Straightforward Probing of Polarization and Spin-Resolved Transmission through Potential Barriers in Silicene

The low-energy physics of silicene is described by Dirac electrons with a strong spin-orbit interaction and its band structure can be controlled by an external electric field Ez. We evaluate the electronic transmission through single and double barriers in silicene as a function of the angle of incidence θ, the electron energy E, and the strength of the field Ez. We find a spin-resolved transmission, reflected in the polarization P, and a conductance qualitatively different in some ranges of E than that in graphene. P exhibits quasiperiodic resonances versus θ in double-barrier structures.


Introduction
The specific two-dimensional (2D) material, a monolayer honeycomb structure of silicon, called silicene, has been recently synthesized [1,2] and attracted, as usual, considerable attention [3].The reason for that is that silicene has Dirac cones mostly similar to those in graphene, which is the genesis of 2D materials achieved in laboratory.In addition, and contrary to graphene in which the spin-orbit interaction (SOI) is very weak, silicene has a strong SOI that leads to a gap of 1.55 meV.This gap can be further controlled by an external electric field   and is facilitated by the buckled structure of silicene resulting from the large ionic radius of silicon.This and its compatibility with silicon-based electronic technology led to important and seminal studies such as the spin-Hall effect [4], the anomalous Hall effect [5,6], and the capacitance of an electrically tunable silicene device [7].
Since the SOI can lead to spin-resolved transport, pertinent to quantum computing and other important issues, it is worth studying SOI further in silicene and eventually contrast these results with graphene's [8,9].We then simply and carefully undertake such a study by evaluating both the electronic transmission and polarization through single and double potential barriers in silicene as a function of the electron's angle of incidence , its energy , and the strength of the external field   .It was suggested that   , solely, might not be enough to lift spin resolution in silicene, and an external magnetic field should be present in order to locally break the time-reversal symmetry [10].But to avoid including further physical parameters such as the magnetic field, which surely brings extra physical nuances to the calculation, we choose to look at one valley only and probe the eventual spin polarization which certainly appears in the presence of magnetic field.In this way, we present a simpler (but scholarly sound) way to probe the spin resolution, which is, by virtue of the SOI, an inherent feature of these novel materials.Furthermore, we calculate the spin polarization for the double-barrier case.This polarization presents structures showing many resonance features which can be also analyzed by virtue of the evaluated polarization measured in laboratory.
In fact, broader studies [11][12][13][14] on 2D Si-based devices have shown important features concerning their timely usage, contribution to the future of both theoretical and experimental investigations, and their correspondent application on technology industry.In these previous works, some interesting issues on magnetic and quantum topological states have been reported.The present work also serves as a primary comparison to such results.

Advances in Condensed Matter Physics
The manuscript is organized as follows.Section 2 is devoted to the theoretical formulation.Sections 3 and 4 explore, respectively, the pertinent cases of single and doublebarrier structures.Our conclusions are given by comparing Sections 3 and 4.

Formalism
The one-electron Hamiltonian for carriers in a single  valley in silicene is given by the following [4]: where V  is the Fermi velocity,  = 0.386 nm the lattice constant, and  ± =   ±   .Also,  ± =  SOI ± ℓ  where  SOI ≡  ≃ 3.9 meV is the strength of the SOI in the lattice and 2ℓ is the distance between the two sublattices  and .It has been noticed that the secondary (Rashba) spin coupling parameters  1 = 10 eV and  2 = 0.7 meV are too small in comparison to the primary one  SOI .Here 1 and 2 stand for two types of the Rashba spin coupling often appearing in the silicene Hamiltonian.These parameters are negligible compared to the spin-orbit one,  SOI .Therefore, we hereafter deliberately assume  −+ =  +− ≃ 0 without any significantly loss of science.The resulting matrix is then block-diagonal with its upper (lower) block pertaining to spin-up (spin-down) electrons.The latter is given by with the Hamiltonian for spin-up electrons  ++ being given by ( 2) with  + replaced with − − .The eigenvectors of ( 2) are straightforwardly written in the forms Ψ = ( ↓  ,  ↓  )  for spindown, and Ψ = ( ↑  ,  ↑  )  for spin-up electrons with  and  labelling the two sublattices and  denoting the transpose.With  = ±, the corresponding eigenvalues are We then leave the details of eigenvectors of (2) for the next sections where we study the single and double barriers scholar problems.

Single Barriers
In this section, we consider a single barrier of a constant height  and of width , along the -axis and infinitely long along the -axis.This barrier is schematically shown in Figure 1.Such a constant potential () is subjected then to the free particle Hamiltonian (2).Along the -direction, the eigenvectors are simple plane waves     , provided the electrons are free from any force there.The same happens for the -direction to the left (region I) and to the right (region II) of the barrier.So, for region I, the eigenvectors are written in the form of plane wave: where  is the reflection amplitude.For region III, it is in the following form: with  being the transmission amplitude.
In the barrier (region II), the eigenvector is written as Here, the parameters , with V  being the Fermi velocity.We mention, concerning the component   , that, through the diagonalization of (2), one can easily find from which the component   can be written for the three barrier regions.
The electronic transmission  is obtained after matching   and   at the interfaces between regions I and II and regions II and III.Within such a procedure, the transmission  =  * takes that standard form obtained whenever the Dirac-like Hamiltonian is used instead of Shröndiger's in calculating .After a straightforward (but quite cumbersome) calculation, the matching conditions lead us to the transmission coefficient for spin-up (+) and spin-down (−) electrons written as with As in the graphene case [16], we see  ± with clear dependence on   , which is absent (  = 0) in usual semiconductor barriers.But in contrast to it, now  depends on  ± as well, and, for  ± = 0, (8) gives the graphene result, naturally, [16] from which one analytically obtains  = 1 for normal incidence (  = 0).So, the electrons are totally transmitted when they are injected in a normal direction with respect to the barrier.In Figure 2(a) we show a (, ) contour plot of the transmission  ↑ for spin-up electrons for a single barrier.The width and height of the barrier are  = 110 nm and  = 100 meV, respectively.Intentionally, we set   = 170 meV/nm because this value of   , together with ℓ = 0.23 Å and  SOI = 3.9 meV, gives  − = 0 and makes the diagonal terms in (2) vanish.In this case we should obtain graphene's results [16] for V  = 10 6 m/s.This is indeed the case.
The plotted results though differ very slightly because we used V  = 5.5 × 10 5 m/s appropriate for silicene.The question then arises what influence changing   has on the results.To partially address that and in view of Figure 3 we focus on the region of the (, ) plane approximately delimited by −2 ≤  ≤ 6 and 90 ≤  ≤ 110 meV, in panels (b), (c), and (d) that correspond, respectively, to   = 50, 170, and 300 meV/nm.As can be seen, for values of   away from the "graphene" value   = 170 meV/nm the perfect transmission weakens considerably.
We now consider spin-down electrons and show in Figure 3(a) the corresponding (, ) contour plot of the transmission.
The results are similar to those in Figure 2(a) for most angles of incidence  except for normal or near-normal incidence, that is, for  ≈ 0. In this case, we have a gap in the transmission approximately for 90 ≤  ≤ 110 meV (Eq. (3) for  = 0 gives  = 0 for  =  ± + and  = +).Together with Figure 2 this means that we have a spin-resolved transmission.
Again, to assess the influence of the field   , we show in panels (b), (c), and (d) a small region containing the gap for the same values of   as in Figure 2. As can be seen, the main effect of increasing   from its "graphene" value   = 170 meV/nm is to increase the gap, while for   less than this value, the gap tends to close.We notice in passing that upon reversing the field   the factor  + in (2) becomes  − and the up spins will be blocked, as in Figure 3, whereas the down spins ( − →  + ) will be transmitted as in Figure 2.
With the results of Figures 2(a) and 3(a) we show in Figure 4(a) a (, ) contour plot of the polarization  defined by  = ( ↑ −  ↓ )/( ↑ +  ↓ ). vanishes in most of the (, ) plane.Notice, however, the regions where it does not and especially its 100% value for near-normal incidence and 90 ≤  ≤ 110 meV, which corresponds to that of the transmission gap in Figure 3(a).This is drastically different from the case of graphene in which  vanishes everywhere due to the smallness of  SOI and the vanishing of the distance ℓ between the two sublattices.This difference remains as sharp when we reduce the height .The plot in Figure 4 shown, the most important change is that the "red" window in (a), centred at  = 0,  = 100 meV, is now centred at  = 0,  = 50 meV, and nearly twice as wide.Again, in there  ↑ , controlled by the "graphene" value of   , is almost 1 and  ↓ ≈ 0. For the same  and   but different , however, the changes are more substantial; details will be given elsewhere.

Double Barriers
In this case, the analytic result for the transmission is very unwieldy and will not be given.We present numerical results for spin-up electrons in Figure 5 with the parameters specified in its caption.Again for   = 170 meV/nm, we have the results for graphene for a (  ,   ) contour plot, since the diagonal terms in (2) vanish.As in that case we see many more resonances than in single barriers.Upon contrasting these results, though, with those of Figure 6 for spin-down electrons, we see again qualitative differences especially for near-normal incidence with several gaps in the transmission.
That is, we have a spin-resolved transmission that is more complex than that for single barriers.Again, this structure of the transmission is reflected in the (, ) contour plot of the polarization  that we show in Figure 7.
Notice the quasiperiodic character of  versus the angle  for energies, say, larger than 80 meV.This contour plot contrasts rather sharply with that of Figure 4 for single barriers as we now see several windows of perfect polarization especially for near-normal incidence.
The results presented so far raise the question of their sensitivity to the parameters.We emphasize that all contour plots for double barriers are sensitive to their height  and the distance  between them.This is expected since more resonances can be created by changing  and .The results for single barriers show a similar but weaker sensitivity (cf.A detailed account of these sensitivities will be reported elsewhere.
Another concern about the results is that experimentally one usually measures the current, which is proportional to a weighted integral of the transmission.One then may wonder to what extent the spin-resolved features survive after such an averaging.We show in Figure 8 the conductance  using the standard formula: The red, dashed curve is for spin-up ( ↑ ,  − = 0) electrons and the blue, solid one for spin-down ( ↓ ,  + ̸ = 0) electrons.As shown in Figure 8(a), near the 2nd minimum of the two curves, we have a gap in  ↓ , as well as a dip for  = 10 meV, in agreement with Figure 3.In general, relative to graphene, the overall silicene results for  are similar but one sees important qualitative differences as well.This is more pronounced for double barriers for which  ↓ , pertinent to Figure 6, has four very deep minima or gaps as Figure 8 As one can notice, this work has not considered substrate effects, which is a very interesting issue to tackle theoretically [13,14].It is well know that the subtracts can even destroy the 2D feature of these materials.Indeed, this happened in the earlier graphene samples, where the Dirac cones vanish if the material was hosted by certain substrates.In some cases a SOI like parameter could mimic the coupling between the Dirac electrons and the substrate, since it recovers the massive Schrödinger behavior for the Dirac electrons in graphene [8,9].As a matter of fact, the concern on how to select the suitable substrate to keep the 2D features intact is actually an open experimental issue which is certainly beyond the scope of this paper.But, we emphasize that our results are very reliable to easily probe the spin polarization strength by simply considering one K valley only in the 2D silicene.
In summary, we evaluated the electronic transmission through single and double barriers in silicene as a function of the angle of incidence , the electron energy , and the

Figure 1 :
Figure 1: (Color online) Schematically potential profile showing regions I, II, and III and their relationship between the incident angle  and the eventual transmission  and reflection  = 1 −  through the barrier [15].

Figure 5 :Figure 6 :
Figure 5: (Color online) (a) (, ) contour plot of the transmission through a double barrier for spin-up electrons.The width and height of the barriers are  = 50 nm and  = 50 meV, respectively, the interbarrier separation  = 100 nm, and the field   = 170 meV/nm.

Figure 7 :
Figure 7: (Color online) Polarization for a double barrier.

Figures 3 (
Figures 3(a) and 3(b)).A detailed account of these sensitivities will be reported elsewhere.Another concern about the results is that experimentally one usually measures the current, which is proportional to a weighted integral of the transmission.One then may wonder to what extent the spin-resolved features survive after such an averaging.We show in Figure8the conductance  using the standard formula: (b) demonstrates.All gaps in Figure 8 could get wider by increasing the value of   (cf.Figures 3(a) and 3(d)).