Download Notes on Topological Insulators and Quantum Spin Hall Effect

Notes on Topological Insulators and
Quantum Spin Hall Effect
Jouko Nieminen
Tampere University of Technology.
Not so much discussed concept in
this session: topology.
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In math, topology discards small details of geometry. Hence
doughnut ↔ coffee mug.
In physics, topology deals with precisely quantized
properties, which remain invariant in smooth deformations.
✔
Gap between valence and conducting bands in insulators.
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Quantum Hall states
✔
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Gapless edge states accompanying the insulating gap in
bulk in the case of topological insulators.
There are exact measures to make topological equivalence
classes.
Some basic measurable or
applicable properties.
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Insulating inside, conducting at surfaces (3D)
or edges (2D), with gapless edge states.
Electron spin and momentum are coupled at
edges → spin polarized conducting channels
along the edges.
Properties tunable by
geometry
Materials – successes and failed
hopes
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2D: not graphene due to too small spin orbit
coupling (SOC), possibly silicene, for sure
HgTe quantum well between CdTe.
3D: Bi2Te3, Bi2Se3 and Sb2Te3, but not Sb2Se3.
In the following, let us concentrate on 2D.
HgTe between CdTe
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Properties tuned by the quantum well width, d.
critical width
dc =6.5nm.
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d<dc → normal
band structure
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d>dc → inverted
Hasan & Kane, Topological Insulators,
Rev.Mod.Phys., 82, (3045)2010
bands and gapless edge states.
A dramatic change in conductivity at
the critical width
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Let us go back to
Bernevig clip:
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d<dc → conductivity
practically zero, G=0!
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d>dc →
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Compare to Integer Quantum Hall Effect (IQHE),
where
What happens in this transition
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The band structure around a certain k-point
(momentum) develops in the following way:
E
Dirac's cone
k
dc
m>0
Note:
d
m<0
Analogous to relativistic E-p dispersion
In addition...
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For gapless edge states, spin orbit coupling is
necessary!
This also leads to
Quantum Spin Hall Effect
(QSHE) and coupling
of electron spin and momentum.
d
Qi and Zhang, Rev. Mod. Phys. 83, 1057(2011).
QHE vs. QSHE and something
about jargon.
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In Quantum Hall Effect, there is an external
magnetic field →
time reversal (TR) symmetry
is broken.
B
TR
Here it is also, where topology
appears:
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Phase of a wave function is path dependent:
In fact, this is related to quantization of
magnetic flux
Time reversal symmetry in QSHE
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The spin polarized channels to both edges are
formed due to spin-orbit coupling. No external
field!
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The conductance for each spin is
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Hence the total conductance is
Hasan & Kane.
Where do the spin resolved
channels come from?
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Origin of spin-orbit coupling: from the frame of
reference of a moving electron, an electric
field is seen as a magnetic field:
At the edges, symmetry is broken and the
potential gradient is significant.
The potential energy felt by a magnetic
moment:
leading to a nice form...
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with electron spin S:
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Hence,
kx
-kx
-kx
kx
y
x
In 3D this is more complicated:
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But it will be a topic of another session:
Qi and Zhang, Rev. Mod. Phys. 83, 1057(2011).
Some further notes about spin-orbit
coupling
An appendix to the previous slides.
Spin is an operator in spinor space
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Spin has the dimension (and properties) of
angular momentum.
For electron, spin is ½, and hence:
and
and they operate on spinors
or spin wave functions.
Dirac matrices
They operate on spinors
or spin wave functions.
… and magnetic moment
In general
and, hence for electrons
And magnetic potential energy
As shown earlier:
We end up to spin-orbit coupling
or equivalently
Spherical symmetry (the case for
core electrons)
and
ergo
for orbital motion
Now the potential is dependent on Z
That is why the spin-orbit coupling is,
in general, the stronger the heavier is the atom.
But this is not necessarily the case for
valence electrons.
Effect of symmetry breaking:
Rashba-effect
This may happen at a surface,
but also due to some other
effect (see, e.g., Silicene).
note also that
What does Rashba carry about with
itself?
First, Spin is transverse to momentum.
Second, spin up and spin down terms
are mixed:
→ spin flip is related to helicity of momentum.
In Silicene SOC and Rashba
appear due to a symmetry break
Onsite SOC is weak
Nearest neighbor is zero due to symmetry
Next nearest neighbor is non-zero
and significant