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

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