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

Authors: Diana Meneses-Gustin; Marcos R. S. Tavares
Journal: Advances in Condensed Matter Physics
 (2018)

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## Abstract

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 fieldEz. 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.


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## Body

## 1. 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 Ez 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 θ, its energy E, and the strength of the external field Ez. It was suggested that Ez, 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 atone valley only and probe theeventual 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 thedouble-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 [11–14] 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.The manuscript is organized as follows. Section2 is devoted to the theoretical formulation. Sections 3 and 4 explore, respectively, the pertinent cases of single and double-barrier structures. Our conclusions are given by comparing Sections 3 and 4.


## 2. Formalism

The one-electron Hamiltonian for carriers in a singleK valley in silicene is given by the following [4]:(1)HK=H++H+-H-+H--=s-ħvFk-iaλR2k-0ħvFk+-s--iλR1-iaλR2k+-iaλR2k+iλR1-s+ħvFk-0iaλR2k+ħvFk+s+,where vF is the Fermi velocity, a=0.386 nm the lattice constant, and k±=kx±iky. Also, s±=λSOI±lEz where λSOI≡λ≃3.9 meV is the strength of the SOI in the lattice and 2l is the distance between the two sublattices A and B. It has been noticed that the secondary (Rashba) spin coupling parameters λR1=10μeV and λR2=0.7meVare too small in comparison to the primary one λSOI. Here R1 and R2 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, λSOI. Therefore, we hereafter deliberately assume H-+=H+-≃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(2)H--=-λ-lEzħvFk-ħvFk+λ+lEz=-s+ħvFk-ħvFk+s+,with the Hamiltonian for spin-up electrons H++ being given by (2) with s+ replaced with -s-. The eigenvectors of (2) are straightforwardly written in the forms Ψ=(φA↓,φB↓)T for spin-down, and Ψ=(φA↑,φB↑)T for spin-up electrons with A and B labelling the two sublattices and T denoting the transpose. With μ=±, the corresponding eigenvalues are(3)Ekμ=μħ2vF2k2+s±21/2.We then leave the details of eigenvectors of (2) for the next sections where we study the single and double barriers scholar problems.


## 3. Single Barriers

In this section, we consider a single barrier of a constant heightU and of width W, along the x-axis and infinitely long along the y-axis. This barrier is schematically shown in Figure 1. Such a constant potential U(x) is subjected then to the free particle Hamiltonian (2). Along the y-direction, the eigenvectors are simple plane waves eikyy, provided the electrons are free from any force there. The same happens for the x-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:(4)φAI=eiηx+re-iηx,where r is thereflection amplitude. For region III, it is in the following form:(5)φAIII=teiηx,with t being thetransmission amplitude.Figure 1
(Color online) Schematically potential profile showing regions I, II, and III and their relationship between the incident angleθ and the eventual transmission T and reflection R=1-T through the barrier [15].In the barrier (region II), the eigenvector is written as(6)φAII=aeiδx+be-iδx.Here, the parameters δ2=[(U-E)2-s±2]/ħ2vF2-ky2, and η2=(E2-s±2)/ħ2vF2-ky2, with vF being the Fermi velocity. We mention, concerning the component φB, that, through the diagonalization of (2), one can easily find(7)φB=iħvF∂φA/∂x-kyφAs±+U-E,from which the component φB can be written for the three barrier regions.The electronic transmissionT is obtained after matching φA and φB at the interfaces between regions I and II and regions II and III. Within such a procedure, the transmission T=tt∗ takes that standard form obtained whenever the Dirac-like Hamiltonian is used instead of Shröndiger’s in calculating T. 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(8)T±=11+Fsin2Wδ,with F=b-b+/[2ηδ(s±-E)(s±+U-E)]2 and b±=[η(s±+U-E)±δ(s±-E)]2+vF2ky2. As in the graphene case [16], we see T± with clear dependence on ky, which is absent (ky=0) in usual semiconductor barriers. But in contrast to it, now T depends on s± as well, and, for s±=0, (8) gives the graphene result, naturally, [16] from which one analytically obtains T=1 for normal incidence (ky=0). So, the electrons are totally transmitted when they are injected in a normal direction with respect to the barrier.In Figure2(a) we show a (E,θ) contour plot of the transmission T↑ for spin-up electrons for a single barrier. The width and height of the barrier are W=110 nm and U=100 meV, respectively. Intentionally, we set Ez=170 meV/nm because this value of Ez, together with l=0.23 Å and λSOI=3.9 meV, gives s-=0 and makes the diagonal terms in (2) vanish. In this case we should obtain graphene’s results [16] for vF=106 m/s. This is indeed the case.Figure 2
(Color online) (a)(E,θ) contour plot of the transmission through a single barrier for spin-up electrons. The width and height of the barrier are W=110 nm and U=100 meV, respectively, the field Ez=170 meV/nm, and vF=5.5×105 m/s. Panels (b), (c), and (d) show the region (E,θ) delimited by -2≤θ≤6 and 90≤E≤110 meV and correspond, respectively, to Ez=50, 170, and 300 meV/nm.
(a) (b) (c) (d)The plotted results though differ very slightly because we usedvF=5.5×105 m/s appropriate for silicene. The question then arises what influence changing Ez has on the results. To partially address that and in view of Figure 3 we focus on the region of the (E,θ) plane approximately delimited by -2≤θ≤6 and 90≤E≤110 meV, in panels (b), (c), and (d) that correspond, respectively, to Ez=50, 170, and 300 meV/nm. As can be seen, for values of Ez away from the “graphene” value Ez=170 meV/nm the perfect transmission weakens considerably.Figure 3
As in Figure2 for spin-down electrons.
(a) (b) (c) (d)We now consider spin-down electrons and show in Figure3(a) the corresponding (E,θ) contour plot of the transmission.The results are similar to those in Figure2(a) for most angles of incidence θ except for normal or near-normal incidence, that is, for θ≈0. In this case, we have a gap in the transmission approximately for 90≤E≤110 meV (Eq. (3) for ky=0 gives T=0 for E=V±s+ and E=s+). Together with Figure 2 this means that we have aspin-resolved transmission.Again, to assess the influence of the fieldEz, we show in panels (b), (c), and (d) a small region containing the gap for the same values of Ez as in Figure 2. As can be seen, the main effect of increasing Ez from its “graphene” value Ez=170 meV/nm is to increase the gap, while for Ez less than this value, the gap tends to close. We notice in passing that upon reversing the field Ez the factor s+ in (2) becomes s- and the up spins will be blocked, as in Figure 3, whereas the down spins (s-→s+) will be transmitted as in Figure 2.With the results of Figures2(a) and 3(a) we show in Figure 4(a) a (E,θ) contour plot of the polarization P defined by P=(T↑-T↓)/(T↑+T↓). P vanishes in most of the (E,θ) plane. Notice, however, the regions where it does not and especially its 100% value for near-normal incidence and 90≤E≤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 P vanishes everywhere due to the smallness of λSOI and the vanishing of the distance l between the two sublattices. This difference remains as sharp when we reduce the height U. The plot in Figure 4(b) is drawn for the same values of W and Ez as in (a) but for U=50 meV. As shown, the most important change is that the “red” window in (a), centred at θ=0, U=100 meV, is now centred at θ=0, U=50 meV, and nearly twice as wide.Figure 4
(Color online) (a) Contour plot of the polarization for a single barrier using the results of Figures2(a) and 3(a). (b) As in (a) with U=50 meV.
(a) (b)Again, in thereT↑, controlled by the “graphene” value of Ez, is almost 1 and T↓≈0. For the same U and Ez but different W, however, the changes are more substantial; details will be given elsewhere.


## 4. 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 Figure5 with the parameters specified in its caption. Again for Ez=170 meV/nm, we have the results for graphene for a (ky,kx) 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 aspin-resolved transmission that is more complex than that for single barriers. Again, this structure of the transmission is reflected in the (E,θ) contour plot of the polarization P that we show in Figure 7.Figure 5
(Color online) (a)(E,θ) contour plot of the transmission through a double barrier for spin-up electrons. The width and height of the barriers are W=50 nm and U=50 meV, respectively, the interbarrier separation d=100 nm, and the field Ez=170 meV/nm.Figure 6
As in Figure5 for spin-down electrons.Figure 7
(Color online) Polarization for a double barrier.Notice the quasiperiodic character ofP versus the angle θ 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 thatall contour plots for double barriers are sensitive to their height U and the distance d between them. This is expected since more resonances can be created by changing U and d. The results for single barriers show a similar but weaker sensitivity (cf. 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 Figure8 the conductance G using the standard formula:(9)G=e2ħ∫TE,θEcos⁡θdθ. The red, dashed curve is for spin-up (G↑, s-=0) electrons and the blue, solid one for spin-down (G↓,s+≠0) electrons.Figure 8
(Color online) (a) Conductance through a single barrier versus energyE. The dashed and solid curves are for spin-up (G↑, s-=0) and spin-down (G↓, s+≠0) electrons, respectively. (b) Conductance through the double barrier of Figures 5–7. The Fermi velocity (energy) is given in the caption of Figure 2.
(a) (b)As shown in Figure8(a), near the 2nd minimum of the two curves, we have a gap in G↓, as well as a dip for E=10 meV, in agreement with Figure 3. In general, relative to graphene, the overall silicene results for G are similar but one sees important qualitative differences as well. This is more pronounced for double barriers for which G↓, pertinent to Figure 6, has four very deep minima or gaps as Figure 8(b) demonstrates. All gaps in Figure 8 could get wider by increasing the value of Ez (cf. Figures 3(a) and 3(d)).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 incidenceθ, the electron energy E, and the strength of the field Ez and obtained aspin-resolved transmission, especially for near-normal incidence, and a conductance qualitatively different in some ranges of E than that in graphene. The strength of the resolution depends on the barrier parameters. This is entirely due to the strong SOI in silicene and does not occur in graphene in which the SOI is very weak. The results may be useful in spintronics.



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*Source: 3165829-2018-03-18.xml*