Spin-related tunneling through a nanostructured electric-magnetic barrier on the surface of a topological insulator
© Wu and Li; licensee Springer. 2012
Received: 30 August 2011
Accepted: 27 January 2012
Published: 27 January 2012
We investigate quantum tunneling through a single electric and/or magnetic barrier on the surface of a three-dimensional topological insulator. We found that (1) the propagating behavior of electrons in such system exhibits a strong dependence on the direction of the incident electron wavevector and incident energy, giving the possibility to construct a wave vector and/or energy filter; (2) the spin orientation can be tuned by changing the magnetic barrier structure as well as the incident angles and energies.
PACS numbers: 72.25.Dc; 73.20.-r; 73.23.-b; 75.70.-i.
The recent discovery of a new quantum state of matter, topological insulator, has generated a lot of interest due to its great scientific and technological importance [1–5]. In a topological insulator, spin-orbit coupling opens an energy gap in the bulk, and results in helical surface states residing in the bulk gap in the absence of magnetic fields. Such surface states are spin-dependent and are topologically protected by time-reversal symmetry [4–7] and distinct from conventional surface states, which are fragile and depend sensitively on the details of the surface geometry and bonding. This discovery sparked intensive experimental and theoretical interests, both for its fundamental novel electronics properties as well as possible applications in a new generation of electric devices.
Very recently, the surface states in Bi-based alloys, Bi1-xSb x , Bi2Se3, Bi2Te3, were theoretically predicted [6, 8] and experimentally observed by using angle-resolved photoe-mission spectroscopy (ARPES) [9–12]. These 3D topological insulators have robust and simple surface states consisting of a single Dirac cone at the Γ point . Note that this Hamiltonian appears similar to graphene , but topological insulators have an odd number of massless Dirac cones on the surface, ensured by the Z2 topological invariant of the bulk, while graphene has twofold massless Dirac cones at the K and K' valleys. Another essential difference is connected with spin-related properties. In the surface, Hamiltonian of the 3D TI σ acts on the real spin of the charge carriers, while for graphene it stands for the pseudo spin, i.e., the A and B sublattices of graphene. Hence, it is natural to manipulate spin transport on the surface of a 3D topological insulator by controlling the electron orbital motion. Based on the topological surface Hamiltonian, it is clear that σ· k is a quantum conserved quantity which implies that spin and momentum of the electron are locked. For instance in a tunneling process, the reflected electron will reverse its spin due to the helical property of the surface states, i.e., the spin-momentum locking . This feature will lead to some interesting phenomena, such as the spin-dependent conductance  and the twisted RKKY interaction .
In this study, we investigate electron tunneling through single electric and magnetic potential barriers which can be created by depositing a ferromagnetic metallic strip on the surface of a 3D topological insulator. We find that the in-plane spin orientation of the transmitted and the reflected electrons can be rotated over certain angles that are determined by the incident angle and energy. Our results demonstrate that the magnetic field of the magnetic barrier bends the trajectory of the electrons, and therefore rotate the spin.
where vF is the Fermi velocity, σ i (i = x, y, z) are the Pauli matrices, V is the gate voltage applied on the magnetic metal strips, and the last term H Z ≡ gμ B σ · B is induced by Zeeman spin spitting. Note that for g = 23 in Bi2Se3, the Zeeman term affects the transmission slightly at low magnetic field. The momentum is π = p + e A, where the vector potential of the inhomogeneous magnetic field generated by the magnetic metal stripe, , is the unit vector normal to the surface. Note that , which implies that is a quantum conserved quantity during the tunneling processes, i.e., spin-momentum locking. If the incident electrons are spin polarized along the direction of the vector , the magnetic field bends the trajectory of the electrons, resulting in a rotation of the spin of the transmitted electrons. The reflected electrons suffer similar spin rotations accompanied by the reversal of the momentum p x . Interestingly, the gate voltage can be used to control the reflection and transmission, and therefore tune the spin polarization of the reflected and transmitted electrons.
with complex coefficients c± and υ ≡ (E F - V)2 - (gμ B B)2. After some lengthy algebra, all of the above coefficients of the wave functions can be obtained from the boundary conditions.
3. Spin and momentum filtering
In summary, we investigated theoretically quantum tunneling processes through a single electric and magnetic barrier on the surface of a 3D topological insulator. Our theoretical results show that the propagating behavior of electrons in such a 2D system can be controlled by electric and magnetic barriers. We have also observed the rotation of the electron spin, depending on the parameters of the magnetic barrier structure as well as the incident angle and energy. This investigation could be helpful to offer a promising functional unit for future wavevector, and/or spin filtering quantum devices.
The study was supported by NSFC Grant No. 11104232.
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