Abstract
Recent angleresolved photoemission spectroscopy measurements on strong spin–orbit coupled materials have shown an inplane orbitaltexture switch at their respective Dirac points, regardless of whether they are topological insulators or ‘trivial’ Rashba materials. This feature has also been demonstrated in a few materials (Bi_{2}Se_{3}, Bi_{2}Te_{3} and BiTeI) though DFT calculations. Here we present a minimal orbitalderived tight binding model to calculate the electron wavefunction in a twodimensional crystal lattice. We show that the orbital components of the wavefunction demonstrate an orbitaltexture switch in addition to the usual spin switch seen in spin polarised bands. This orbitaltexture switch is determined by the existence of three main properties: local or global inversion symmetry breaking, strong spin–orbit coupling and nonlocal physics (the electrons are on a lattice). Using our model, we demonstrate that the orbitaltexture switch is ubiquitous and to be expected in many real systems. The orbital hybridisation of the bands is the key aspect for understanding the unique wavefunction properties of these materials, and this minimal model helps to establish the quantum perturbations that drive these hybridisations.
Introduction
In quantum systems, the key piece of information that describes the physics involved is the Hamiltonian and the wavefunctions of the system’s constituents.^{1–3} Typically, we are interested in the electron energies, momenta and spin states, i.e., the electronic band structure. However, with the recent interest in strongly spin–orbit coupled systems and topological materials, it is becoming clear that there is additional critical information, i.e., that pertaining to the orbital wavefunctions and symmetries, their relative phases and how they couple with the spin degrees of freedom of the material.
In threedimensional topological insulators (TIs), the inversion of an odd number of bands per unit cell leads to the necessity of a topological surface state with Diraclike dispersion, and a frequently described momentumlocked helical spin structure that is lefthanded above the Dirac point and righthanded below.^{3,4} However, such a description ignores the fact that the J states and not the spin states are the relevant eigenstates of the spin–orbit coupled system, so there must be a richer manifold of entangled spin and orbital states than described in this simplistic picture. This was shown by detailed angleresolved photoemission spectroscopy (ARPES) experiments in a prototypical TI Bi_{2}Se_{3}.^{2,3,5,6} As part of this physics, different orbital states directly couple with specific spins.^{7} In the case of Bi_{2}Se_{3}, the different orbitals can couple with spins that do not follow the net helicity of the spin bands.^{1,2,5} This has ramifications when considering the bands as being entirely spin polarised, as in reality they are a superposition of opposing spins coupled with different orbitals.
A similar situation exists for Rashba states, in which the conventional picture is the spin–split parabolic band.^{8,9} In this picture, the electron wavefunction is simply the two split bands with the spin component pointing in opposing directions. The opposite spins couple with the magnetic field (or broken inversion symmetry at the surface) and raise or lower the electrons energy. More precisely, a recent work^{1,10} has shown a complicated orbital and spin texture that is highly reminiscent of that of the TIs. In particular, there are spins of both helicities in the inner and outer Rashba bands, and these spins may couple with orbitals of different types. Elucidating the origin and underlying symmetry requirements for the spin behaviour and especially the orbitaltexture switch is the goal of the present paper.
In an earlier work,^{11} we used Density Functional Theory (DFT) to study the effects of spin–orbitinduced hybridisation in multiband solids, including both TIs with band inversion as well as Rashba bulk solids. In that work, we showed quite generally that Spin Orbit Coupling (SOC)induced hybridisation of different azimuthal orbital momenta leads to a truncation of the spin magnitude in each band below its maximal value of ±1, with different levels of spin truncation in different bands arising from different orbital textures in those bands. Distilling the minimal ingredients that drives such physics is, however, difficult to access from these DFT calculations.
Here we use an orbitally intuitive minimal model of Rashba states at the outset, both for solving the electronic structure problem and for explaining the crucial couplings responsible from the main effects, focusing especially on the crucial orbitaltexture switch, which has been observed to occur exactly at the Dirac points.
We show that the orbitaltexture switch is determined by the existence of three main properties: local or global inversion symmetry breaking, strong spin–orbit coupling and nonlocal physics (the electrons are on a lattice). Using our model, we demonstrate that the orbitaltexture switch is ubiquitous and to be expected in many real systems. The orbital hybridisation of the bands is the key aspect for understanding the unique wavefunction properties of these materials, and this minimal model helps to establish the quantum perturbations that drive these hybridisations.
Results
The Hamiltonian of equation (6) has three main components: orbital hybridisation ω and δ, spin–orbit coupling α, and the outofplane symmetrybreaking field γ. Figure 1 shows the band solution to the model Hamiltonian with parameter choices $\mathit{\alpha}=2.5$, $\mathit{\delta}=1.5$, $\mathit{\omega}=0.5$ and $\mathit{\gamma}=1$ that are reasonable for a typical ‘strong’ spin–orbit compound on a hexagonal lattice. Six bands are observed, as equation (1) begins with a sixstate basis. All bands, but especially the lowest pair of bands, exhibit a typical Rashbalike band structure corresponding to an ‘inner’ and ‘outer’ set of bands that are degenerate at Γ. All bands are made of a combination of various orbitals and spins, with the mixing ratios of the spins and orbitals determined when the Hamiltonian is diagonalised. The colouring of the bands in both Figure 1a,b indicate the orbital decomposition of the wavefunctions, including all three orbitals (Figure 1a) or the inplane orbitals only (Figure 1b). The upper four bands have principally inplane character (p_{x}, p_{y} or p_{rad}, p_{tan}) at the gamma point (blue/green), whereas the lower two bands have principally outofplane character at the gamma point (p_{z} or red). Ignoring minor splittings, these would nominally correspond to the J_{3/2} states (two upper branches) and the J_{1/2} states (lower branch), although from the diagrams it is clear that this nomenclature is only reasonable near the zone centre.
Figure 2 shows more details of the orbital and spin contributions of the lower pair of Rashbasplit states near the zone centre, over the kspace range shown by the rectangular box in Figure 1b. The left panels of Figure 2 show the breakdown for the outer states (bold, Figure 2a) and the right panels show the breakdown for the inner states. It can be seen from Figure 2b that at gamma, the p_{z} orbital (red) dominates the wavefunction of both inner and outer states, although this dominance quickly decays as one moves away from the gamma point. In addition, we can see that at gamma, the radial and tangential orbitals have a small and equal contribution to the wavefunction. As we move far away from gamma, the radial component quickly grows and the tangential and outofplane components decrease. In the inner bands, the tangential component initially raises in contribution, whereas the radial component initially decreases. This is the fundamental aspect of the orbitaltexture switch in these Rashba bands—one band picks up a radial contribution, whereas the other picks up a tangential one. Next, as it applies to spin (Figure 2c), we can see that in the outer bands, both the outofplane and the radial components, have righthanded helicity, whereas the tangential component carries a lefthanded spin. In the inner bands, the situation is reversed and the radial and outofplane components carry a lefthanded spin, and the now stronger tangential bands carry a righthanded spin. An important aspect here is that the p_{z} and radial states carry the same spin helicity, whereas the tangential states carry an opposite helicity, with all helicities switching when going from the inner to the outer Rashba band. This is identical to the situation discovered empirically for the Dirac state in the TIs Bi_{2}Se_{3} and Bi_{2}Te_{3},^{12} although here we show how it comes directly from a simplistic model.
The superposition of these opposing helicities in these bands can create unique spin polarisations and reduce the overall net magnitude of spin measured in experiments. This has been an issue in many TI experiments, and we demonstrate here that this feature should be expected to be present in nearly all Rashba materials (even if the effect is small). For the case of carefully selecting the light’s electric fields to be in the plane of the material, it is possible to ignore the outofplane orbital in the photoemission process, and therefore measure the spin of purely these inplane orbitals.^{2} These inplane orbitals have spin components that oppose each other, giving rise to complete control over the photoelectron spin. By coming in with normal incidence light (Efield in the plane of the sample so selecting only inplane orbital states), and changing the polarisation from linear horizontal, vertical, +sp, −sp, +circular and −circular, it should be possible to controllably and reproducibly eject photoelectrons with their spin along any arbitrarily chosen direction (x, y, z or anywhere in between). This as a technically feasibility has been demonstrated multiple times in recent ARPES measurements.^{6,7}
Figure 3 compares another aspect of this Rashba simulation with calculations and experimental data from the prototypical threedimensional TI, Bi_{2}Se_{3}. We can characterise the strength of the orbital polarisation through the orbital polarisation parameter λ, originally defined for the TIs Bi_{2}Se_{3} and Bi_{2}Te_{3} in ref. 2:
where I_{0} and I_{90} are the photoemission intensities along two orthogonal highsymmetry directions when using properly polarised incident photons, or equivalently, the projected orbital polarisations. Figure 3a shows the kdependence of the λ term for the two lower bands, inner and outer, in the Rashba system calculated here, whereas Figure 3(b) shows the kdependence of λ for the upper and lower Dirac cones in Bi_{2}Se_{3} and Bi_{2}Te_{3} calculated from DFTprojected intensities. Clearly, the trends of the two systems are extremely similar, with the main difference that the model Rashba system has a more marked inplane orbitaltexture switch (with magnitude approaching unity) than the TIs, which have maximum magnitude of ~0.5. In addition, we can simulate an expected ARPES spectrum of these Rashba bands. As expected, we see two concentric circles in kspace at a constant energy surface if we come in with ppolarisation (selecting outofplane orbitals). However, if we instead come in with a light polarisation in the plane of the material, we select the inplane orbitals and see arcs of ARPES intensity that have opposing directions. The outer Rashba band shows arcs topbottom, whereas the inner one shows arcs leftright. This can be compared directly with the measurement of Bi_{2}Se_{3} reproduced in Figure 3d, which shows that the upper Dirac cone exhibits the leftright arc pattern, whereas the lower Dirac cone exhibits a topbottom spectral intensity pattern.
Figure 4c,d shows cartoons that summarise our findings for both the nominal J_{1/2} Rashba bands (top) and surface Dirac bands from the TI compounds Bi_{2}Se_{3} and Bi_{2}Te_{3}. It can be seen for both materials that the bands actually built out of a superposition of orbitals. Shown in the cartoon, they are composed of 90% outofplane porbitals with a 10% contribution of inplane orbitals (coupled with their own spins). The outofplane orbitals couple with the traditional spin helicity expected in both Rashba and TI bands. Separately, the inplane orbitals actually couple with spin in a unique manner, giving a righthanded spin texture to both the inner and outer Rashba bands (upper and lower Dirac cones). The orbitals themselves are also not uniquely radial or tangential, and in fact, switch their dominance at the gamma point in both materials. For the Rashba bands, the inner band is dominated by tangential porbitals close to the gamma point, whereas the outer band is dominated by radial orbitals.
Discussion
The inversion symmetrybreaking term γ expanded to first order in crystal momentum k shows a linear dependence near gamma. This term hybridises the inplane and p_{z} orbitals, breaking the usual assumption where the bands would not interact at all. This interaction term additionally can cause an avoided crossing in the band structure, where at the anticrossing the two bands are strongly hybridised and demonstrate the most mixing.
When spin–orbit coupling is turned on, the degeneracy of the bands is lifted due to further mixing among each spin state. These spin states, however, are also coupled to orbital angular momentum, so the orbitals themselves must also mix. It is through this coupling that the bands can develop unique features such as orbitaltexture switches centred around various highsymmetry points. The most striking of these is at the gamma point, where the orbitaltexture switches from a radial to tangential texture, having direct consequence to experiments on the materials.
It is also possible to extend this model to nontwodimensional materials as well. By further extending the model in the standard tight binding approach, it will be possible to calculate the orbital texture of bands in materials with more complicated atomic bases. These materials may show unique orbital textures for each atom type in the material, as the basis would be a summation of porbitals on each likeatom in the material. This will help further understand experiments that are sensitive to the depth of the material.
Here we presented a simple model with few restrictions: local or global inversion symmetry breaking, strong spin–orbit coupling and nonlocal physics (electrons on a lattice). This model shows the orbitaltexture switch seen in ARPES studies.^{1,2,5} This model also predicts that this feature is not unique to these materials, and in fact should be present in all strong spin–orbit coupled materials with broken inversion symmetry. This would suggest that this feature is as ubiquitous as the classic spin splitting seen in spin–orbit coupled models. This feature is also not restricted to materials on a hexagonal lattice, and we predict orbitaltexture switches to be present on square or rectangular lattice materials as well. Through this simple model, it is possible to understand the underlying physics of seemingly exotic experimental observations that happen in strong spin–orbit coupled materials.
Materials and methods
We shall model Rashba bands with a tight binding model of a twodimensional sheet of hexagonal or square lattice atoms. Each site will have the atomic states p_{x}, p_{y} and p_{z} orbitals centred on them, where the z axis is perpendicular to the plane of atoms. We chose to neglect sorbitals because they have no angular momentum, therefore not contributing to spin–orbit coupling. The basis set chosen is that used by many other DFT projections in the field.^{5} As there is strong spin–orbit coupling, the basis set cannot be separated into spin and orbital components separately. The basis set must instead contain a full set of spins and orbitals assuming there is coupling of each orbital with any arbitrary spin. To account for this, the basis is
where p_{i} are porbitals in the three Cartesian directions (i=x, y, z), and σ_{z} is the spin component in the outofplane direction.
We take the Hamiltonian from Peterson and Hedergard.^{9}
Where
We then add spin–orbit coupling in the atomic basis form:
with
Where the basis set is ${p}_{x},{\sigma}_{z}^{+}\u3009,{p}_{x},{\sigma}_{z}^{}\u3009,{p}_{y},{\sigma}_{z}^{+}\u3009,{p}_{y},{\sigma}_{z}^{}\u3009,{p}_{z},{\sigma}_{z}^{+}\u3009,{p}_{z},{\sigma}_{z}^{}\u3009$
The γ term in the Hamiltonian allows for the hopping of an electron from an inplane porbital to a neighbouring atom’s p_{z} orbital, and in this simplest twodimensional model will only be present if there is an outofplane distortion or field. In a bulk threedimensional system, this will usually come from a surface term (the classic Rashba effect) though it can also come from an intrinsic symmetrybreaking field or distortion.^{8,13–15}
This hopping shows up in the Hamiltonian as offdiagonal elements between inplane and p_{z} orbitals. These hopping elements of the Hamiltonian develop a momentum dependence, having no interaction at k=0 (gamma point). These terms mix the basis states further than just the offdiagonal SOC terms, which are kindependent.
This entire Hamiltonian has some symmetries by design. First is the crystal symmetry, chosen here as a hexagonal, rectangular or square lattice. This lattice allows for the electrons to hop, therefore bringing in a nonlocal momentumdependence despite being built out of localised atomic orbitals. Next, there is outofplane inversion symmetry breaking, i.e., the γ terms. Last, there is spin–orbit coupling in the form explained previously. As we will show, it is with the combination of all three of these ingredients that we produce the unique orbitaltexture switch observed in the experiments and the DFT calculations. Other interesting features of these states such as ‘backwards’ and/or ‘partial’ spin polarisation are also readily duplicated and understood using these simple terms.
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Acknowledgements
This work was funded by NSF DMREF project DMR1334170 to the University of Colorado and the University of Kentucky. The Advanced Light Source is supported by the Director, Office of Science, Office of Basic Energy Sciences, of the US Department of Energy under contract no. DEAC0205CH11231.
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Waugh, J., Nummy, T., Parham, S. et al. Minimal ingredients for orbitaltexture switches at Dirac points in strong spin–orbit coupled materials. npj Quant Mater 1, 16025 (2016). https://doi.org/10.1038/npjquantmats.2016.25
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