TopologicalInsulator.
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Topological Insulators
What is this?
No conduction through
interior of material
Current flows along surfaces,
not terribly sensitive to defects
With spin-orbit interaction,
similar to intrinsic Spin Hall
effect, yet without magnetic field
Often called Quantum Spin Hall state
C. L. Kane (UPenn) and E. J. Mele PRL 2005
König et al, Science 318, 766 (2007), Hasan 2010,...
Topological Insulators: Features and requirements
There are still many misconceptions around. Here some important facts:
Single-electron effect and therefore sensitive to chemistry
Edge states in the gap occur independently of dimensionality
The basic effect is independent of spin and spin-orbit interaction
Effect is very common but not within fundamental gap
Interesting cases require inverted band structure (overlapping s & p-bands)
The effect requires sufficient distance between the material‘s boundaries
„Topological“ example:
defect levels in polyacetylene (CH)x
Short-Long-…
Long-Short-…
p*
p*
p
p
C-p
C-p
Bound state in gap center
1-D Tight Binding model of Topological Insulators
…
p
s
p
s
p
s
Normal band structure:
Large s-p energy separation
s
p
p
s
p
s
Inverted band structure:
Small s-p energy separation
Tss
s
p
Tpp
Semiconductor
Metal
…
1-D Tight Binding model of Topological Insulators
…
p
s
p
s
p
s
Normal band structure:
NN-coupling has little effect
s
p
Tss
p
s
p
s
…
Inverted band structure:
NN-coupling opens gap and …
Tsp
Tsp
s
p
Tpp
Semiconductor
Semiconductor
1-D Tight Binding model of Topological Insulators
s
p
s
p
s
Normal band structure:
NN-coupling has little effect
p
s
Tpp
p
s
p
Inverted band structure:
… and boundary produces states in the gap
Tsp
Tsp
p
s
Tss
Semiconductor
Semiconductor
1-D Tight Binding model of Topological Insulators
y
s
p
Inverted band structure:
Band gap opens + 2 bound states
s
p
s
p
s
Tsp
p
Tsp
NN-coupling has no effect
on boundary since y = 0.
Leads to gap states !
Semiconductor with gap states
2D topological insulator HgxCd1-xTe:HgTe:HgxCd1-xTe
Cartoon - without spin-orbit interaction
Quantum
Wire
HgTe
2-DEG
HgTe
lh
Gate
Bulk HgTe
zero-gap
e
Fermi Energy
hh
hh
e
lh
k3D
k2D
Overlapping bands
produce HOMO-LUMO gap
k1D
Edges produce bound states
in gap
2D topological insulator HgxCd1-xTe:HgTe:HgxCd1-xTe
Cartoon - with spin-orbit interaction
Spin-orbit interaction adds another twist for the edge states in the gap:
Spin-up and spin-down edge states within the gap get split
For k1D > 0, only spin-up/spin-down electrons can propagate in right/left channel
Spin-orbit resolved gap states
left-
right-
E
left-
right-
k1D
2D topological insulator HgxCd1-xTe:HgTe:HgxCd1-xTe
Barrier Hg.3Cd.7Te
HgTe quantum well thickness 7.8 nm
Carrier density ~ 1×1011 cm-2
HgTe quantum wire width 240 nm
Gate
Relativistic 4-band Envelope Function Calculations
Band structure E(k1D)
Spin-split band states
(k-linear spin-orbit splitting,
occurs in all ZnS semiconductors)
Energy [meV]
10
5
Spin-split gap states
(comes with inverted band structure)
0
-5
-10
-0.06
-0.03
0.00
k1D [1/nm]
0.03
0.06
2D topological insulator HgxCd1-xTe:HgTe:HgxCd1-xTe
Relativistic 4-band Envelope Function Calculations
100
50
0
0
-50
-100
0
k1D < 0 (VSD <0)
80
160
Gate
Spin Polarization [%]
Spin Polarization across Quantum Wire
240
Spin Polarization [%]
Wire Crossection [1/nm]
100
50
±V
k1D > 0 (VSD >0)
0
-50
-100
0
80
160
240
Wire Crossection [1/nm]
NEGF Application: All-Electric Spin Analyzer based on
Inverse Quantum Spin Hall Effect
HgTe 2DEG
T = 100 mK
VDS = 100 mV
DVgate = 18 mV
QSH Normal conducting
QSH
200 nm
Probe
Spin Density
Source
1
V
-1
Drain
injector
region
Proposal by H. Buhmann
0
gate
region
Probe
Resulting V: 8 mV
collector
region