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 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

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 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

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 Divergence of a curl is 0

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 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!

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

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 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!

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

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 Effect König 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 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

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