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Notes on Topological Insulators and Quantum Spin Hall Effect Jouko Nieminen Tampere University of Technology. Not so much discussed concept in this session: topology. ● ● 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. ✔ Quantum Hall states ✔ ● 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. ● ● ● 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 ● ● 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 ● Properties tuned by the quantum well width, d. critical width dc =6.5nm. ● d<dc → normal band structure ● 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 ● Let us go back to Bernevig clip: ● d<dc → conductivity practically zero, G=0! ● d>dc → ● Compare to Integer Quantum Hall Effect (IQHE), where What happens in this transition ● 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... ● ● 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. ● 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: ● ● Phase of a wave function is path dependent: In fact, this is related to quantization of magnetic flux Time reversal symmetry in QSHE ● The spin polarized channels to both edges are formed due to spin-orbit coupling. No external field! ● The conductance for each spin is ● Hence the total conductance is Hasan & Kane. Where do the spin resolved channels come from? ● ● ● 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... ● with electron spin S: ● Hence, kx -kx -kx kx y x In 3D this is more complicated: ● 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 ● ● 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