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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

- 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

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

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

VH

“Pressing effect” much greater in ferromagnetic materials

Additional term predicts Hall voltage in the absence of a magnetic field

Where ρxx is the longitudinal resistivity and β is 1 or 2

Maxwell’s Equations can also be written in terms of vector potentials A and φ

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

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!

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

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!

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!

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 (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:

- 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,…

- Rk=h/e2

- 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)

König et, al

Bi2Se3

Insulating bulk, conducting surface

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

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

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- http://phy.ntnu.edu.tw/~changmc/Paper/wp.pdf
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