Giant spin rotation under quasiparticle-photoelectron conversion: Joint effect of sublattice interference and spin-orbit coupling

Spin- and angular-resolved photoemission spectroscopy is a basic experimental tool for unveiling spin polarization of electron eigenstates in crystals. We prove, by using spin-orbit coupled graphene as a model, that photoconversion of a quasiparticle inside a crystal into a photoelectron can be accompanied with a dramatic change in its spin polarization, up to a total spin flip. This phenomenon is typical of quasiparticles residing away from the Brillouin zone center and described by higher rank spinors, and results in exotic patterns in the angular distribution of photoelectrons.

Spin-and angular-resolved photoemission spectroscopy is a basic experimental tool for unveiling spin polarization of electron eigenstates in crystals. We prove, by using spin-orbit coupled graphene as a model, that photoconversion of a quasiparticle inside a crystal into a photoelectron can be accompanied with a dramatic change in its spin polarization, up to a total spin flip. This phenomenon is typical of quasiparticles residing away from the Brillouin zone center and described by higher rank spinors, and results in exotic patterns in the angular distribution of photoelectrons. Establishing spin polarization of quasiparticles in crystals is of crucial importance for semiconductor spintronics, physics of metallic surfaces, and the emerging field of topological insulators. The latter typically comprises narrow-gap systems with strong spin-orbit coupling. Graphene, a zero-gap conductor, attracts attention due to its linear Dirac-Weyl energy spectrum and prospects for applications. Its quasirelativistic spectrum, manifesting itself in unconventional quantum Hall effect 1,2 and Klein tunneling 3 , is essentially a nonrelativistic phenomenon originating from two equivalent sublattices, A and B; their effect is conveniently accounted for by pseudospin. However, the relativistic effects that entangle their spin and pseudospin degrees of freedom 4 lift the spin degeneracy of the energy spectrum. Intrinsic spin-orbit interaction in graphene 5 is weak, not exceeding tens of µeV 6,7,8,9 , but breaking the up-down symmetry by a substrate can result in a substantial extrinsic spin-orbit coupling. Such a coupling of the scale of 10 meV was discovered by Varykhalov et al. 10 by spin-and angular-resolved photoemission spectroscopy (SARPES) techniques. Despite its modest magnitude, this coupling modifies essentially the zero-gap nonrelativistic spectrum. This makes spin-orbit coupled graphene an excellent platform for unveiling nontrivial effects of spin-orbit coupling on SARPES spectra.
In graphene the zeros of the gap are achieved in two nonequivalent corners of the Brillouin zone (BZ), K and K [ Fig. 1c]. In the vicinity of these points the nonrelativistic quasiparticles (electrons and holes) possess a quasirelativistic energy spectrum E(k) = ±γk, k being a quasimomentum counted relative to the K(K ) point 11 . Near the K point, the nonrelativistic Hamiltonian is H 0 = γ(σ · k), where the pseudospin σ = (σ 1 , σ 2 ) is a vector of Pauli matrices acting on (A, B) sublattices [ Fig. 1(a)]. The leading term in the extrinsic spin-orbit coupling is H so = 1 2 λ(σ × s) z , where s are Pauli matrices of the real spin and λ is the spin-orbit coupling constant 4,12 ; it couples spin to the pseudospin. We disregard the breaking of the (A, B) symmetry by the staggering potential of the substrate; for some substrates it ћν electron detector E,kx,ky,S is weak and graphene behaves as quasifreestanding 10,13 . The two-sublattice structure of graphene is known to result in an interference effect in the nonrelativistic ARPES spectrum that produces strong photoemission anisotropy but does not distort the shape of the isoenergy surfaces 14 .
In relativistic spectra, the joint effect of interference and the spin-pseudospin entanglement produces giant rotations of electron spins during photoemission resulting in drastic differences in the spin polarization of quasiparticles inside the crystal and photoelectrons in vacuum. This phenomenon is the focus of this paper.
where ν, µ = ±1. The spectrum consists of two ungapped and two gapped hyperbolas shifted by λ that are shown in Fig. 1(b) with their quantum numbers. The spectrum is similar to unbiased bilayer graphene 11 , but with different nature of eigenstates and narrower gap. In the new basis, matrices of the quasiparticle spin areŜ = σ 1 s, and their mean values in the (ν, µ) eigenstates are z being a unit vector perpendicular to the substrate. We note in passing that Ŝ is proportional to the group velocity ∂E νµ / ∂k. The sign of the chirality of eigenspinors (ϕ νµ , φ νµ ) is determined by ν, the sign of spin polarization by µ, and the sign of the energy by the product νµ.
The polarization of all branches is depicted in Fig. 1(b). Such a spin-polarized spectrum with |λ| ≈ 13 meV was deduced from SARPES data taken from graphene on a Au/Ni(111) substrate 10 for |E νµ (k)| |λ|; data for lesser energies are not available. According to Eq. (2), quasiparticles are in-plane polarized perpendicular to the momentum k, and the magnitudes of their spins depend on k and can even vanish. This is well compatible with the fact thatŜ 2 = 3, because spin-orbit coupling results in large nondiagonal components ofŜ in the 2×2 spin-pseudospin space, hence restrictions on its diagonal components ("crystal" spin) are relaxed. The scale of the anticipated effect follows from the ratio of H 0 and H so that is about k/k λ , where k λ = |λ|/2γ is a spin-orbit momentum. The term H 0 , Zeeman energy of the pseudospin in an effective magnetic field k, describes the k-dependence of the interference of Bloch waves scattered by two equivalent sublattices, and this mechanism affects the spin sector through the term H so . This spin renormalization is strongest for k/k λ 1 [ Fig. 1(b)] and leads to vanishing spin at the symmetry points K and K as a result of lattice interference. We emphasize that this is impossible for simple lattices, where ŝ 2 = 1 is maintained either by in-plane polarization 16 or by out of plane spin rotation as recently reported 17 .
While the effect of sublattice interference on the angular dependence of ARPES, unveiled by Shirley et al. 14 , is well documented for monolayer 10,13 and bilayer 13,18 graphene and for graphite 14,19,20,21 , the recent progress in studying quasiparticles by SARPES techniques, the novelty of results, and the large magnitude of spin-orbit coupling found in a number of systems 10,17,22,23,24,25,26,27,28,29,30 , all call for proper understanding of interference-generated spin patterns in photoemission. They are specific for the single-step transformation of Bloch spinors of quasiparticles into photoelectron plane waves with large in-plane (Brillouin) momentum and manifest themselves in singular k dependencies in the k → 0 limit. The Fano effect in atomic photoionization 31 and polarization-dependent interband transitions 32 never show such singularities. They also are absent in the single-step normal photoemission showing remarkable polarization properties depending on the incidence conditions 33 .
For calculating photoemission, one needs to employ detailed wave functions that are products of the components of envelope spinors Ψ νµ and Bloch basis functions. At the K point of the BZ [ Fig. 1(c)], the |B functions differ from the symmetrically equivalent to them |A functions by the phase factor (−ω), with ω = exp (i2π/3) 34,35 . The ω factor reflects the effect of sublattice interference on photoemission in terms of the envelope functions, while the specific form of |A is of no importance as long as small spin-orbit corrections depending on atomic form-factors 22 factorize out and can be disregarded. For nonrelativistic electrons (λ = 0) the interference factor in the photoemission intensity reduces to [(k − /k)(E/|E|) − ω] near the K point, k K; here k − = k x − ik y . The first term in the brackets originates from H 0 , and the second from the interference of outgoing photoelectrons (exp [iK · (R A − R B )] = ω, with R A,B for sublattice coordinates). The interference factor is related to the quasiparticle pseudospin whose mean value equals σ x − iσ y = −ω(E/|E|) for a quasiparticle with a momentum k along K. The resulting photoemis-sion is highly anisotropic as described by the large k limit of Eq. (5) and displayed in Fig. 2(b). For relativistic electrons (λ = 0) the envelope spinor Φ νµ (k), describing the flux of spin-polarized photoelectrons from the (ν, µ) eigenstate, can be found by adding the components of Ψ νµ (k) corresponding to the same spin polarizations of the basis spinors, supplied with the proper phase factors originating from |A and |B functions where superscripts indicate the upper and lower components of the spinors ϕ and φ. This spinor, defined with the accuracy to a factor depending on the atomic matrix element, intensity of the source, etc., describes the dependence of the spin flux on the azimuth θ [ Fig. 1(c)]. The final form of Φ νµ (k) is determined by the explicit form of the quasiparticle spinors normalized as Ψ νµ (k)|Ψ νµ (k) = 1. In particular, the total flux equals I νµ (k, θ) ≡ Φ νµ (k)|Φ νµ (k) I νµ (k, θ) = 1 + 2νµγk cos θ/ λ 2 + 4γ 2 k 2 .
In the large k limit, k k λ , it coincides with the well known result 13,14 , 1 + νµ cos θ, with its strong θ dependence originating from the sublattice interference. On the contrary, in the small k region, k k λ , emission is isotropic due to a strong spin-pseudospin entanglement.
Spins of photoelectrons coming from different branches are compared in Fig. 3(a-d) in a wider k range. For k k λ , the difference between gapped and ungapped branches strikes the eye: for the former spins follow a simple one-sided flow, while for the latter their direction strongly depends on k and θ. For k k λ , polarization is transverse only inside the bright sector. While the transverse polarization follows from the k → ∞ expansions of Eqs. (6) and (7), the convergence is nonuniform, and asymptotic expansions diverge along the "dark corridor". Near it, the signs of all s ⊥ νµ are opposite for quasiparticles and photoelectrons [ Fig. 3(a) and 3(c)]. Near the corridor, spins rotate at the scale of the azimuths of (1 + νµ cos θ) ∼ (λ/2γk) 2 that grows narrower with increasing k/k λ . The intensity is low and vanishes when k → ∞, but the regions of the anomalous spin polarization are wide enough to be accessible for experiment. Figure 4 displays energy resolved photoemission spectra that can be anticipated. Left panels of (a) and (b) show Lorentz-widened (width 0.2|λ|) spectra from all spectrum branches in two central sections, vs k ⊥ /k λ (with k = 0) and k /k λ (with k ⊥ = 0). Due to λ = 0, finite intensity persists in panel (b) even inside the "dark corridor". Right panels exemplify the energy dependence  Fig. 4(a)] are quite general. The width of the spectra originating from the lifetime of the holes left after the photoemission is the main obstacle for resolving spin polarized spectra. Because it decreases with decreasing hole energy due to the elimination of scattering channels 13 , performing measurements close to the emission threshold should increase the spectral resolution. From this standpoint, small λ provides advantages.
For in-plane symmetry preserving substrates, the restrictions imposed by competing mechanisms of SO coupling are essential only within a narrow vicinity of the degeneracy point controlled by intrinsic SO coupling [about 10 µeV∼ 10 −3 λ (Refs. 6,7,8,9)] and trigonal SO correc- It is seen from comparing Figs. 1(b) and 3(a-d) that the difference between the spins of quasiparticles and photoelectrons ( Ŝ νµ (k) and s νµ (k, θ)) differs between three 2π/3 sectors around the K point; for panels (a) and (b) of Fig. 3 it is minimal inside the internal sector that is bright. These data allowed reconstructing the general pattern of spin polarization in the k k λ range 10 . Unveiling the spin polarization in the k k λ range is more challenging. Actually, it is an open question whether reconstructing Ŝ νµ (k), related to four-component spinors Ψ νµ (k), from s νµ (k, θ), related to two-component spinors Φ νµ (k), is a well posed mathematical problem, and how much spin information is lost during photoemission. In any case, reconstruction of the k k λ region should essentially include the data from external sectors.
All calculations were performed for spin-orbit coupled graphene. However, the qualitative conclusions are general and are based on (i) a narrow-gap spectrum of quasiparticles with strong spin-orbit coupling described by four-component spinors, and the presence of the ω phase factors that requires (ii) existence of (nearly) equivalent sublattices and (iii) residence of quasiparticles away from the BZ center. These requirements are fulfilled for the Bi 1−x Sb x 26,29 and Pb 1−x Sn x Te 36 compounds. Therefore, we expect that their SARPES spectra of bulk electrons and the electrons from their Tamm-Shockley surface bands should manifest similar anomalies.
In conclusion, we demonstrated a dramatic effect of the spin-pseudospin entanglement on the SARPES spectra of spin-orbit coupled graphene. Similar effects are predicted for different narrow-gap spin-orbit coupled materials.
We thank H. Churchill, J. Williams, C. Marcus and O. Rader for discussions, and acknowledge support in part by NSF, DARPA, DOD, and Harvard Center for Nanoscale Systems.