Source: https://www.groundai.com/project/rash-spin-orbit-interaction-in-graphene-and-graphene-zigzag-nano-ribbons/
Timestamp: 2019-04-24 14:23:49+00:00

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We investigate the effects of Rashba spin-orbit (RSO) interactions on the electronic band-structure and corresponding wavefunctions of graphene. By exactly solving a tight-binding model Hamiltonian we obtain the expected splitting of the bands -due to the SU(2) spin symmetry breaking- that is accompanied by the appearance of additional Dirac points. These points are originated by valence-conduction band crossings. By introducing a convenient gauge transformation we study a model for zigzag nanoribbons with RSO interactions. We show that RSO interactions lift the quasi-degeneracy of the edge band while introducing a state-dependent spin separation in real space. Calculation of the average magnetization perpendicular to the ribbon plane suggest that RSO could be used to produce spin-polarized currents. Comparisons with the intrinsic spin-orbit (I-SO) interaction proposed to exist in graphene are also presented.
Understanding the mechanism for the generation and manipulation of spin polarized currents is one of the greatest challenges for the development of spin-based devices. Much of the advance in the field in latest years (1) has been achieved by studying semiconductor materials which make up the bulk of current electronic circuitry. Among the mechanisms proposed to induce spin-polarized currents, the spin Hall effect (SHE) appears as the most efficient one. The SHE refers to the phenomenon in which a spin-polarized current is created when an external bias voltage is applied to the system. The effect is based on a coupling between spin and momentum degrees of freedom, and usually the existence of some kind of spin-orbit (SO) interaction in the particular system under study is invoked. For bulk semiconductor materials for example, the SO interaction, has been proposed to lead to two different manifestations of SHE: a) the intrinsic SHE (2); (3) , in which the material inherits a strong SO interaction from its atomic constituents or due to its crystalline symmetries (lack of inversion symmetry) and b) the extrinsic SHE (4); (5) , in which spin-polarized currents appear as a consequence of electron scattering by a SO dependent scattering potential. In this last situation the scattering potential may be caused for instance, by magnetic impurities that couple via a spin-orbit term to the conducting electrons or by defects that produce spin-dependent scattering. Among these scenarios one possibility is when interfaces or surfaces are considered. In this case, the existence of the interface/surface introduces inversion symmetry breaking and thus, materials that do not fall into the categories cited above can also exhibit SHE. The effective SO interaction generated in this situation is known as the Rashba spin-orbit (RSO) interaction responsible for the Rashba effect (6) . Among the many materials in which RSO interactions could be exploited to obtain spin-polarized currents, graphene presents a unique and intriguing case. The material, first isolated as a single layer of graphite in 2004 (7) , gives access to a crystalline surface with linear dispersion around two independent points in its Brillouin zone, the Dirac points. The special dispersion plus its crystal structure (two-atom base triangular lattice) makes possible to calculate low-energy properties using Dirac-type models as the vast literature in recent years shows (8) . Furthermore, the two sublattices of the honeycomb structure makes appropriate the use of two-valued wave-functions, or spinors, for calculations of various properties. The relativistic description for low energy properties has also been used to argue for the relevance of additional terms in the standard Dirac Hamiltonian, including an intrinsic spin-orbit (I-SO) interaction represented by a second-neighbor spin-dependent hopping term that respects all the symmetries of the graphene plane (9) . One of the consequences of the I-SO interaction is to make possible the existence of spin-polarized edge states in a new phase of matter, the quantum Spin Hall (QSH) phase (10) . In previous works we have studied the physics introduced by this interaction in narrow graphene ribbons with armchair and zigzag edge terminations and in the presence of electron-electron interactions (11); (12) . We have shown that the I-SO interaction does not change the metallic behavior of armchair nanoribbons in contrast with the predicted result for graphene sheets. Moreover, the interaction produces spin-filtered states localized along the edges of the ribbon, (independent of edge termination) and, as a consequence, the current induced by an applied low external voltage is spin polarized. These results are in good agreement with several numerical and analytic studies (13); (14); (16); (15); (17); (18); (19); (20); (21); (22) that point to various magnetic instabilities that graphene ribbons may sustain, leading to some kind of magnetic order along the edges. Unfortunately, all numerical estimates for the strength of the I-SO interaction remain, although still controversial, rather small, in the range of 0.05−0.0011meV(600−13mK) (23) .
However, as a pure two-dimensional material, a graphene flake on a substrate lacks inversion symmetry and it is natural to expect that a RSO interaction may introduce important changes to the material properties. The RSO coupling λR is controlled by the applied bias and although predicted to be in the range of λR∼1meV (Refs. (24); (25); (26) ) recent experiments have shown that it can reach values up to λR∼200meV for graphene deposited on a Ni substrate (27) . Furthermore, experiments also have shown short spin relaxation times that suggest an important effect of spin-orbit interactions in graphene (28); (29); (30); (31) .
These new developments highlight the need for a better understanding of the role played by the RSO interaction on various properties of graphene and graphene ribbons. In this work we address the questions raised by the presence of RSO in graphene sheets and zigzag ribbons. The RSO interaction strongly affects the dispersion relation near the two-independent Dirac points as well as the nature of the corresponding wave-functions as we will show below. As a SU(2) breaking symmetry interaction, it favors a spatial spin ordering but in contrast to the I-SO interaction introduced above, the spin order is not originated on each individual state having the same spatial spin distribution but it emerges from averaging over several states.
where a=2.4 \AA is the lattice constant [see Fig. (1)].
Figure 1: Left panel: graphene lattice. Lengths along the x and y directions are measured in units of a and b=a√3/2 respectively. Right panel: First Brillouin zone. K=(4π/3a,0) and K′=(2π/3a,π/b) are Dirac points.
with φ=|φ|eiα0 and N the normalization factor. For neutral graphene, Ψ+ (Ψ−) represents solutions with E>0 (E<0) and refers to electron (hole) conduction (valence) bands. In this language, the particle-hole symmetry implies that for each electron state with energy E=ε and eigenstate characterized by α0, there is a hole state with E=−ε and eigenstate given by α0+π.
where →E is the applied electric field in the direction perpendicular to the graphene sheet, d=a/√3 is the distance between the two adjacent sites (i,j) and →δij=δj−δi is a vector on the graphene plane.
where (ξ,η) stand for spin up and down.
Note that ν is independent of the RSO coupling.
Equation (12) shows explicitly that E→−E, representing the particle-hole symmetry, is preserved by the RSO interaction. Using this property, in the rest of the paper we will focus on conduction bands only.
In Fig. (2) and Fig. (3) we plot few bands for the infinite graphene plane in the presence of the RSO interaction. Two new features appear: a) due to the breaking of the SU(2) symmetry, each of the degenerate bands (in the absence of spin orbit interaction) splits into two. For a given (kx,ky) the energies of the newly separated upper (Eu) and lower bands (El) are related by E2u−ε20−ε2±=−(E2l−ε20−ε2∓). b) The inset of Fig. (2) shows the change in the band with ky=π which originally touches the corresponding valence band at a Dirac point located at K=(2π/3a,π/b).
In the linear (Dirac) approximation of the Hamiltonian and for small values of the interaction strength λR, the splitting of the Dirac points is missed, and the low-energy effective Hamiltonian describes graphene with RSO interactions as a zero-gap semiconductor as reported in previous works (9) .
It is important to remark that the RSO interaction does not open a gap in the spectrum at the Dirac point, in contrast to the I-SO interaction mentioned above.
Figure 2: Energy bands as function of kx of an infinite graphene plane without (small dotted lines) and with (filled circles and empty diamonds) RSO. The strength of the RSO interaction is λR=0.2t. The plotted bands correspond to ky=π (lower set) and ky=π/2 (upper set). The inset shows the new lower energy band with ky=π and the splited Dirac point. A finite gap separates all other conduction and valence bands.
Figure 3: Energy bands as function of ky of an infinite graphene plane without (dotted lines) and with (full lines). The strength of the RSO interaction is λR=0.2t. The plotted bands correspond to kx=0.6π. As described in the text there are four degenerate states at ky=±k1 and ky=±k2 for each kx value.
Figure 4: The positions of new zero-energy (Dirac) points in momentum space for different values of the RSO interaction strength λR. The distribution of the new Dirac points around the original ones has the 2π/3 rotational symmetry of the graphene lattice.
In the limit λR→0, √E+E−→1 and α→α0. The expressions above (Eq.(16)) correspond to a state with energy E in the split upper band and momentum (kx,ky). The sate with the same momentum components in the split lower band is obtained by the replacement ν→ν+π. The remaining particle-hole symmetric states (in the split valence bands) are obtained by taking α→α+π.
Wave function (16) has the property of uA↑=¯uB↓ and uB↑=¯uA↓. This reflects that, in the presence of the SU(2) symmetry breaking RSO interaction the probability of finding an electron in the spin up state i.e |uA↑|2+|uB↑|2 is equal to the probability of finding it in the spin down state i.e |uA↓|2+|uB↓|2. This is a direct consequence of the fact that the RSO interaction does not break time reversal symmetry. However, as we will see below, this dos not exclude the possibility of separating the spin-up and spin-down electron states and localizing them at different positions in the sample while preserving a zero net magnetization.
To study the interplay between confinement and the RSO interaction we analyze the case of zigzag graphene (ZGR) nanoribbons, defined according to Fig. (5).
Figure 5: Hard-wall boundary conditions for ZGR imposes uA=0 on the lower edge and uB=0 just before the upper edge (green dotted line). The deformed lattice shown on the right side, corresponds to a gauge transformation (see text) and it is equivalent of labeling both A and B sites in each zigzag line with the same y-coordinate. The horizontal oval shows our choice of the unite cell and the vertical oval shows the choice of the unit cell which corresponds to Eq.(20).
which correspond to choosing a unit cell along the vertical link in the right-side panel of Fig. (5).
Figure (6) shows the conduction bands of a ribbon with W=4b. Zigzag ribbons present two remarkable features as compared to graphene sheets: the momentum across the ribbon ky can take complex values between two Dirac points (34); (35); (36); (38); (37) , producing edge states and their band-structure depends on the width W or the number of zigzags chains N=W/b−1. It can be shown (39); (40); (41); (42) that in zigzag ribbons with odd number of chains N, the so-called ’zigzag/zigzag’ configuration, conduction and valence edge bands cross at kxa=π. In contrast, ribbons with even number of chains N, in the ’zigzag/anti-zigzag’ configuration, edge bands do not cross albeit remain degenerate at kxa=π.
Figure 6: Energy bands of a zigzag ribbon with W=4b in the absence of the RSO interaction. Each band is doubly degenerate due to the SU(2) spin symmetry. The edge band (diamonds) corresponds to an imaginary value for the label k in Eq.(22).
As seen in the previous section, ZGRs present the peculiar feature of edge states which remain highly quasi-degenerate at low-energies for wide ribbons. These states are expected to be strongly affected by the presence of a RSO interaction. Below we proceed to obtain the exact expressions for the band-structure and corresponding eigenstates for the ZGR with RSO interactions.
where N is the normalization factor. From these expressions it can be shown that ΨZGRA↑(y)=iΨZGRB↓(−y) and ΨZGRA↓(y)=iΨZGRB↑(−y).
Figure 7: Energy bands of a ZGR with W=4b and λR=0.4t. Different regions correspond to values of k1 and k2: I) both real, II) one real and the other one imaginary III) both imaginary, IV) complex with k1=k∗2.
where k0x is defined in Eq.(27).
It is interesting to notice that RSO interactions preserve the power law energy dispersion and edge bands crossing/anticrossing feature as shown in Figs. (8).
Figure 8: The upper panel shows the edge bands of a ribbon with W=3b (N=2) and λR=0.2t. Conduction and valence edge bands do not cross at the band center. In contrast, the lower panel shows a W=4b (N=3) ribbon with crossing bands. The full lines are fits using the expression in Eq.(28).
With the expressions obtained for the wave-functions, it is possible to calculate various quantities. In particular, Fig. (9) shows the spatial probability distribution for Sz, the z-component of the spin operator defined as <Sz>=|uA↑|2+|uB↑|2−|uA↓|2−|uB↓|2 for the lowest energy conduction band of a ribbon with W=4b and λR=0.4t.
The figure highlights the fact that the RSO interaction produces a clear spin polarization on the edge states of the ZGR. The non-homogeneous spin distribution across the ribbon is, however, highly dependent on the state considered. This is in contrast to the effect produced by the I-SO interaction where each state becomes spin-polarized with the same spatial spin distribution (12) .
Graphene ribbons show unique and interesting transmission properties due to its band-structure and the pseudo-spin nature of its wave-functions (39); (40) . The relativistic nature of the description normally used makes it necessary to understand further other relativistic effects that could alter their transport properties. In this work we have investigated the consequences of one of such interactions: the Rashba spin-orbit interaction that is expected to be relevant under applied external bias voltages. We have shown that in graphene sheets, the RSO removes the SU(2) spin degeneracy as expected while it does not open a gap in the spectrum. It does, however, introduce additional Dirac points in the Fermi surface at low energies due to crossings between valence and conduction bands.
Because of its peculiar edge band, zigzag graphene ribbons are potential candidates for spintronic applications. The edge bands are expected to be magnetically unstable and as such to be strongly affected by SO interactions. We have shown that the RSO in particular produces states that have spin polarization and are strongly localized along the edges. These states present opposite polarization at opposite edges and the spatial spin distribution is strongly dependent on the state under consideration. Without external fields the net spin polarization of the ribbon remains null as a natural consequence of the conservation of time reversal symmetry under the RSO interaction. However, these results suggest the possibility to obtain spin polarized currents if the states selected by an applied external voltage sustain an average non-zero spin polarization.
We acknowledge S. E. Ulloa and G. Diniz for useful discussions. This work was partially supported by NSF under grants N0 DMR 0710581 and PHY05-51164 and Ohio University BNNT funds.
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