Majorana Bound States In Inhomogeneous Topological Superconductors

In 1937,the Italian physicist Ettore Majorana proposed a symmetric solution of Dirac equation.This solution describes a fermionic particle which is its own antiparticle,namely the Majorana fermion.Majorana's work had a profound impact on elementary particle physics,particularly on neutrino physics.Now,the question of the existence of a Majorana fermion as an elementary particle remains unanswered.In the last decade,the concept of Majorana fermion stepped out the area of particle physics.The equation derived by Majorana appears naturally in the description of electronic sates in superconductors.Some superconductors with one or two spatial dimension may contain zero-energy quasiparticles with similar properties of Majorana fermions.These zero-energy quasiparticles are also called Majorana bound states or Majorana fermions.Due to their nonAbelian statistics,they have many potential applications in quantum computation.In this thesis,we focus on Majorana bound states in a one dimensional semiconductor nanowire with finite length.The model we considered has been widely studied and adopted in many experiments.This model requires a semiconductor nanowire with a strong spinorbit coupling proximately coupled to a s-wave superconductor in the presence of the external magnetic field.Varying the external magnetic field and the chemical potential,we can realize the topological phase transition and generate the Majorana bound states at the two ends of a nanowire.Using the trial wave function method to analyze the Majorana bound states in a semiconductor nanowire has its advantages over other methods,such as the variational method.It can numerically give us the exact eigenvalues and eigenfunctions of Bd G equation.The main results of this thesis are organized as follows:1.Full details of the trial wave function method applied in solving the Bd G equation of a nanowire with finite length is first presented.Numerically,we obtain the energy splitting of Majorana bound states as a function of the Zeeman field and the chemical potential for different length of a nanowire.We also get the distribution of the total probability density,probability density of different spin directions and charge density along the nanowaire for some chosen parameters.An analytical expression of energy splitting as a function of the length of a nanowire is derived.At the same time,we numerically find a similar relation.As an application of the trial wave function method,we explore the energy splitting of the Majorana Kramer pairs.2.In the inhomogeneous case where two segments of a nanowire have different lengths and different chemical potentials,the low-energy spectra of the full system as a function of the chemical potential of the shorter part of the nanowire is calculated after imposing two more boundary conditions at the junction.The total probability density,probability density of different spin directions and charge density as functions of the position in the entire nanowire show that a chemical potential well will restrict the leakage of wave functions of Majorana bound states into the shorter part of the nanowire.