Topological Insulators

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Topological Insulators

• Jason Lambert
• University of Tennessee
• Department of Physics

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Outline

• Topological Invariants and State of matter
• The Quantum Hall Effect ( QHE )
• The Quantum Spin Hall Effect ( QHSE )
• Experimental Realization QHSE
• Exotic Physics

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Prediction

• I predict that some worker in this field will win
  the big prize!!!

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Topological States of Matter
M. Z. Hasan and C. L. Kane, arXiv(Febuary 2010)
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Topological States of Matter
Chiral Symmetry
Particle Hole Symmetry
Topological classification
Dimensionality
Time Reversal Symmetry
QHE
QHSE
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The Quantum Hall Effect

• Discovered in 1980 by Klaus von Klitzing.
• Nobel Prize awarded in 1985.
• Extremely accurate resistance is now used as a
  NIST standard

K. v. Klitzing, G. Dorda, and M. Pepper, Phys.
Rev. Lett. 45, 494 (Aug. 1980)
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Quantum Hall Effect

• The resistance in the y direction quantized.
• Resistance in the xy direction goes to zero

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Topological Invariant of QHE

• Topological invariant referred as the TKNN
  invariant.
• Determined from the integral of the Brillouin
  zone.
• The topology of the manifold of occupied states
  distinguishes the quantum hall state and
  conventional insulator

M. Z. Hasan and C. L. Kane, arXiv(Febuary 2010)
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Quantum Spin Hall effect

• The QSHE effect has quantized conduction bands
  due to surface states.
• The electrons are spin polarized depending upon
  direction of flow.

C. Day, Physics Today 61, 19 (2008)
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Quantum Spin Hall Effect

• Requires a linearly dispersing edge state
• also known as a dirac cone
• Bulk Valence and conduction bands are connected
  by the surface conducting states

C. Day, Physics Today 61, 19 (2008)
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QSHE

• The surface states of the quantum spin hall
  effect are protected by time reversal symmetry
• There must be an odd number of dirac cones in the
  Brillouin zone.

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Is this a strong topological insulator?

• In three dimensional systems you can have both
  strong and weak topological insulators
• We want strong topological insulators.
• Determining the nature of the topological
  insulator is not trivial
• Easiest method works when material has inversion
  symmetry.

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Is this a strong topological insulator?
L. Fu and C. L. Kane, Phys. Rev. B 76, 045302
(Jul. 2007)
H. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang,
and S.-C. Zhang, Nat Phys 5, 438 (Jun. 2009)
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Experimental Realization

• Looking for materials that have a bulk insulating
  gap with strong spin orbit coupling.
• The spin orbit coupling should be strong enough
  to invert the highest valence band and lowest
  conduction band.
• Need to look toward heavy elements on periodic
  table spin orbit coupling is strong

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CdHgTe Quantum wells

• First topological insulator HgTe quantum wells.

M. König et al., Science 318, 766 (2007).
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Second Generation Topological insulators
H. Zhang et al., Nat. Phys. 5, 438 (2009).
Y. Xia et al., Nat. Phys. 5, 398 (2009) D. Hsieh
et al., Science 323, 919 (2009).
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Potential Applications

• Spintronics
• Room temperature dissipation-less
  conductionBecause of the perfectly conducting
  edge state
• Possible applications topological quantum
  computing
• Table Top testing of the Standard Model

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Physics Exotic

• Majorana Fermions
• Unobserved in nature
• Nuetrinos?
• Predicted to exist in s-wave superconductor TI
  boundary
• Image Magnetic Monopoles
• Because of unique EM field theory that describes
  TI

S.-C. Z. Xiao-Liang Qi, Physics Today 63, 33
(2010)
F. Wilczek, Nature 458, 129 (2009) Nat. Phys. 5,
614 (2009).
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Conclusions

• Topological States of matter including the
  topological insulator have become a very Hot
  topic.
• The second generation materials have a lot of
  promise.
• Some topological insulator properties are very
  applicable
• Testing ground exotic physics and the standard
  model