1. Thermal and electrical quantum Hall effects in ferromagnet — topological insulator — ferromagnet junction V. Kagalovsky 1 and A. L. Chudnovskiy 2 1 Shamoon College of Engineering Beer-Sheva, Israel 2 Institut für Theoretische Physik, Universität Hamburg, Germany Delocalisation Transitions in Disordered Systems July 24 ~ August 02 2015

2. 사무국 휴무 Context Topological Insulators Integer quantum Hall effect vs. quantum spin Hall effect Ferromagnet – Topological Insulator - Ferromagnet Quantum Hall Effects in FM – TI – FM system What do we measure?

3. 사무국 휴무 Topological insulators Insulator in the bulk Conducting states on the surface Realized as 2D and 3D systems Physical mechanism — quantum spin Hall effect

4. 사무국 휴무 2D Topological Insulator (TI) Insulator in the bulk, conductor at the edge Experimentally CdTe-HgTe-CdTe heterostructure Two counterpropagating spin-polarized states at the edge

5. 사무국 휴무 Integer quantum Hall effect (IQHE) Topological Insulator (TI) in quantum spin Hall (QSH) state Strong external magnetic field Each spin projection feels ist own effective magnetic field

6. 사무국 휴무 IQHE

7. 사무국 휴무 TI QSH

8. 사무국 휴무 How to use TI in practice? FM — TI — FM junction Ferromagnet has better electrical contact to the edge state with spin direction parallel to the magnetization  Selective contacting of edge states with differect spin-polarization  Control over the spin- and charge flow by changing the direction of magnetization

9. 사무국 휴무 Theory of FM-TI-FM junction (two spin channels)

10. 사무국 휴무 Coupling to the majority and minority bands of FM (i=1,2) Partial conductances between FM bands (+/-) and spin-polarized edge states (↑,↓) Contact polarization of the FM Total conductance Total conductances of the spin-up and spin-down channels

11. 사무국 휴무 Back-scattering by magnetic impurities Anderson localization length  B. L. Altshuler, I. L. Aleiner, V. I. Yudson, PRL 111, 086401 (2013) Scattering time  Edge channels are robust as long as

12. 사무국 휴무

13. 사무국 휴무 Boundary conditions Solutions

14. 사무국 휴무 Effective chemical potential and temperature

15. 사무국 휴무 i, q Solutions

16. 사무국 휴무 Predictions for experiment i, q Hall resistances: Dimensionless electrical & thermal Hall resistances:

17. 사무국 휴무

18. 사무국 휴무 Robustness to disorder independent of x and 

19. 사무국 휴무 Dependence on spin scattering strength

20. 사무국 휴무 Dependence on the position x of the measurement contact

21. 사무국 휴무 nonmagnetic electrodes Knez I., et al., Phys. Rev. Lett. 112, 026602 (2014) electrodes are attached asymmetrically Dependence on the position y of the second electrode

22. 사무국 휴무 Dependence on the angle  when electrodes are attached asymmetrically nonmagnetic electrodes Knez I., et al., Phys. Rev. Lett. 112, 026602 (2014)

23. 사무국 휴무 IQHE

24. 사무국 휴무 TI QSH Two-terminal measurement (NO terminals 2,3,5,6) Terminals 2,3,5,6 mix spin-up and spin-down states leading to longitudinal resistance So, no problem for Hall conductance/resistance – they are both ZERO! But in zero temperature limit and one can speak about RESISTANCE ONLY!

25. 사무국 휴무 Conclusions FM-TI-FM junction — control over the electrical and heat flow by changing the magnetization direction — current and heat switch in spintronic devices FM-TI-FM junction is robust to disorder in TI, in particular to the scattering by magnetic impurities Symmetric FM-TI-FM junction produces Hall voltage independent of position of measurement contacts and Hall resistance R H independent of spin-scattering strength N.B. Independent of both: bulk time-reversal symmetric disorder and time-reversal symmetry breaking surface disorder (magnetic impurities) For nonmagnetic asymmetric electrodes we reproduce experimental results by fitting the strength of the spin-scattering Resistance vs. conductance measurements