Unveiling Giant Hidden Rashba Effects in Two-Dimensional Si$_2$Bi$_2$

Recently, it has been known that the hidden Rashba (R-2) effect in two-dimensional materials gives rise to a novel physical phenomenon called spin-layer locking (SLL). However, not only has its underlying fundamental mechanism been unclear, but also there are only a few materials exhibiting weak SLL. Here, through the first-principles density functional theory and model Hamiltonian calculation, we reveal that the R-2 SLL can be determined by the competition between the sublayer-sublayer interaction and the spin-orbit coupling (SOC), which is related to the Rashba strength. In addition, the orbital angular momentum distribution is another crucial point to realize the strong R-2 SLL. We propose that a novel 2D material Si$_2$Bi$_2$ possesses an ideal condition for the strong R-2 SLL, whose Rashba strength is evaluated to be 2.16~eV{\AA}, which is the greatest value ever observed in 2D R-2 materials to the best of our knowledge. Furthermore, we reveal that the interlayer interaction in a bilayer structure ensures R-2 states spatially farther apart, implying a potential application in spintronics.


I. INTRODUCTION
The spin-orbit coupling (SOC) combined with an asymmetric crystal potential at surfaces or interfaces induces spin-polarized electronic states, called as Rashba (R-1) spin splitting. [1][2][3] The Rashba states exhibit Mexican hat-like band dispersion with spin-momentum locking, which can be described by where α R , σ, and k represent a Rashba strength coefficient, a Pauli spin matrix vector, and a crystal momentum; andẑ indicates a direction of the local electric field induced by the asymmetric crystal potential. The unique physical properties of the Rashba states have been utilized to realize some crucial concepts in the spintronics, 4,5 such as spin field transistor 6,7 and intrinsic spin Hall effects. 8 It has, however, been reported that the Rashba spin splitting is strongly affected by local orbital angular momentum (OAM) L in a system with a strong SOC, [9][10][11][12] which can be described by the orbital Rashba Hamiltonian where p = γL × k is electric dipole moment produced by the asymmetric charge distribution, 9 and γ is a proportional coefficient. Since the Rashba effects from these two model Hamiltonians Eqs.
(1) and (2) may not be distinguished in band calculations, Eq. (1) may be used to extract the Rashba strength α R from the Rashba states.
Since the centrosymmetry guarantees the spin-degenerate electronic structures, only noncentrosymmetric system have been considered as candidates for the R-1 based spintronics applications. Recently, however, new insights were suggested that local symmetry breaking may induce "local Rashba" (R-2) spin splitting even in centrosymmetric materials 13,14 . In such systems, intriguingly, degenerate spin states protected by the centrosymmetry are spatially separated into each inversion partner, which can be experimentally detected by spinand angle-resolved photoemission spectroscopy in both bulk and two-dimensional (2D) materials [14][15][16][17][18] . Among materials exhibiting the R-2 effects, bulk systems are not suitable for utilizing the spatially-separated states because their localized spin states would be canceled out by their adjacent inversion partners. In the van der Waals (vdW) 2D materials, on the other hand, opposite spins in the degenerate states can be split into the top and bottom layers (or atomic sub-layers). Such a spatially-separated spin splitting is called spin-layer locking (SLL). [14][15][16] Even though a few experimental observations have shown clear evidences for the existence of the R-2 effects, the following important questions still remain unanswered. 1) Why do some R-2 materials exhibit parabolic band structure rather than the Mexican hat-like dispersion?
2) How can the R-2 SLL effect be distinguished from unavoidable substrate effects in the experiments? 19 3) Why does the degree of spin segregation depend on the band index of an R-2 material? In addition, some of R-2 materials display an energy gap between the upper and lower R-2 bands, which cannot be described by the conventional Rashba (R-1) model Hamiltonian given by Eq.
(1). Therefore, a new model Hamiltonian is required to correctly describe the R-2 SLL. Furthermore, to utilize the R-2 Rashba states in the spintronics applications, it is essentially demanded not only to understand the fundamental physics of the R-2 effects but also to search for 2D materials exhibiting strong R-2 SLL.
To answer and resolve these questions and issues, in this paper, we propose a novel vdW 2D material Si 2 Bi 2 with strong R-2 SLL and explore the physical origins combining the first-principles density functional theory and a model Hamiltonian. We found that the strong spin-orbit coupling (SOC) restricts wavefunction overlap between local inversion partner and enables the OAM to contribute to the band-selected Rashba effects, leading to the giant R-2 SLL. The Rashba strength of Si 2 Bi 2 was calculated to be 2.16 eVÅ, which is the greatest value ever observed in 2D R-2 materials to the best of our knowledge. In addition, we suggest that multilayer configuration may enhance the spatial segregation of spin splitting occurring only at the outermost surfaces, while diminishing almost completely at the inner ones due to the interlayer interactions. Such a stacking process eventually leads to the evolution from the R-2 to R-1 spin splitting.

II. COMPUTATIONAL DETAILS
To understand the underlying physics of the R-2 SLL in 2D Si 2 Bi 2 , we performed firstprinciples calculations based on density functional theory 20 as implemented in Vienna ab initio simulation package (VASP) 21 . The electronic wavefunctions were expanded by plane wave basis with kinetic energy cutoff of 500 eV. We employed the projector-augmented wave pseudopotentials 22,23 to describe the valence electrons, and treated exchange-correlation (XC) functional within the generalized gradient approximation of Perdew-Burke-Ernzerhof (PBE) 24 with noncollinear spin polarization. 25 For bilayer calculations, in which interlayer interaction cannot be neglected, Grimme-D2 Van der Waals correction 26 was added. To mimic 2D layered structure in periodic cells, we included a sufficiently large vacuum region in-between neighboring cells along the out-of-plane direction. The Brillouin zone (BZ) of each structure was sampled using a 30×30×1 k-point mesh according to the Monkhost-Pack scheme. 27 To describe and visualize the R-2 SLL, we included spin-orbit interaction in the all calculations, and evaluated the angular momentum-resolved spinor wavefunctions by projecting the two-component spinor into spherical harmonics Y α lm centered at ion index α with angular momentum quantum numbers (l, m). Here n and k are band index and crystal momentum, and the arrows ↑ and ↓ represent spin up and down. Such projected components were further manipulated to understand the contribution of each orbital angular momentum to the band structures and to generate the atom-resolved spin texture map.
To verify the SLL in our system, we quantify the spatial spin separation by introducing the degree of wavefucntion segregation (DWS) D(ψ σ k ) defined as 28 , with where σ =↑ and ↓, n is band index, and S i indicates the real space sector for the upper Bi-Si , for example, represents the wavefunction ψ ↑ n,k localized on the upper SL sector S α .

III. RESULTS AND DISCUSSION
Our earlier study reported that group 4 element X (X=C, Si, Ge, and Sn) can combine with group 5 element Y (Y=N, P, As, Sb, and Bi) to form stable layered compounds X 2 Y 2 . 29 .
These layered compounds can be classified into two groups by the crystal symmetries, one In the real I-Si 2 Bi 2 material, however, the underlying physics becomes much more complicated because SL-SL interaction also takes part in determining its electronic structure.  the spatially-separated R-2 spin splitting, but the inversion symmetry still guarantees the spin-degeneracy, which are schematically summarized in Fig. S2 in SI. There is, however, an inevitable interaction between the top and bottom SLs, which lifts their four-fold degeneracy via the wavefunction overlap even without SOC, as shown in Fig. 2 (e). The splitting energy ∆E I due to such interaction was calculated to be 0.13 eV at the Γ point. One could expect that SOC would split these bands further into two sets of Rashba bands, which is, however, contradictory to degeneracy guaranteed by the inversion symmetry. It was instead surprisingly found that turning on SOC converted two separated doubly-degenerate parabolic bands In view of previous results observed in other 2D R-2 materials, such as PtSe 2 16 or bilayer WeSe 2 14 , which have revealed the SLL phenomena, but still with parabolic bands similar to those shown in Fig. 2 (e), our hidden Rashba bands shown in Fig. 2 (f) are exceptionally unusual since they look like the Rashba-like bands shown in Fig. 2 (c). To answer what causes such distinction, we examined the pathway from the parabolic bands (Fig. 2 (e)) to the Rashba-like ones (Fig. 2 (f)) while manipulating the SOC strength λ/λ 0 ∈ (0, 1), where λ 0 is the real SOC strength of our I-Bi 2 Si 2 system. As λ increases, two split bands tend to form a Rashba-like bands through continuous change as shown in Fig. 3 (a). This guaranteed by Kramer's degeneracy. Here m * is the effective mass, E 0 I and E 1 I the SL-SL interaction coefficients. To discover how to compete the SL-SL interaction with Rashba spin splitting, the DFT bands (black dots) were fitted to the model bands (red lines) given in Eq. (5) resulting in almost perfect agreement as shown Fig. 3 (a). The fitted parameters E int 0 and α R as a function of λ are shown in Fig. 3 (b). As expected, SOC weakens the SL-SL interaction, but strengthens α R , which was calculated to be 2.16 eVÅ at λ = lambda 0 . This value is much larger than those observed in metal surfaces, for example Au(111) (0.33 eVÅ), 32 Bi(111) (0.55 eVÅ), 33 as well as other materials exhibiting the R-2 SOC such as BaNiS 2 (0.24 eVÅ), 28 and is also comparable with conventional giant Rashba system, such as hybrid perovskites (1.6 eVÅ), 34 BiSb monolayer (2.3 eVÅ), 35 or BiTeI (3.8 eVÅ), 36 . Therefore, we may classify our system into the first "giant hidden" Rashba material.
This result was further confirmed by |ψ ↓ CB1 (r)| 2 , obtained from the spin-resolved wavefunction yielded near the Γ point. As shown in Fig. 3 (c), it evolves from an even distribution on both SLs at λ = 0 toward a complete spatial segregation at λ = λ 0 , which is quantified by D(ψ) defined by Eq. (3), shown in Fig. 3 (d). For every λ value, we also reckoned the spatially-resolved spin map on the Bi top SL to verify the degree of the SLL, which shown in Fig. 3 (e). We emphasize that no sharp phase transition was observed and thus even for λ < λ 0 , the system exhibits the SLL while maintaining two parabolic bands due to appreciable SL-SL interaction. When λ becomes λ 0 eventually, all three features clearly reveal complete Rashba-like bands, wavefunction segregation and SLL implying that our system, 2D I-Si 2 Bi 2 possesses vastly strong SOC minimizing the SL-SL interaction.
On the other hand, we noticed that there is no Rashba spin splitting at the highest valence band (VB1) unlike at CB1, perusing the band structures shown in Fig. 1 (c). As shown in the inset of Fig. 1 (c) and Fig. ?? (a), there is nearly no p z orbital contribution at VB1, resulting in no OAM distribution to produce Rashba spin splitting even with strong SOC, which is clear from Eq. (2). This non Rashba feature observed in the VB1 was further confirmed by the spin texture and the wavefunction segregation computed on the VB1 shown in Fig. S3 (b) and (c) in SI. Therefore, to utilize an R-2 material in the spintronics application, its hidden Rashba SLL should be induced by the bands near the Fermi level, which possess the OAM perpendicular to the local electric field.
At this time, it is worth mentioning that the spin splitting was also observed in M-Si 2 Bi 2 with the broken inversion symmetry, which additionally lifts the degeneracy protected in the I-counterpart and guarantees the Dresselhaus spin splitting. 37 Intriguingly, we also observed a strong SLL in a few lowest conduction bands near the Γ point, as shown in Fig. S4 in SI.
Such strong SLL is also attributed to the OAM similar in its inversion counterpart. Here, we again emphasize that the OAM is an important factor to determine the R-2 SLL.
Since 2D materials usually form multilayers rather than monolayers, it is also of importance to understand the effect of the interlayer-or vdW interaction on the SLL phenomenon.
To do this, we constructed a bilayer of I-Si 2 Bi 2 with "AA" stacking which is still maintains the inversion symmetry, and investigated its electronic properties. Figure 4  Rashba effects are suppressed, as shown in Fig. 4 (b). To verify if the SLL in the bilayer is indeed from the CB2 and CB3, we represented the spinor wavefunctions squared, |ψ σ n | 2 in real space. In Fig. 4 (c), those for σ = ↑ and ↓ calculated on the n = CB2 clearly display spatially-segregated spin states demonstrating the strong SLL. It is worthy of mentioning that each band is still doubly degenerate since the bilayer configuration also possesses the inversion symmetry. In other word, it is the interlayer interaction that removes the R-2 effect at the inner surfaces, but the Rashba states survive only at the outer surfaces of the bilayer.
This spatial segregation would be utilized in some spintronics applications since one could control the spin behaviors only on the top surface without being influenced by those on the bottom one. We also noticed that when the layered R-2 materials become a bulk structure, Rashba spin splittings inside bulk region may be removed as seen in the bilayer, and only the surface Rashba states survive, implying that the R-2 SLL automatically changes to the R-1 spin splitting. It is, therefore, the local symmetry breaking that is the physical origin causing not only the R-2 SLL, but also R-1 effects.

IV. CONCLUSION
Using first-principles density functional theory, we predicted a new 2D material, which is layered Si 2 Bi 2 exhibiting the giant R-2 SLL. To understand an underlying physical origin of R-2 SLL, we performed first-principles calculations as well as solved a devised model Hamiltonian to describe the R-2 SLL. Through this model calculation, we found that there is a competition between the SOC and SL-SL interaction to reveal the R-2 SLL. As the former, as it increases, weakens the latter and strengthens α R leading to the giant hidden Rashba spin splitting. Furthermore we found that the R-2 SLL is also closely related to the OAM distribution. The Rashba strength in Si 2 Bi 2 was calculated to be 2.16 eVÅ, which is the greatest value ever observed in R-2 materials to the best of our knowledge. We also revealed from a bilayer case that the R-2 SLL can be removed at the inner surfaces due to the interlayer interaction, but remained spatially farther apart at the outer surfaces. This eventually leads to a conclusion that the R-1 effect is also originated from the same local symmetry breaking causing the R-2 SLL. Our findings may not only uncover the fundamental physics of R-2 SLL, but also provide a guidance for searching novel R-2 materials. where α R and k are Rashba strength parameter and crystal momentum. On the other hand, H I can be expanded, without losing generality, as

Unveiling Giant Hidden Rashba Effects in
Since we consider the Hamiltonian for small k, the matrix elements of H I becomes up to the 2nd order, Note that the opposite spin states on the different SLs may not give repulsive SL-SL inter-actions, and thus T, ↑ |H I |B ↓ = 0, and so on. Thus, the total Hamiltonian (H) becomes, To obtain its energy eigenvalues, we solved its characteristic equation |H − EI| = 0 to get which are two doubly-degenerate solutions satisfying the degenerate condition guaranteed by the inversion symmetry. These two equations correctly reproduce two asymptotic band behaviors, such as Mexican hat-like bands for H I ≈ 0 and two parabolic bands for H R ≈ 0.
The parameters in Eq. (S1) were determined by fitting the lowest conduction band (CB1) and second lowest one (CB2) computed by our first-principles calculation shown in Fig. ??
(a) in the main text as follow. From bands without SOC, m * and E 1 I were first determined to be m * = 0.15m e and E 1 I = 12.83 eVÅ 2 , respectively. With these values fixed, α R and E 0 I were then determined for nonzero SOC cases and summarized in