Ab-initiocalculations of spin tunneling through an indirect barrier
© Chantis et al. 2008
Received: 26 November 2007
Accepted: 06 June 2008
Published: 06 June 2008
We use a fully relativistic layer Green's functions approach to investigate spin-dependent tunneling through a symmetric indirect band gap barrier like GaAs/AlAs/GaAs heterostructure along  direction. The method is based on Linear Muffin Tin Orbitals and it is within the Density Functional Theory (DFT) in the Local Density Approximation (LDA). We find that the results of our ab-initio calculations are in good agreement with the predictions of our previous empirical tight binding model [Phys. Rev. B, 075313 (2006)]. In addition we show the k||-dependence of the spin polarization which we did not previously include in the model. The ab-initio calculations indicate a strong k||-dependence of the transmission and the spin polarization due to band non-parabolicity. A large window of 25–50% spin polarization was found for a barrier of 8 AlAs monolayers at k|| = 0.03 2π/a. Our calculations show clearly that the appearance of energy windows with significant spin polarization depends mostly on the location of transmission resonances and their corresponding zeros and not on the magnitude of the spin splitting in the barrier.
PACS Codes: 71.70.Ej, 71.15.Mb, 71.55.Eq
The possibility of spin-polarized transmission through a zinc-blende semiconductor symmetric barrier was investigated in a few previous articles [1–3]. In Ref.  only direct band gap materials were considered for the barrier. Nevertheless, many zinc-blende wide band gap semiconductors are indirect, therefore the indirect tunneling through a barrier must also be considered. This feature was addressed in Ref. . The authors argued that the linear-k spin-orbit splitting at X point will induce larger effect than the k3-term at Γ point . They found an energy window around the top of the barrier, where the effect arising from the linear-k term at X point is larger than the effect stemming from the k3-term at Γ point.
Mishra et al. made no reference to the importance of the interaction between the discrete states at the X valley in the barrier and the continuum . In their work, the authors considered the electrons close to the top of the indirect barrier, where the indirect tunneling becomes dominant . In our previous work  we have shown that, at these energies, the Γ-X mixing plays a greater role than Mishra et al. suggested . We have also considered the Fano resonances that occur because of the interaction between the discrete states at the X point in the barrier and the continuum at the Γ point in the contacts. To be more precise, in such systems an incident electron can tunnel either directly through evanescent Γ state in the barrier or through quasi-bound X state in the barrier. The resonance occurs due to resonant tunneling through quasi-bound X state, while the anti-resonance occurs due to destructive interference between the two channels of electron transmission . Below or above the resonance the two channels are out of phase. The zero in the transmission occurs whenever the amplitude magnitudes of the two channels become equal. In our previous paper  we have used a realistic empirical tight binding model to give a unified description of the spin dependent tunneling through an indirect symmetric barrier with the GaAs/AlAs/GaAs heterostructure as an example. We showed several interesting aspects of such process, with particular emphasis on the large energy windows of spin polarization that can be obtained when appropriate conditions are satisfied.
Although our study presented quantitative results, several simplifications were made. The tight-binding Hamiltonian assumes parabolic conduction bands and it does not distinguish between X1 and X3 states , whereas the conduction band minimum is at Δ, along Γ-X line. Moreover, the complex structure of spin-dependent evanescent states in the band gap was completely ignored. This complicated structure of spin-dependent evanescent states might play an important role in the spin-dependent transport due to the meV scale on which the spin splittings occur [5–7]. In the present paper we extend the previous work  by reporting the results of the first ab-initio multiband study of spin-dependent tunneling through a zinc-blende barrier. The calculations are performed at the LDA level of the DFT with its well-known shortcomings regarding the band gaps and the effective masses. From the point of view of device modeling this may constitute a handicap; tools like sp3d5s* full band semi-empirical tight-binding would perform a better job in reproducing band gaps, effective masses, and complex and imaginary bands. In this work we are checking if the main points made in the previous work  (namely, the criteria needed for obtaining large energy windows with significant spin polarization) are still valid in the context of truly multi-band calculations. The current calculations take into account the full band structure of the GaAs/AlAs/GaAs  heterostructure (including the spin dependency of the imaginary bands in the barrier) and, unlike our previous two-band tight-binding calculations, the ab-initio multiband calculations show that the nonparabolicity should be also considered in order to obtain large spin polarization.
Results and discussion
Here, Ω κμ () are the spin spherical harmonics, μ is the projection of the total angular momentum and κ is the relativistic quantum number: κ2 = J(J + 1) + .
B p (k||, E) = i [Γ p (k||, z+) - Γ p (k||, z-)]
The surface Green's functions of the electrodes are constructed scalar-relativistically, which allows us to decompose the conductance into spin-conserving and spin-flip components . g1,Nand gN,1 are the upper right corner and lower left corner components of the auxiliary Green's function matrix
g = [P-S]-1
Here, P is the fully relativistic potential function (6) and S the tridiagonal matrix of scalar relativistic structure constants .
The heterostructure is 'grown' in the  direction. We have tried two different geometries: one where the left semi-infinite GaAs is separated from the right semi-infinite GaAs by 4 monolayers of AlAs and another where the separation is 8 monolayers. The self-consistent charge distribution is achieved within scalar-relativistic TB-LMTO calculations for GaAs/AlAs heterostructures treated using supercells with 6 monolayers of GaAs separated by the number of AlAs monolayers that correspond to the above mentioned geometries. The transmission is calculated at zero bias with lateral k|| taken in the  direction.
We make a comment on the ab-initio method that we use in this paper. The surface Green functions of the left/right semi-infinite GaAs regions are scalar-relativistic, thus the Dresselhaus term is not taken into account in there. We consider also GaAs layers in the active region where the Hamiltonian is fully relativistic, but the transmission is for an electron that starts at the left semi-infinite and ends at the right semi-infinite region. However, we already questioned the importance of the Dresselhaus term in contacts . The related calculations  show that this term has a minor effect on the overall spin polarization with a much greater role played by the current operator. In our ab-initio calculations the current operator is properly taken into account.
Finally, we comment about experimental aspects of the present subject. An experimental setup would measure currents with contributions from all other energy and momentum values that, in principles, will wash out the spin polarization. For instance, electronic states with the same energy, corresponding to k|| and -k||, will have opposite spins, thus the total current will carry the same amount of spin-up and spin-down. To fix these aspects we already suggested  that the electrons should be injected from a resonant tunneling diode in order to focus the electrons in the k||-plane . Moreover, by applying a voltage bias in the k||-plane of the emitter [27, 28], the isotropy of k|| will be further broken, thus a net spin polarization will be induced in the energy window.
In conclusion, we applied a fully-relativistic first-principles transport method to the spin-dependent tunneling through a GaAs/AlAs/GaAs  heterostructure. The method is based on the Green's function representation of the Tight-Binding Linear Muffin-Tin Orbital basis in the Atomic Spheres Approximation. The calculations were performed in the Local Spin Density Approximation within the Density Functional Theory.
Considering the full band structure of the GaAs/AlAs/GaAs  system, these calculations confirm previous general predictions made with a simplified empirical tight-binding method. Namely, in order to have windows with large spin polarizations, two conditions need to be satisfied: the first is to have well separated resonances such that their corresponding anti-resonances do not interact with each other and the second is that the energy order of the resonances in the spin channels have to be the same as the energy order of their corresponding zeros.
We found a large energy window of 25–50% spin polarization for a barrier of 8 AlAs monolayers at k|| = 0.03 2π/a, but there is no such energy window at k|| = 0.06 2π/a. Our study suggests that, in order to find energy windows with large spin polarization, a detailed knowledge of the energy dependence on k|| and spin must be considered.
The authors gratefully acknowledge the financial support from the Office of Naval Research and from Romanian Ministry of Education and Research.
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