1. Spin filtering in mesoscopic systems Shlomi Matityahu, Amnon Aharony, Ora Entin-Wohlman Ben-Gurion University of the Negev Shingo Katsumoto University of Tokyo

2. Spintronics (spin electronics) Spintronics – Study of spin degrees of freedom in condensed matter systems Typical questions – How to polarize and manipulate effectively a spin system? How long is the system able to remember its spin orientation? How can spin be detected? Potential applications – Giant magnetoresistance (GMR) Disk read/write head Quantum computer??

3. Spin filtering Generate spin-polarized current out of an unpolarized source Can we find a system which generates a full polarization in a tunable direction? Unpolarized beam Polarized beam Spin filter

4. The Aharonov-Bohm (AB) effect An electron travelling from point A to point B in a region with zero magnetic field, but non- zero vector potential, acquires a phase The phase acquired in a close loop is - The magnetic flux through the surface enclosed by the loop - Flux quantum

5. Spin-Orbit interaction (SOI) Non-relativistic limit of the Dirac equation Rashba SOI - In a 2DEG confined to a plane by an asymmetric potential along This is equivalent to an effective (momentum-dependent) magnetic field The strength of the Rashba term can be tuned by a gate voltage!

6. The Aharonov-Casher (AC) effect Magnetic moment in an electric field also acquires a quantum mechanical phase In contrast to the AB phase, the AC phase is given by an SU(2) rotation matrix

7. Quantum networks – Tight- binding approach Discrete Schrödinger equation tight- binding Hamiltonian - 2-component spinor at site m - site energy - hopping amplitude (a scalar) - unitary matrix representing AB and AC phases

8. Our spin filter – A simple exercise in quantum mechanics

9. Derivation of spin filtering In general, one has to solve for the transmission matrix of the quantum network and then look for the filtering conditions The main conclusion – we can achieve full spin filtering in a tunable direction provided we use both SOI and AB.

10. Additional conclusions The main conclusion – we can achieve full spin filtering in a tunable direction provided we use both SOI and AB The spin filter can also serve as a spin reader Spin filtering is robust against current leakage