The Hall States and Geometric Phase
Jake Wisser and Rich Recklau
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
I. The Ordinary and Anomalous Hall Effects
Hall, E. H., 1879, Amer. J. Math. 2, 287
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
The Anomalous Hall EffectVH
“Pressing effect” much greater in ferromagnetic materials
Additional term predicts Hall voltage in the absence of a magnetic field
Anomalous Hall Data
Where ρxx is the longitudinal resistivity and β is 1 or 2
II. The Aharonov-Bohm Effect and Berry Phase Curvature
Vector Potentials
Maxwell’s Equations can also be written in terms of vector potentials A and φ
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
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!
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
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!
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
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
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:
III. The Quantum Trio and Topological Insulators
The Quantum Hall Trio
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,…
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)
The Quantum Spin Hall Effect
The Quantum Spin Hall EffectKönig et, al
What is a Topological Insulator (TI)?
Bi2Se3
Insulating bulk, conducting surface
V. The Quantum Anomalous Hall Effect
Breaking TRS
• Breaking TRS suppresses one of the channels in the spin Hall state
• Addition of magnetic moment• Cr(Bi1-xSbx)2Te3
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
VI. Future Directions
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
Top Related