Topological Insulators
Jason Lambert
University of Tennessee
Department of Physics
Outline

Topological Invariants and State of matter

The Quantum Hall Effect ( QHE )

The Quantum Spin Hall Effect ( QHSE )

Experimental Realization QHSE

Exotic Physics
Prediction

I predict that some worker in this field will win
the big prize!!!
Topological States of Matter
M. Z. Hasan and C. L. Kane, arXiv(Febuary
Topological States of Matter
Topological
classification
Particle
Hole
Symmetry
Chiral
Symmetry
Dimensionality
Time
Reversal
Symmetry
QHE
QHSE
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)
Quantum Hall Effect


The resistance in the
y direction quantized.
xx xy


  
 yx yy
Resistance in the xy
direction goes to zero
RH =
h 25812.807557
=
i
e2 i
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
n=
1
2
u
⟨
∫
bz
i
2
k ∣∇ k∣u k ⟩ d k
M. Z. Hasan and C. L. Kane, arXiv(Febuary
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)
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)
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.
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.
Is this a strong topological insulator?
∏
m
a
=
−1 =∏
a
a
4
where = 0∨ 1
a= 1
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)
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
CdHgTe Quantum wells

First topological
insulator HgTe
quantum wells.
M. König et al., Science 318, 766
(2007).
Second Generation Topological
insulators
Bi 2 Se 3
Sb2 Te 3
Bi 2 Te 3
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).
Potential Applications




Spintronics
Room temperature dissipation-less conduction
Because of the perfectly conducting edge state
Possible applications topological quantum
computing
Table Top testing of the Standard Model
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).
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