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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

The Hall States and Geometric Phase

Jake Wisser and Rich Recklau


Outline

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

I. The Ordinary and Anomalous Hall Effects

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


The ordinary hall effect

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

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

Anomalous Hall Data

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


Ii the aharonov bohm effect and berry phase curvature

II. The Aharonov-Bohm Effect and Berry Phase Curvature


Vector potentials

Vector Potentials

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


Schr dinger s equation for an electron travelling around a solenoid

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

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

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

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

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

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 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

III. The Quantum Trio and Topological Insulators


The quantum hall trio

The Quantum Hall Trio


The quantum hall effect

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

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 Effect


The quantum spin hall effect1

The Quantum Spin Hall Effect

König et, al


What is a topological insulator ti

What is a Topological Insulator (TI)?

Bi2Se3

Insulating bulk, conducting surface


V the quantum anomalous hall effect

V. The Quantum Anomalous Hall Effect


Breaking trs

Breaking TRS

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

  • Addition of magnetic moment

  • Cr(Bi1-xSbx)2Te3


Observations

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


Vi future directions

VI. Future Directions


References

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


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