Spin orbit interaction and Majorana fermions in carbon nanotubes and Rashba nanowires

Spin physics plays a fundamental role in many fields of modern condensed matter,
notably for spintronics and spin-based quantum computation. At the heart of this lies
the spin orbit interaction (SOI), a relativistic effect that is responsible for many fascinating
phenomena discovered recently, such as topological insulators, Majorana fermions, spin
effects in strongly correlated materials and in optical lattices. The main focus of this
PhD work is on spin phenomena and, in particular, on the microscopic origin of SOI in
graphene nanostructures and on phenomena connected to it.
The last decades have seen remarkable experimental progress in fabrication of high-
purity carbon nanotubes (CNTs). Similar developments have taken place in the field
of graphene. This progress paved the way to use carbon-based materials for spin-based
effects that are fundamental both for quantum information processing and spintronics.
One goal in spintronics is the control of spin by electric fields or gates since this is fast
and local, and thus much superior over magnetic field control of spins. The key to such
electric control is spin orbit interaction, as it couples charge and spin degrees of freedom.
We have studied SOI effects in carbon nanotubes and other carbon-based materials.
The theory we have developed allowed us to explain several experiments on SOI in CNT
quantum dots. Moreover, we have proposed a more efficient spin-manipulation in quantum
dots by means of e.g. electron dipole spin resonance (EDSR). The SOI also opens up new
possibilities for striking effects such as helical modes, modes which transport opposite
spins in opposite directions, and Majorana Fermions (MFs), particles that are their own
antiparticles, in carbon nanotubes, single- and bilayer graphene.
Majorana fermions in semiconducting nanowires have attracted wide attention over
the past years, partially due to their non-Abelian statistics, which is of great interest
for topological quantum computation. In our work, we have studied the wavefunction
structure and shown that the various MF wavefunctions have different localization lengths
in real space and interference between them leads to pronounced oscillations of the MF
probability density, which can serve as a signature of MFs in experiments. In the case of a
transparent normal-superconducting junction, the MF leaks out from the superconducting
into the normal section of the wire and is delocalized over the entire normal section. The
interplay between a uniform and a spatially periodic magnetic fields in Rashba nanowires
leads to a competition of phases with two topological gaps closing and reopening, resulting
in unexpected reentrance behavior. Besides the topological phase with localized Majorana
fermions (MFs) we find new phases characterized by fractionally charged fermion (FF)
bound states of Jackiw-Rebbi type. The system can be fully gapped by the magnetic fields
alone, giving rise to FFs that transmute into MFs upon turning on superconductivity.
Spin orbit interaction is also of a great use for the manipulation of spin states in
quantum dots. For example, we have developed a scheme for implementing the CNOT
gate over qubits encoded in a pair of electron spins in a double quantum dot. The scheme
is based on exchange and spin orbit interactions and on local gradients in Zeeman fields.
The switching times for the proposed CNOT gate can be as fast as a few nanoseconds for
realistic parameter values in GaAs semiconductors.
Spin-orbit interaction is not the only interesting hallmark of spin physics. Another
quantity of interest is the spin susceptibility, which is connected to the Rudermann-
Kittel-Kasuya-Yosida (RKKY) interaction - indirect exchange interaction between spins
over relatively large distances mediated by itinerant carriers, determines the magnetic
properties of the system. Moreover, it can provide a mechanism for manipulation of spin
over larger distances. This is of great interest for the field of spin qubits in order to build
scalable quantum computing architectures. We have studied RKKY interaction in CNTs
and graphene nanoribbons in the presence of both spin orbit interactions and magnetic
fields. In metallic CNTs the RKKY interaction depends strongly on the sublattice and,
at the Dirac point, is purely ferromagnetic (antiferromagnetic) for the localized spins on
the same (different) sublattice, whereas in semiconducting CNTs the spin susceptibility
depends only weakly on the sublattice and is dominantly ferromagnetic. The spin orbit
interactions break the SU(2) spin symmetry of the system, leading to an anisotropic
RKKY interaction of Ising and Moryia-Dzyaloshinsky form, besides the usual isotropic
Heisenberg interaction. Quite remarkably, parameter regimes could be identified that
show strong anisotropies. This opens the door for magnetism in these low-dimensional
carbon systems that can be controlled by electric fields.