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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.

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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

### I. The Ordinary and Anomalous Hall Effects

Hall, E. H., 1879, Amer. J. Math. 2, 287

### The Ordinary Hall Effect

VH

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 Effect

VH

“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

### 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

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

### 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

Critical condition:

Due to magnetic flux tapering in the whisker, we expect to see a tilt in the fringes

Useful to measure extremely small magnetic fluxes

### 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!

### A Classical Analog

Non-Zero Berry Curvature

Zero Berry Curvature

Parallel transport of a vector on a curved surface ending at the starting point results in a phase shift!

### Anomalous Velocity

VH

E

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

### 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:

### 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?

• Increasing B, decreases r

• As collisions increase, Hall resistance increases

• Pauli Exclusion Principle

• Orbital radii are quantized (by de Broglie wavelengths)

König et, al

### What is a Topological Insulator (TI)?

Bi2Se3

Insulating bulk, conducting surface

### Breaking TRS

• Breaking TRS suppresses one of the channels in the spin Hall state

• Cr(Bi1-xSbx)2Te3

### 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

### 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