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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,...

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**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

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**„Topological“ example: defect levels in polyacetylene (CH)x**

Short-Long-… Long-Short-… p* p* C-p C-p p p Bound state in gap center

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**… 1-D Tight Binding model of Topological Insulators Tss s s p p Tpp s**

Normal band structure: Large s-p energy separation Inverted band structure: Small s-p energy separation Tss s s p p Tpp Semiconductor Metal

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**… 1-D Tight Binding model of Topological Insulators Tss Tsp Tsp s s p**

Normal band structure: NN-coupling has little effect Inverted band structure: NN-coupling opens gap and … Tss Tsp Tsp s s p p Tpp Semiconductor Semiconductor

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**1-D Tight Binding model of Topological Insulators**

Normal band structure: NN-coupling has little effect Inverted band structure: … and boundary produces states in the gap Tpp Tsp Tsp p p s s Tss Semiconductor Semiconductor

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**1-D Tight Binding model of Topological Insulators**

s p s p s p s p Tsp Tsp Inverted band structure: Band gap opens + 2 bound states NN-coupling has no effect on boundary since y = 0. Leads to gap states ! Semiconductor with gap states

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**Edges produce bound states**

2D topological insulator HgxCd1-xTe:HgTe:HgxCd1-xTe Cartoon - without spin-orbit interaction Quantum Wire HgTe Bulk HgTe zero-gap 2-DEG HgTe Gate Gate lh e Fermi Energy hh hh e lh k3D k2D k1D Overlapping bands produce HOMO-LUMO gap Edges produce bound states in gap

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**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 E left- left- right- right- k1D

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**2D topological insulator HgxCd1-xTe:HgTe:HgxCd1-xTe**

Relativistic 4-band Envelope Function Calculations Barrier Hg.3Cd.7Te HgTe quantum well thickness 7.8 nm Carrier density ~ 1×1011 cm-2 HgTe quantum wire width 240 nm Gate Gate Band structure E(k1D) Spin-split band states (k-linear spin-orbit splitting, occurs in all ZnS semiconductors) Spin-split gap states (comes with inverted band structure)

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**2D topological insulator HgxCd1-xTe:HgTe:HgxCd1-xTe**

Relativistic 4-band Envelope Function Calculations Spin Polarization across Quantum Wire Gate Gate ±V

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**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 Spin Density Resulting V: 8 mV Proposal by H. Buhmann

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