Cubic Rashba Effect in the Surface Spin Structure of Rare-Earth Ternary Materials

D. Yu. Usachov , I. A. Nechaev , G. Poelchen , M. Güttler , E. E. Krasovskii , S. Schulz , A. Generalov , K. Kliemt , A. Kraiker , C. Krellner , K. Kummer, S. Danzenbächer, C. Laubschat, A. P. Weber , J. Sánchez-Barriga, E. V. Chulkov, A. F. Santander-Syro , T. Imai , K. Miyamoto, T. Okuda , and D. V. Vyalikh 4,6,* St. Petersburg State University, 7/9 Universitetskaya Naberezhnaya, St. Petersburg, 199034, Russia Department of Electricity and Electronics, FCT-ZTF, UPV-EHU, 48080 Bilbao, Spain Institut für Festkörperphysik und Materialphysik, Technische Universität Dresden, D-01062 Dresden, Germany Donostia International Physics Center (DIPC), 20018 Donostia/San Sebastián, Basque Country, Spain Departamento de Física de Materiales UPV/EHU, 20080 Donostia/San Sebastián, Basque Country, Spain IKERBASQUE, Basque Foundation for Science, 48013, Bilbao, Spain Max IV Laboratory, Lund University, Box 118, 22100 Lund, Sweden Kristallund Materiallabor, Physikalisches Institut, Goethe-Universität Frankfurt, Max-von-Laue Strasse 1, D-60438 Frankfurt am Main, Germany European Synchrotron Radiation Facility, 71 Avenue des Martyrs, Grenoble, France Helmholtz-Zentrum Berlin für Materialien und Energie, Elektronenspeicherring BESSY II, Albert-Einstein-Strasse 15, D-12489 Berlin, Germany Centro de Física de Materiales CFM-MPC and Centro Mixto CSIC-UPV/EHU, 20018 Donostia/San Sebastián, Basque Country, Spain Tomsk State University, Lenina Avenue 36, 634050, Tomsk, Russia Université Paris-Saclay, CNRS, Institut des Sciences Moléculaires d’Orsay, 91405, Orsay, France Graduate School of Science, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima 739-8526, Japan Hiroshima Synchrotron Radiation Center, Hiroshima University, 2-313 Kagamiyama, Higashi-Hiroshima 739-0046, Japan

Spin-orbit interaction and structure inversion asymmetry in combination with magnetic ordering is a promising route to novel materials with highly mobile spin-polarized carriers at the surface. Spin-resolved measurements of the photoemission current from the Si-terminated surface of the antiferromagnet TbRh 2 Si 2 and their analysis within an ab initio one-step theory unveil an unusual triple winding of the electron spin along the fourfold-symmetric constant energy contours of the surface states. A two-band k · p model is presented that yields the triple winding as a cubic Rashba effect. The curious in-plane spinmomentum locking is remarkably robust and remains intact across a paramagnetic-antiferromagnetic transition in spite of spin-orbit interaction on Rh atoms being considerably weaker than the out-of-plane exchange field due to the Tb 4f moments. DOI: 10.1103/PhysRevLett.124.237202 Spin-orbit interaction (SOI) combined with exchange [1] or Kondo interactions [2] in noncentrosymmetric twodimensional (2D) systems is the basis for novel magnetic materials with spin-polarized carriers at the surface. The underlying phenomena are the momentum-dependent spinorbit splitting of 2D states governed by the Rashba effect and the Zeeman-like exchange splitting, whose simultaneous action brings about an exotic spin structure very different from textbook examples [1][2][3][4].
The Rashba effect significantly influences the spin properties of the carriers in magnets even when the exchange interaction is much stronger than SOI [3]. The SOI-induced splitting and in-plane spin-momentum locking often deviate from the prediction of the classical [linear, R ð1Þ ] Rashba model with its helical effective magnetic field (EMF). In reality, the k dependence of the splitting is far from linear, and the spin-momentum locking is not orthogonal [5,6].
Vivid examples are semiconductor quantum wells [7,8] and oxide surfaces and interfaces [9][10][11][12][13], which prominently feature the so-called cubic, R ð3Þ , Rashba effect responsible for the nonlinear (∝ jkj 3 ) dependence of the spin-orbit splitting of 2D heavy-hole states [14][15][16]. Remarkably, the EMF in the R ð3Þ model has a different symmetry from the R ð1Þ model, so that the in-plane pseudospin of the heavy hole rotates three times faster in moving around the Fermi contour and is no longer orthogonal to k everywhere [9,[17][18][19][20].
Here, using spin-and angle-resolved photoelectron spectroscopy (SR-ARPES), we unveil a system that realizes a R ð3Þ effect for the true spin: the Si-terminated surface of the antiferromagnetic (AFM) TbRh 2 Si 2 (TRS). It belongs to a family of RET 2 Si 2 materials (RE and T are rare-earth and transition metal atoms, respectively) of the ThCr 2 Si 2 type [21]. We report the first observation of the exotic spin structure predicted ab initio for the RET 2 Si 2 family [3], which has the distinctive triple winding of the in-plane spin [1] along the fourfold-symmetric constant energy contour (CEC). In the present context, the term spin refers to the expectation value of the spin operator rather than to a pseudospin. We corroborate our observation with a calculation of the spin photocurrent within an ab initio one-step theory [22,23]. Furthermore, we derive from the full microscopic Hamiltonian a minimal relativistic k · p model that proves the observed spin structure to be due to the R ð3Þ effect.
The angle-resolved photoelectron spectroscopy (ARPES) experiments were performed at the I05 beam line of the Diamond Light Source, while the SR-ARPES measurements were conducted at the ESPRESSO instrument (BL-9B) of the HiSOR facility [24] and at the SR-ARPES instrument of RGBL-2 beam line at BESSY-II. These instruments are equipped with VLEED and Mott detectors for spin analysis, respectively. All the data were measured at ℏω ¼ 55 eV and linear polarization. Single crystalline samples of TRS were grown using a high-temperature indium-flux method [25]. The calculations in Fig. 1 were performed with the full-potential local orbital method [26] within density functional theory (DFT) (see Supplemental Material [27]). The ARPES spectra were calculated within the one-step theory [32], in which the outgoing electron is described by a (scalar-relativistic) time-reversed LEED state [33]. The initial states are eigenfunctions of a thick slab obtained from a large-size relativistic k · p Hamiltonian (see Supplemental Material [27]).
Antiferromagnetic TRS has the Néel temperature of ∼90 K. In the AFM phase, the Tb 4f moments in the ab plane are ferromagnetically ordered with out-of-plane orientation [34]. The neighboring planes of Tb are separated by silicide Si-Rh-Si blocks and the ordered Tb 4f moments couple antiferromagnetically along the c axis. Upon cleavage, the resulting surface exhibits either Tb or Si termination. The latter has surface states in a large projected band gap around theM point. Figure 1(a) shows Fermi contours in PM (paramagnetic) and AFM phases. The surface states are seen as four-point stars around the corners of the surface Brillouin zone (SBZ) with a strong splitting typical of the AFM-ordered Si-terminated RET 2 Si 2 materials [2, [35][36][37][38].
The DFT band structure alongside the ARPES from the Si-terminated surface of TRS is shown for PM in Fig. 1(b) and for AFM in Fig. 1(c). The three surface states are labeled α, β, and γ. The occupied states α and β are clearly seen in the experiment in perfect agreement with the theory. For the PM phase, the states α, β, and γ have only in-plane spin components S x and S y . The splitting obviously stems from the SOI of Rh in the noncentrosymmetric Si-Rh-Si-Tb surface block [2,3,38]. The inset in theM-X graph of Fig. 1(b) shows a magnified ARPES map of α with a well-resolved k-dependent splitting, which reaches 35 meV at the Fermi level (see the black arrows). In Black arrows in (c) indicate the 140 meV splitting of the α state in AFM phase. Violet-brown palette shows calculated surface-projected bulk states. For the surface states the in-plane spin polarization is shown in red and blue, while yellow and green highlight the positive and negative spin component S z , respectively. Spin-resolved ab initio CECs for the state α in the PM (d) and AFM (e) phase calculated at the binding energy of 0.23 eV. The in-plane spin orientation, S k , is indicated by pink (purple) arrows for the inner (outer) contour. In AFM phase, the color of the spin vector S ¼ S k þ S zẑ shows the sign of S z as in Fig. 1(c).
Figs. 1(b) and 1(c), the color indicates the orientation of the in-plane spin: clockwise (red) or anticlockwise (blue) assuming the origin atM [see the sketch of the SBZ in Fig. 1(a)]. Thus, the lineM-Γ is related toM-X by an anticlockwise rotation aroundM by π=4, whereby β and γ preserve their chirality (a classical Rashba behavior), whereas the chirality of α reverses: alongM-Γ the inner branch becomes red and the outer becomes blue [ Fig. 1(c)] indicating the rotation of spin by 3π=4.
The comparison of Figs. 1(b) and 1(c) suggests that the in-plane spin-momentum locking of the states α, β, and γ survives the transition from PM to AFM phase, i.e., the emergence of the ferromagnetic order within the Si-Rh-Si-Tb surface block. The experimentally observed large Zeeman-like splitting [ Fig. 1(c)] and the perfect symmetry of the split contours [ Fig. 1(a)] points to the out-of-plane orientation of the Tb 4f moments [38]. This gives rise to a sizable out-of-plane spin component S z of α and β and strongly enhances their splitting, cf. Figs. 1(b) and 1(c), i.e., the exchange field felt by these states is much stronger than the SOI. The splitting of γ is only slightly affected by the magnetization because of the negligible overlap with Tb.
The calculated spin orientation for the state α over the CECs aroundM for both PM and AFM phases is shown in Figs. 1(d) and 1(e). For the PM phase, the spin lies in plane with a curious triple winding around the fourfoldsymmetric contours. In Fig. 1(d), the points between which the spin rotates by 2π are shown by the encircled arrows. For the AFM phase, the sizable S z is due to the magnetization, Fig. 1(e). In spite of a relatively weak SOI on Rh atoms, the unique in-plane spin-momentum locking survives and remains practically unaltered in the AFM phase.
The essence of the triple winding is a fast rotation of the in-plane spin S k ¼ S xx þ S yŷ over a symmetry-irreducible fragment of CEC between the directionsM-X andM-Γ. Experimentally, we are limited by the fact that the spinquantization axes X, Y, and Z of the spin analyzer depend on the emission direction J: X and Y are perpendicular, and Z is parallel to J, the respective components of the photocurrent being J X , J Y , and J Z . Thus, the k points must be selected such that the actual direction of S k be close to X or Y axis and, at the same time, unambiguously manifest the effect. In particular, in Fig. 2(a), the energy distribution curves (EDCs) for the PM phase at two characteristic k points show the flip of the J X polarization, i.e., of the net-spin X photocurrent.
In the AFM phase, for each branch of α or β, S z is large in magnitude and does not change sign alongX-M-X and Γ-M-Γ. This is expected to manifest itself in a sizable J Z polarization of the same sign (see Supplemental Material [27]). In the SR-ARPES measurements, depending on the tilt and polar angles, the large S z also contributes to J X and J Y . In turn, J Z may be influenced by S k . An additional contribution to the J Z polarization may be caused by the light breaking the symmetry of the experiment. This is a matrix element effect (MEE), which depends on the light incidence and on the final state [23], and the present photoemission theory is instrumental in separating the different sources of J Z . However, the observed behavior of J Z cannot be reconciled with the assumption that the signal comes from a single magnetic domain: J Z has opposite polarization in points 1 (4) and 3 (6), the J Z polarization is weak, and atM it changes sign depending on the experimental geometry, cf. points 2 [X-M-X setup, Fig. 2 Fig. 2 Fig. 2(c). According to our theory, Fig. 2(d), this implies a presence of oppositely magnetized domains (see Supplemental Material [27] for details). If we assume that in the ground state they counterbalance each other, then the averaging over magnetization would destroy the out-ofplane spin and eliminate its contribution to J Z . This leads to the overall good agreement both at points 1 (4) and 3 (6) and atM. Also, the S k contribution to J Z turns out to be rather weak in the AFM phase, not only in the vicinity ofM but also far from it. Thus, the J Z EDCs in Fig. 2(c) are a manifestation of the MEE, since the influence of the initialstate spin is here unimportant.

(f)] in
Another manifestation of the MEE is that over a rather wide range aroundM the J Y polarization strongly deviates from the spin S k of the initial state, cf. Fig. 2(d) and 1(c), while sufficiently far fromM it follows S k , both in the theory and in the experiment. The good agreement between the theoretical and measured J Y and J Z in Fig. 2(c) fully supports the triple-winding interpretation of the measured spin structure.
To prove that the observed spin pattern is a manifestation of the R ð3Þ effect, let us focus on the PM phase. In the literature, the term "cubic" implies a specific form of the two-band Rashba effective Hamiltonian for the total angular momentum states jjm j i with the z projection m j ¼ AE3=2 (in particular for 2D heavy holes) [14,39,40]. There, the cubic spin-orbit splitting of these states is due to the term H ð3Þ , and σ ρ (ρ ¼ x, y, and z) are Pauli matrices referring to a pseudospin since they operate on the jjm j i states, as emphasized in Refs. [19,41]. To highlight the impact of the R ð3Þ effect on the pseudospin, the k-cubic contribution is expressed as a Zeeman-like term H ð3Þ and φ k being the polar angle of k. Thus, the EMF drives pseudospin to be collinear with B ð3Þ R at a given k {see Fig. 3(a)} [15]. As a result, the triple winding of the pseudospins with a complete 2π rotation of k is a hallmark of the cubic effect. In contrast, the R ð1Þ twoband Hamiltonian is H Fig. 3(a)].
In order to understand whether the observed triple winding of the true spin is a manifestation of the R ð3Þ effect, we draw on the microscopic approach of Refs. [3,4,42] to derive a two-band k · p Hamiltonian of the form [27] Here, ϵ is the surface-state energy atM. The effective mass MðkÞ ¼ M ð2Þ þ M ð4Þ k 2 includes a second order correction, and W describes the fourfold warping of the CECs. In Eq. (1), the parameterαðkÞ ¼α ð1Þ þα ð3Þ k 2 accounts for the conventional (orthogonal) in-plane spin-momentum locking governed by B R . The term withγ is responsible for the R ð3Þ effect we are interested in. Note that the respective EMF B ð3Þ ¼ k 3 ðsin 3φ k ; cos 3φ k ; 0Þ has a fourfold symmetry, whereas the "heavy-hole" field B ð3Þ R has a twofold symmetry [see Fig. 3(a)].
The spin polarization of the states α and β rapidly grows in moving away fromM. Being practically unpolarized in the vicinity ofM, they become almost completely polarized far from it [ Fig. 1(c)]. This is due to the interaction between α and β, and, naturally, the 2 × 2 Hamiltonian (1) does not have this property. However, if we associate the σ matrices in Eq. (1) with the spin [27] [Fig. 3(b)], the Hamiltonian turns out to accurately describe both the triple winding in the α contour [ Fig. 3(c)] and the single winding in the β contour [ Fig. 3(d)]. The spin of the state α follows the winding of B ð3Þ [Fig. 3(c)], revealing the dominance of the k AE -cubic term over the k AE -linear one. Actually, for this stateγ is almost twice as large asα ð3Þ (the microscopically obtained parameters are listed in Table I). By contrast, for the state β, the k AE -linear term dominates because hereγ is two times smaller thanα ð3Þ (Table I). This results in a single spin winding around the CEC [ Fig. 3(d)]. Thus, we conclude that the observed spin structure is an interference of the R ð1Þ and R ð3Þ effects that yields the winding of surface-state spin in Fig. 1(d).
In summary, with SR-ARPES we unveiled an unusual inplane spin structure of surface states with a triple winding of the spin along the constant energy contours in the exemplary antiferromagnet TbRh 2 Si 2 , a representative of a wide class of rare-earth ternary materials. The unique spin structure appears to be rather robust and persists when a strong out-of-plane magnetic order of the Tb moments sets in. This property is due to a cubic Rashba effect, whose strength can be tuned by a proper choice of the transition metal atom. Combined with a different orientation and strength of the exchange field near the surface, it can create a variety of fascinating spin patterns. A fundamentally important finding is that relatively light atoms like Rh may give rise to a distinct spin-momentum gyration stable in an exchange field much stronger than the spinorbit field.