Based on a proper definition of the current operators for non-quadratic Hamiltonians, we derive the expression for the transport current which involves the derivative of the imaginary part of the free-electron current, highlighting peculiarities of the extra terms. The expression of the probability current, when Spin-Orbit Interaction (SOI) is taken into account, requires a reformulation of the boudary conditions. This is especially important for tunnel heterojunctions made of non-centrosymmetric semiconductors. Therefore, we consider a model case: tunneling of conduction electrons through a -oriented GaAs barrier. The new boundary conditions are reduced to two set of equations: the first one expresses the discontinuity of the envelope function at the interface while the other one expresses the discontinuity of the derivative of the envelope function.
We propose a novel set of boundary conditions, based on the continuity of a generalized velocity and on the continuity of the probability current at the interface of heterojunctions, which is well suited to construct the solution of the tunneling problem when spin-orbit interaction is taken into account. We illustrate this procedure in a model case: tunneling of conduction electrons through a -oriented GaAs barrier. In that case, the new boundary conditions reduce to two set of equations: the first one expresses the discontinuity of the envelope function at the interface while the other one is close to the standard condition on the derivative of the envelope function.
New boundary conditions are derived for tunnel-heterojunctions, where the effective Hamiltonian is a generic
power of the momentum-operator. A novel expression of probability-current operator, which can be also applied
in presence of the D'yakonov-Perel (DP) Hamiltonian, has to be used. We test our technique on the interface
between two semi-infinite media, with on one side a free-electron-like material and on the other side the -
oriented GaAs barrier.
We propose a procedure, based on momentum power series expansion Hamiltonian of nth general order, to
obtain a coherent expression of the probability current operator that is valid also when Spin-Orbit Interaction
(SOI) terms are included. We prove that we recover the standard definition when the free electron-like term
is included in the Hamiltonian, but when taking into account higher order Spin-Orbit Interaction (SOI) terms,
a more general definition of the probability current operator is mandatory, due to its different symmetrization,
compared to the Hermitian velocity operator expression.
In non-centrosymmetric semiconductors with zinc-blende structure grown along the  crystallographic direction,
electrons with up and down spins undergo different quantum phase shifts upon tunneling, which can be
wieved as resulting from spin precession around a complex magnetic field. There is no spin filtering but a pure
spin dephasing. The phase shift of the transmitted wave is proportional to the overall barrier-material thickness.
We show that a device incorporating a number of resonant tunnel barriers constitutes an efficient quantum-phase
We consider spin-dependent tunneling through a gallium arsenide barrier, a material which has no inversion symmetry. We are dealing with free electrons, with one effective mass and a spin-splitting in the barrier material. When we take into account both the spin-orbit interaction and the absence of the inversion symmetry, the evanescent states in the barrier are spin split and the tunneling process can become rather involved: Depending on the crystallographic direction, the incident wave experiences spin filtering during the tunneling or a spin precession around an effective magnetic field. These results open stimulating perspectives for spin manipulation in tunnel devices.
In a crystal without inversion center, when the spin-orbit interaction is taken into account, it has been shown that the evanescent branches inside the forbidden band gap correspond to complex wave vectors. Here, we discuss possible tunneling phenomena associated with complex wave vectors. We demonstrate that in a case where the wave vector has orthogonal real and imaginary components, an almost standard tunneling process can be restored under crude approximations, analoguous to off-normal tunneling through a potential barrier. Any more accurate analysis is an open problem. In one-dimensional tunneling with a complex wave vector, no solution can be calculated under simple hypotheses.
Using third and forth order perturbation, we have derived main GaAs band parameters (such as EP) from both experimental results (m*, g*, ...) and theoretical results (overall band structure from Cohen-Chelikowski pseudo-potential calculations). The analysis of the set of data leads to a drastic change to the "admitted" value of some parameters (E'P) from the only experimental results and show inconsistency if theoretical results are furthermore taken into account.
In the model case of a III-V semiconductor, we calculate the evanescent waves and the associated energies throughout the forbidden band gap, taking into account the electron spin. Starting with simple pictures, step by step we include more bands and, finally, the calculation is performed using a k.p technique within a 30-band model. We show that no evanescent state associated with a purely imaginary wave vector may exist in some simple directions. In general, the evanescent wave functions have to involve complex wave vectors associating non-collinear propagation and attenuation directions. Such waves only exist in limited wave-vector and energy domains and these properties have deep consequences on tunneling phenomena.