Download Topological Insulators and the Quantum Anomalous Hall Effect

The Hall States and Geometric
Phase
Jake Wisser and Rich Recklau
Outline
I. Ordinary and Anomalous Hall Effects
II. The Aharonov-Bohm Effect and Berry Phase
III. Topological Insulators and the Quantum Hall
Trio
IV. The Quantum Anomalous Hall Effect
V. Future Directions
I. The Ordinary and Anomalous Hall
Effects
Hall, E. H., 1879, Amer. J. Math. 2, 287
The Ordinary Hall Effect
VH
I
B
VH µ r xy µ
n
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
I
rxy = R0 Bz + Rs M z
“Pressing effect” much greater in ferromagnetic materials
Additional term predicts Hall voltage in the absence of a magnetic field
Anomalous Hall Data
rxy = R0 Bz + Rs M z
Rs µ r
b
xx
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
Where
Solution:
Fm
A=
F̂ For a solenoid
2p r
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:
A=
There is an additional phase difference!
Fm
F̂
2p r
Experimental Realization
Critical condition:
Interference fringes due to biprism
Due to magnetic flux tapering in the whisker, we
expect to see a tilt in the fringes
Useful to measure extremely small
magnetic fluxes
hc
= 4.135´10-7 G × cm 2
e
Berry Phase Curvature
For electrons in a periodic lattice potential:
ynk = Aeikr unk (r)
The vector potential in k-space is:
A(k) = i ynk iÑk ynk
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
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
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?
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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
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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_0810/09AnomalousHallEffect_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