1. The Hall States and Geometric Phase
Jake Wisser and Rich Recklau

2. 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

3. I. The Ordinary and Anomalous Hall Effects
Hall, E. H., 1879, Amer. J. Math. 2, 287

4. The Ordinary Hall EffectVH
Right hand rule
Evidence of negative charge
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

5. The Anomalous Hall EffectVH
“Pressing effect” much greater in ferromagnetic materials
Additional term predicts Hall voltage in the absence of a magnetic field

6. Anomalous Hall Data
If regular hall, expect linear resistivity relationship
Resistivity increases rapidly at small B, then saturates at a value that depends on the magnetization
Where ρxx is the longitudinal resistivity and β is 1 or 2

7. II. The Aharonov-Bohm Effect and Berry Phase Curvature

8. Vector Potentials
Maxwell’s Equations can also be written in terms of vector potentials A and φ
Divergence of a curl is 0

9. Schrödinger’s Equation for an Electron travelling around a Solenoid
Where
For a solenoid
Solution:
Where
ψ’ solves the Schrodinger’s equation in the absence of a vector potential
Key: A wave function in the presence of a vector potential picks up an additional phase relating to the integral around the potential

10. Vector Potentials and Interference
If no magnetic field, phase difference is equal to the difference in path length
Vector potential becomes non zero in the presence of a magnetic field
If we turn on the magnetic field:
There is an additional phase difference!

11. Experimental Realization
Interference fringes due to biprism
Critical condition:
S is source (electron microscope)
E are biprisms to bend beam around
F is a quartz filament
A is iron whisker
Useful for measuring extremely small fluxes
Due to magnetic flux tapering in the whisker, we expect to see a tilt in the fringes
Useful to measure extremely small magnetic fluxes

12. Berry Phase Curvature For 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!

13. A Classical Analog
Zero Berry Curvature
Non-Zero Berry Curvature
Parallel transport of a vector on a curved surface ending at the starting point results in a phase shift!

14. Anomalous Velocity E VH
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

15. 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:

16. III. The Quantum Trio and Topological Insulators

17. The Quantum Hall Trio

18. 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,…

19. 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)

20. The Quantum Spin Hall Effect

21. The Quantum Spin Hall Effect
König et, al

22. What is a Topological Insulator (TI)?
Bi2Se3
Insulating bulk, conducting surface

23. V. The Quantum Anomalous Hall Effect

24. Breaking TRS
Breaking TRS suppresses one of the channels in the spin Hall state
Addition of magnetic moment
Cr(Bi1-xSbx)2Te3

25. Observations No magnetic field!
As resistance in the lateral direction becomes quantized, longitudinal resistance goes to zero
Vg0 corresponds to a Fermi level in the gap and a new topological state

26. VI. Future Directions

27. References http://journals.aps.org/pr/pdf/10.1103/PhysRev.115.485