS
• Breaking TRS suppresses one of the channels in the spin Hall state
• Addition of magnetic moment• Cr(Bi1-xSbx)2Te3
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Observations
As resistance in the lateral direction becomes quantized, longitudinal resistance goes to zero
No magnetic field!
Vg0 corresponds to a Fermi level in the gap and a new topological state
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VI. Future Directions
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References• http://journals.aps.org/pr/pdf/10.1103/PhysRev.115.485• http://phy.ntnu.edu.tw/~changmc/Paper/wp.pdf• http://mafija.fmf.uni-lj.si/seminar/files/2010_2011/seminar_aharonov.pdf• https://www.princeton.edu/~npo/Publications/publicatn_08-10/
09AnomalousHallEffect_RMP.pdf• http://physics.gu.se/~tfkhj/Durstberger.pdf• http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.5.3• http://journals.aps.org/rmp/abstract/10.1103/RevModPhys.25.151• http://www-personal.umich.edu/~sunkai/teaching/Fall_2012/chapter3_part8.pdf• https://www.sciencemag.org/content/318/5851/758• https://www.sciencemag.org/content/340/6129/167• http://www.sciencemag.org/content/318/5851/766.abstract• http://www.physics.upenn.edu/~kane/pubs/p69.pdf• http://www.nature.com/nature/journal/v464/n7286/full/nature08916.html• http://www.sciencemag.org/content/340/6129/153