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