⇦Chinese Physics Letters, 2020, Vol. 37, No. 8, Article code 087105Express Letter Symmetry-Assisted Protection and Compensation of Hidden Spin Polarization in Centrosymmetric Systems Yingjie Zhang (张英杰)1†, Pengfei Liu (刘鹏飞)1†, Hongyi Sun (孙红义)1†, Shixuan Zhao (赵世轩)1, Hu Xu (徐虎)1, and Qihang Liu (刘奇航)1,2,3* Affiliations 1Shenzhen Institute for Quantum Science and Technology and Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China 2Guangdong Provincial Key Laboratory for Computational Science and Material Design, Southern University of Science and Technology, Shenzhen 518055, China 3Center for Quantum Computing, Peng Cheng Laboratory, Shenzhen 518055, China Received 23 June 2020; accepted 14 July 2020; published online 21 July 2020 This work was supported by the National Natural Science Foundation of China (Grant No. 11874195), the Guangdong Provincial Key Laboratory of Computational Science and Material Design (Grant No. 2019B030301001), and the Center for Computational Science and Engineering of SUSTech.
†These authors contributed equally to this work.
*Corresponding author. Email: liuqh@sustech.edu.cn Citation Text: Zhang Y J, Liu P F, Sun H Y, Zhao S X and Xu H et al. 2020 Chin. Phys. Lett. 37 087105    Abstract It was recently noted that in certain nonmagnetic centrosymmetric compounds, spin–orbit interactions couple each local sector that lacks inversion symmetry, leading to visible spin polarization effects in the real space, dubbed “hidden spin polarization (HSP)”. However, observable spin polarization of a given local sector suffers interference from its inversion partner, impeding material realization and potential applications of HSP. Starting from a single-orbital tight-binding model, we propose a nontrivial way to obtain strong sector-projected spin texture through the vanishing hybridization between inversion partners protected by nonsymmorphic symmetry. The HSP effect is generally compensated by inversion partners near the ${\varGamma}$ point but immune from the hopping effect around the boundary of the Brillouin zone. We further summarize 17 layer groups that support such symmetry-assisted HSP and identify hundreds of quasi-2D materials from the existing databases by first-principle calculations, among which a group of rare-earth compounds LnIO (Ln = Pr, Nd, Ho, Tm, and Lu) serves as great candidates showing strong Rashba- and Dresselhaus-type HSP. Our findings expand the material pool for potential spintronic applications and shed light on controlling HSP properties for emergent quantum phenomena. DOI:10.1088/0256-307X/37/8/087105 PACS:71.70.Ej, 85.75.-d, 61.50.Ah © 2020 Chinese Physics Society Article Text Spin polarization, generally measured by the difference between spin-up and spin-down electrons for a given direction $S=(I_{\uparrow}-I_{\downarrow})/(I_{\uparrow}+I_ {\downarrow})$, was thought to be limited to magnetic materials such as Fe or Ni. The current paradigm, however, shows that spin polarization is possible also in nonmagnetic crystals with strong spin-orbit coupling (SOC) and broken inversion symmetry.[1] Recently, it was shown that in certain centrosymmetric crystals where the energy bands are at least two-fold degenerate, the inversion-asymmetric sectors also manifest Rashba- or Dresselhaus-type spin textures.[2] For a local sector (named A), such as a monolayer of 2H-MoS$_{2} $[3,4] and a BiS$_{2}$ sublayer of LaOBiS$_{2} $,[5,6] such a form of “hidden spin polarization” (HSP) is calculated by projecting the Bloch wavefunctions $\psi_{n}({\boldsymbol k})$ of the doubly degenerate bands on sector A, which can be written as follows: $$\begin{alignat}{1} {\boldsymbol S}_{\mathbb{N}}^{\rm A}({\boldsymbol k})=\sum\limits_{i\in A} \sum\limits_{n\in \mathbb{N}} \langle \psi_{n}({\boldsymbol k})\vert (\mathbf{\boldsymbol \sigma}\otimes\vert i \rangle \langle i \vert)\vert \psi_{n}({\boldsymbol k})\rangle,~~ \tag {1} \end{alignat} $$ where ${\boldsymbol k}$, $\mathbf{\boldsymbol \sigma}$, and $\vert i \rangle$ denote the wavevector, Pauli operator, and the localized orbitals belonging to sector A. The band index $\mathbb{N}$ implies a summation over the two bands with degenerate energy. As an observable, ${\boldsymbol S}_{\mathbb{N}}^{{\rm A}({\rm B})}({\boldsymbol k})$ is independent of the choice of any unitary basis transformations of the two-dimensional (2D) wavefunction space, holding the potential for detectability. While HSP is an intrinsic property widely existing in crystals with inversion-asymmetric sectors, the word “hidden” implies that in general HSP cannot be directly measured without inversion symmetry breaking, but can be detected by local-probe techniques that distinguish different sectors. By using spin- and angle-resolved photoemission spectroscopy (ARPES) measurements, the HSP effect has been experimentally confirmed in a number of centrosymmetric bulk materials.[7–10] Very recently, HSP has been reported in traditional cuprates and thus prompts the open question of how high-temperature superconductivity correlates with such a nontrivial spin pattern.[11] The discovery of HSP considerably broadens the range of materials for potential spintronic applications and brings exotic physical insights into the existing fields, such as spin field effect transistors,[5] spin switching of antiferromagnets,[12] topological insulators[13] and topological superconductivity.[14] Furthermore, the concept of HSP has triggered a broader field of “hidden polarization”, where various physical effects have been recognized to be determined by the local symmetry breaking of a system, albeit with a higher global symmetry that seemingly prohibits the effect from happening. Examples include orbital polarization,[15] optical activity,[16] circular polarization,[3,17] Berry curvature,[18] Ising superconductivity,[19,20] etc. Based on the symmetry analysis of the atomic sites or sectors,[2] most centrosymmetric materials would be expected to manifest HSP, especially for quasi-2D layered materials with finite thickness. However, materials with strong HSP that hold realistic potential for applications indeed form a much smaller subgroup. For example, silicon, germanium and tin in diamond structure are nominal Dresselhaus-type HSP systems because each $X$ ($X =$ Si, Ge, or Sn) atom has an inversion-asymmetric $T_{\rm d}$ site point group, leading to opposite Dresselhaus spin textures localized at the two $X$ sublattices. However, the strong coupling between the two $X$ sublattices (hopping term through the $X$–$X$ bond) usually leads to strong hybridization between the two sublattices and thus significantly compensates for the HSP localized on each $X$ atom. Therefore, after completing “proof of existence” for HSP, the next important question is which specific physical features within centrosymmetric crystals actually control the magnitude of such HSP effect, i.e., “where to look”.[21] In addition to strong SOC that favors heavier elements, the design principle of HSP also requires to minimize the hybridization between sectors. One obvious but trivial way is to separate the two sectors as far as possible. This is analogous to the thick slab of a topological insulator with inversion symmetry, in which the top and bottom surfaces manifest energy-degenerate Dirac cones and spin polarizations with opposite chirality.[22] Such a type of HSP is fundamentally two individual sets of spin polarization in quite different places, which is difficult to manipulate and integrate in one system. Here, we explore a nontrivial, symmetry-assisted approach to minimize the hopping effect between the inversion-asymmetric sectors and thus protect the HSP in each sector. Our tight-binding model reveals that when the two sectors connect with each other by nonsymmorphic crystalline symmetry, the electron hopping between sectors vanishes along the Brillouin zone (BZ) boundary and is strongly suppressed in the vicinity of the time-reversal invariant (TRI) momenta. As a result, a spin map constructed throughout the full BZ shows that nearly perfect HSP survives even though the two sectors are close to each other in a quasi-2D lattice. We further perform symmetry analysis and identify 17 quasi-2D layer groups (out of 80) in total that support such symmetry protected HSP. Among the realistic materials in the existing quasi-2D material databases[23,24] with the selected layer groups, we perform first-principle calculations and choose a rare-earth family LnIO (Ln = Pr, Nd, Ho, Tm, and Lu) as representative candidates showing strong Rashba and Dresselhaus HSP, which is in excellent agreement with our tight-binding model. Our finding offers a general principle for extensively exploring strong HSP materials and provides an ideal platform to control HSP for emergent physical properties. One possible example is to tune the hybridization gap of such an HSP system by tensile strain to realize time-reversal-invariant topological superconductivity.[14,25,26] HSP from a Single-Orbital Tight-Binding Model. We begin with a simple model of a nonsymmorphic 2D lattice to illustrate how HSP in one sector survives under the existence of its inversion partner. Considering a square lattice with only two identical atoms (A and B) in a unit cell with a small displacement between them along the $z$ direction [see Fig. 1(a)], such a buckling structure has a $p4/nmm$ layer group with glide mirror reflection {$M_{z}$|(1/2,1/2,0)} and screw axis operations {$C_{2x}$|(1/2,0,0)} and {$C_{2y}$|(0,1/2,0)}. In addition, the buckling also creates opposite local polar fields felt by each sector (here, each atom forms a sector) along the $\pm z$ direction, rendering a Rashba-type HSP system. Therefore, we can easily construct the single-orbital (e.g., $s$, $p_{z}$ or $d_{z^{2}}$) tight-binding model (four bands) as follows: $$\begin{alignat}{1} H({\boldsymbol k})={}&t_{1}{\cos}\frac{k_{x}}{2}{\cos}\frac{k_{y}}{2}\tau_{x}\sigma_{0}+t_{2}(\cos k_{x}+\cos k_{y})\tau_{0}\sigma_{0}\\ &+\lambda_{\rm R}(\sin k_{x}\sigma_{y}-\sin k_{y}\sigma_{x})\tau_{z},~~ \tag {2} \end{alignat} $$ where $\mathbf{\tau}$ and $\mathbf{\sigma}$ are Pauli matrices describing the sector and spin degrees of freedom, respectively. The derivation of Eq. (2) is provided in Section A of Supplementary Material. The first (second) term describes the nearest (next nearest) neighbor hopping, while the third term presents Rashba SOC caused by local polar fields. The band structures of Eq. (2) is shown in Fig. 1(b). In the absence of SOC ($\lambda_{\rm R} = 0$), only the second term of Eq. (2) survives along the BZ boundary, leading to double degeneracy (excluding spin), i.e., the “band sticking” effect due to nonsymmorphic symmetry.[27] When including SOC, the energy bands along the BZ boundary generally split into two doubly degenerate bands, except at the high-symmetry TRI momenta $X$ and $M$. This is because at these points, a nonsymmorphic symmetry fulfills the anticommutation relationship with the inversion operator, leading to an extra two-fold degeneracy between two pairs of Kramers degeneracy, i.e., four-fold degeneracy.[28,29] Nonsymmorphic symmetry has recently drawn extensive attention due to such band degeneracy effects, leading to new types of quasiparticles, such as Dirac nodes,[29–32] nodal chains,[33] and nodal surfaces.[34] However, its impact on the spin textures as well as Bloch states of different sublattices has not yet been substantially explored.[35] To reveal this effect, we analytically solve Eq. (2) with the basis of $\vert A\uparrow \rangle$, $\vert A\downarrow \rangle$, $\vert B\uparrow \rangle$, $\vert B\downarrow \rangle$ (see Supplementary Section B). The resulting eigenstates are $$\begin{alignat}{1} &\phi_{1}=\frac{1}{\sqrt 2 } \left({\begin{array}{c} {\begin{array}{c} 1\\ D_{-}/F\\ \end{array} } \\ -M/F \\ 0 \\ \end{array}} \right),~ \phi_{2}=\frac{1}{\sqrt 2 }\left({ \begin{array}{c} { \begin{array}{c} 0\\ -M/F\\ \end{array} } \\ {-D}_{+}/F \\ 1 \\ \end{array}} \right),\\ &\phi_{3}=\frac{1}{\sqrt 2 }\left({ \begin{array}{c} {\begin{array}{c} 1\\ {-D}_{-}/F\\ \end{array}} \\ M/F \\ 0 \\ \end{array}} \right), ~\phi_{4}=\frac{1}{\sqrt 2 }\left({ \begin{array}{c} { \begin{array}{c} 0\\ M/F\\ \end{array} } \\ D_{+}/F \\ 1 \\ \end{array}} \right),~~ \tag {3} \end{alignat} $$ where $M=t_{1}\cos(k_{x}/2)\cos(k_{y}/2)$, $N=t_{2}(\cos k_{x}+\cos k_{y})$, $D_{+}=\lambda_{\rm R}(\sin k_{y}+i\sin k_{x})$, $D_{-}=D_{+}^{† }$, and $F=\sqrt {M^{2}+D_{+}D_{-}}$. In order to quantify and minimize the mixture of wavefunctions between different sectors, we then define a quantity named sector polarization $P_{n}^{{\sec}}({\boldsymbol k})=(\rho_{n}^{\rm A}-\rho_{n}^{\rm B})/(\rho_{n}^{\rm A}+\rho_{n}^{\rm B})$, where $\rho_{n}^{{\rm A}/{\rm B}}=\sum_{i\in {\rm A}/{\rm B}} \langle \phi_{n}({\boldsymbol k})\vert i \rangle \langle i \vert \phi_n({\boldsymbol k})\rangle$ is the module squared wavefunction projected onto sector A or B. Note that in quantum mechanics, if two eigenstates are degenerate, any linear combination of the two states is also an eigenstate of the system. Therefore, $P_{n}^{{\sec}}$ is gauge variant for a single branch of doubly degenerate bands, depending on the unitary transformation of basis that mixes $\phi_{1}$ and $\phi_{2}$ (while HSP is gauge invariant, see below). When a certain form of inversion-symmetry breaking is introduced, such as a tiny electric field along the $z$ direction, a definitive gauge (in this case {$\phi_{1}$, $\phi_{2}$} and {$\phi_{3}$, $\phi_{4}$}) is picked up. The sector polarization $P_{n}^{{\sec}}({\boldsymbol k})$ of the lowest doubly degenerate bands and eigenstates {$\phi_{1}$, $\phi_{2}$} are shown in Fig. 1(c). The most important feature is that at the BZ boundary $X$–$M$, $P_{n}^{{\sec}}$ is pinned to its maximum value $\pm$1, indicating a vanishing hopping effect between sectors A and B. In sharp contrast, the sector polarization at the $\varGamma$ point is exactly zero. These observations can be explained by the model Hamiltonian, where the off-diagonal matrix elements are contributed solely by the first term of Eq. (2) containing $\tau_{x}$. At the BZ boundary, the first term vanishes, and the Hamiltonian is the direct sum of two matrices separately exerted on two subspaces spanned by two sectors. Therefore, the two eigenstates are naturally chosen to be located in either sector A or B, leading to maximum sector polarization. This is analogous to the fact that two bands with different group representations do not hybridize when they meet each other, leading to a degenerate band-crossing point. In either case, the representation space of this special point can be spanned as a direct sum of two subspaces from the two crossing bands. On the other hand, when the wavevector moves to $\varGamma$ from the BZ boundary, the $\tau_{x}$ term of Eq. (2) becomes more predominant, leading to descending $P_{n}^{{\sec}}$ with the electron density finally distributed equally in A and B sectors at $\varGamma$. Note that if we do not consider any symmetry requirements but just separate the two sectors very far away from each other, the $\tau_{x}$ term of Eq. (2) also vanishes because of the negligible hopping parameter $t_{1}$. However, such a strategy is just a trivial overlap of two individual inversion-asymmetric systems in $k$-space and is not useful for real material design. The direct consequence of full sector polarization is that the HSP also reaches its maximum value at the BZ boundary. The sector-projected spin polarization is calculated by projecting the wavefunction on sector A or B, as written in Eq. (1). Remarkably, ${\boldsymbol S}_{\mathbb{N}}^{{\rm A}({\rm B})}({\boldsymbol k})$ is independent of the choice of unitary basis transformation, indicating an observable (see Section B of Supplementary Material). The sector-projected spin textures on sector A and B are shown in Fig. 1(d). At the BZ boundary, both inversion-asymmetric sectors retain large but opposite HSP. Such spin polarization signals can be measured by spin-ARPES when the local probe is located at sector A or B. In sharp contrast, around the $\varGamma$ point, the HSP localized at one sector is almost fully compensated by its inversion partner because the corresponding wavefunction contains a substantial mixture of the two sublattices, leading to vanishing $P_{n}^{{\sec}}$. Interestingly, we note that around the TRI momenta, the projected spin textures are Rashba-type at the $M$ point but Dresselhaus-type at the $X$ point, validating that the symmetry requirement of the Rashba effect always suggests an accompanying Dresselhaus effect.[2] Specifically, the low energy effective $k\cdot p$ Hamiltonians expanded by the SOC term of Eq. (2) takes the form of $(k_{x}\sigma_{y}-k_{y}\sigma_{x})\tau_{z}$ at $M$, while $(k_{x}\sigma_{y}+k_{y}\sigma_{x})\tau_{z}$ at $X$, indicating Rashba- and Dresselhaus-type spin patterns, respectively.
cpl-37-8-087105-fig1.png
Fig. 1. (a) A square lattice with two identical atoms (A and B) in one unit cell. The nonsymmorphic symmetry is caused by the atomic displacement along the $z$ direction. $t_{1}$ ($t_{2}$) is the nearest (next nearest) neighbor hopping parameter. (b) Band structure of this square lattice without and with SOC ($\lambda_{\rm R}/t_{1} = -0.2$). (c) Sector polarization ($P_{n}^{{\rm sec}}$) of the lowest two doubly-degenerate bands $E_{1}$ and $E_{2}$ for different strength of Rashba SOC $\lambda_{{\rm R}}$ induced by the local polar field. (d) Projected spin texture onto sector A (red) and B (blue) for the lowest two bands $E_{1}$ and $E_{2}$ ($\lambda_{{\rm R}}/t_{1} = -0.2$).
The sector polarization for different SOC strengths (quantified by $\lambda_{{\rm R}}/t_{1}$) is also shown in Fig. 1(c). The main variation lies in the descending trends from $M$ and $X$ to $\varGamma$. Along these directions, the sector polarization $P_{n}^{{\rm sec}}$ increases with $\lambda_{{\rm R}}/t_{1}$, indicating that in Eq. (2) the SOC term containing $\tau_{z}$ enhances the polarization, while the intersector coupling term containing $\tau_{x}$ suppresses the polarization. The results shown in Fig. 1(c) clearly indicate that in certain nominal HSP materials, such as silicon, the magnitude of HSP would be very small because of the large hopping parameter between two silicon atoms and the small SOC strength. Consequently, it is desirable to have the protected HSP effect forming a large domain throughout the BZ by tuning the SOC strength relative to the intersector coupling. Symmetry Consideration and Other Prototypical Nonsymmorphic Structures. For the realization of realistic materials that manifest strong Rashba HSP, we next construct some symmetry requirements as design principles and scan the stable quasi-2D material databases, including Materialsweb[23] and AiiDA,[24] to select ideal candidates with our target property. The reason we choose such layered materials is to build a straightforward connection with our tight-binding model. Since Rashba HSP requires a principle axis that is favored by layered structures, our screening process can be easily generalized to 3D Rashba hidden spin systems (such as LaOBiS$_{2} $[2] and BaNiS$_{2} $[36,37]). We choose layer groups (80 in total) to classify the symmetry of quasi-2D materials, which greatly narrows the range of 3D space groups (230 in total) that need to be considered. The correspondence between all the layer groups and space groups is provided in the process of classification, as shown in Section C of Supplementary Material.
Table 1. List of layer groups with inversion symmetry and nonsymmorphic symmetry and the corresponding quasi-2D materials.
Point group Layer group Space group # of materials Representative materials
$C_{\rm 2h}$ L15 $p2_{1}/m11$ 11 $P2_{1}/m$ 49 AgBr
L16 $p2/b11$ 13 $P2/c$ 8 Pr$_{2}$I$_{5}$
L17 $p2_{1}/b11$ 14 $P2_{1}/c$ 31 AgF$_{2}$
$D_{\rm 2h}$ L38 $pmaa$ 49 $Pccm$ 0 -
L39 $pban$ 50 $Pban$ 1 BaP$_{2}$(HO)$_{4}$
L40 $pmam^{\ast}$ 51 $Pmma$ 10 CuHgSeBr
L41 $pmma^{\ast}$
L42 $pman$ 53 $Pmna$ 7 P
L43 $pbaa$ 54 $Pcca$ 0 -
L44 $pbam$ 55 $Pbam$ 2 WBr$_{2}$
L45 $pbma$ 57 $Pbcm$ 1 SiO$_{2}$
L46 $pmmn$ 59 $Pmmn$ 43 ZrTe$_{5}$
L48 $cmme$ 67 $Cmme$ 3 TlF
hline $C_{\rm 4h}$ L52 $p4/n$ 85 $P4/n$ 4 MoPO$_{5}$
$D_{\rm 4h}$ L62 $p4/nbm$ 125 $P4/nbm$ 0 -
L63 $p4/mbm$ 127 $P4/mbm$ 1 MoBr$_{2}$
L64 $p4/nmm$ 129 $P4/nmm$ 64 LuIO
$^{*}$Layer group $pmam$ (L40) and $pmma$ (L41) are distinguished from each other by the direction of the glide mirror, and these two layer groups correspond to the same 3D space group Pmma.
cpl-37-8-087105-fig2.png
Fig. 2. Top view of the prototypical structures for (a) layer group $p4/nmm$ ($D_{{\rm 4h}}$, No. L64), (b) layer group $pmmn$ ($D_{{\rm 2h}}$, No. L46), (c) layer group $p2_{1}/m11$ ($C_{{\rm 2h}}$, No. L15) and (d) layer group $p4/n$ ($C_{{\rm 4h}}$, No. L52). Balls with different colors denote identical atoms but have different coordination along the $z$ direction.
The initial symmetry conditions are inversion symmetry and nonsymmorphic symmetry, which have been shown to protect HSP at the BZ boundary in our model calculation. This search yields 17 layer groups from 4 point groups $C_{{\rm 2h}}$, $D_{{\rm 2h}}$, $C_{{\rm 4h}}$ and $D_{{\rm 4h}}$ and 225 material candidates from the chosen databases (see Table 1). For the $C_{{\rm 4h}}$ point group, there is only one layer group $p4/n$, which requires at least eight atoms to construct. On the other hand, we note that $p2_{1}/m11$, $pmmn$, and $p4/nmm$ (belonging to $C_{{\rm 2h}}$, $D_{{\rm 2h}}$, and $D_{{\rm 4h}}$, respectively) are the three layer groups with the largest number of materials. They can be constructed by only two identical atoms, representing a total of 156 materials. The prototypical structures of the four point groups are shown in Fig. 2. The model Hamiltonian of the structure for layer group $p4/nmm$ ($D_{{\rm 4h}}$) has been introduced in Eq. (2). The prototypical structure $p4/n$ ($C_{{\rm 4h}}$) is built by putting a set of general positions in the lattice because any simpler structures that include a set of special positions in the lattice turn out to be structures with space groups of higher symmetry such as $p4/nmm$. Then, by taking each of the four clustered atoms as a sector, we can still build a $4 \times 4$ effective Hamiltonian similar to Eq. (2). Similarly, the prototypical structure for layer group $pmmn$ ($D_{{\rm 2h}}$) can be easily constructed by using the same analysis as Eq. (2), except using a rectangular lattice rather than a square lattice, in the following form: $$\begin{align} H({\boldsymbol k})={}&t_{1}{\cos}\frac{k_{x}}{2}{\cos}\frac{k_{y}}{2}\tau_{x}\sigma_{0}\\ &+(t_{21}\cos k_{x} +t_{22}\cos k_{y})\tau_{0}\sigma_{0}\\ &+\lambda_{\rm R}(\sin k_{x}\sigma_{y}-\sin k_{y}\sigma_{x})\tau_{z}.~~ \tag {4} \end{align} $$ Then, from Eqs. (A5) and (A9) in Section A of Supplementary Material, we can obtain the one-orbital Hamiltonian of the $p2_{1}/m11$ structure with $C_{\rm 2h}$ symmetry in Fig. 2(c). By substituting the concrete values of ${\boldsymbol d}_{jl}$ into this expression, we obtain $$\begin{align} H={}&t_{1}\cos \left(\frac{k_{x}}{2} \right)\cos \left(\frac{k_{y}}{2} \right)\tau_{x}\sigma_{0}\\ &+\left[ t_{21}\cos k_{x}+t_{22}\cos k_{y} \right]\tau_{0}\sigma_{0}\\ &+t_{1}'{\sin}\left(\frac{k_{x}}{2} \right)\cos \left(\frac{k_{y}}{2} \right)\tau_{y}\sigma_{0}\\ &+\lambda_{\rm R}[\sigma_{y}\sin k_{x}-\sigma_{x}\sin k_{y}]\tau_{z}\\& +\lambda_{\rm R}'\sin k_{y}\tau_{z}\sigma_{z},~~ \tag {5} \end{align} $$ where $t_{1}={3\,t}_{11}-t_{12}$, $t_{1}'=t_{12}-t_{11}$, and $\lambda_{\rm R}'/\lambda_{\rm R}\sim d_{z}/b$, which can influence the relative ratio of the $z$ component relative to the $xy$ component of HSP, while the total HSP is determined by $\lambda_{\rm R}/t_{1}$. Among the four prototypical structures, $p2_{1}/m11$ ($C_{\rm 2h}$) has the lowest symmetry. Due to the absence of a screw axis along the $x$ direction, the four-fold degeneracy at the $X$ point lifts, and the HSP along $M$–$X$ is not enforced to its maximum value, as shown in Fig. 3(a). Interestingly, the spin polarization now has sizable $s_{z}$ components peaked along $M$–$X$, in contrast to the other three representative layer groups in which the spin orientation is totally in-plane. This is because the reduced symmetry of $p2_{1}/m11$ leads to an additional term $\sin k_{y}\sigma_{z}\tau_{z}$ in the model Hamiltonian, as shown in Eq. (5). Thus, as shown in Fig. 3(b), the total HSP around $M$ is still significant but experiences a visible out-of-plane canting from the $k_{x}-k_{y}$ plane (37$^{\circ}$ for the tight-binding parameters in Fig. 3).
cpl-37-8-087105-fig3.png
Fig. 3. (a) Projected spin texture onto sectors A (red) and B (blue) for the lowest two bands ($\lambda_{\rm R}/t_{1} = -0.5$) of the $p2_{1}/m11$ structure ($C_{\rm 2h}$). (b) Total HSP and the $s_{z}$ component of the two sectors. The crystal structure is shown in the inset of panel (b).
Materials Realization. We next apply density functional theory (DFT) calculations, with the presence of SOC, to the material candidates selected by the symmetry principles described above. Our first-principle calculations use the projector-augmented wave (PAW) pseudopotentials[38] with the exchange-correlation of Perdew-Burke-Ernzerhof (PBE) form[39] as implemented in the Vienna ab initio Simulation Package (VASP).[40] The energy cutoff is chosen to be 1.5 times as large as the values recommended in relevant pseudopotentials. The $k$-point-resolved value of the Brillouin zone sampling is $0.02\times 2\pi$ /Å. Total energy minimization is performed with a tolerance of 10$^{-6}$ eV. The crystal structure is fully relaxed until the atomic force on each atom is less than 10$^{-2}$ eV/Å. For the calculations with strain, atoms are relaxed with fixed lattice constants. SOC is included self-consistently throughout the calculations.
cpl-37-8-087105-fig4.png
Fig. 4. (a) Crystal structure and two local sectors of LuIO. (b) Band structure (with SOC) with projection onto the Lu-$d_{z^2}$ orbital. (c) Projected spin texture onto sectors A (red) and B (blue) for the lowest two conduction bands. (d) Hybridization gap at the $M$ point induced by tensile strain along the [110] direction (left) and its evolution along with the monoclinic distortion $\gamma$ (right).
We find that a family of rare-earth compounds LnIO (Ln = Pr, Nd, Ho, Tm, and Lu) manifests strong HSP throughout the majority of the BZ, in excellent agreement with the predictions based on our single-orbital model. As an example, LuIO crystallizes in a tetragonal nonmagnetic structure with a nonsymmorphic layer group $p4/nmm$ (No. L64), containing 6 atoms per unit cell, as shown in Fig. 4(a). The oxygen plane containing the inversion center separates the unit cell into two LuI sectors (A and B) that are connected by the inversion symmetry. The two sectors feel opposite polar fields generated by their local environments that lack inversion symmetry, indicating Rashba HSP. Figure 4(b) shows the band structure of LuIO with SOC; as discussed above, each band is at least doubly degenerate, while the nonsymmorphic symmetry guarantees the four-fold band crossing at the $M$ point. The conduction band minimum is in the vicinity of point $M$, with the projected atomic orbitals dominated by the $d_{z^{2}}$ character of Lu atoms, fitting with our single-orbital model very well. Therefore, we would expect the lowest two doubly degenerate conduction bands to form a pair of Rashba bands with opposite helical spin textures. Similar to Eq. (1), the HSP is calculated by projecting the wave functions $\psi_{n}({\boldsymbol k})$ with plane-wave expansion on the orbital basis (spherical harmonics) of each atomic site and summing for a given sector A or B that contains a number of sites (here is Lu and I). Again, the $k$-resolved hidden spin density ${\boldsymbol S}_{\mathbb{N}}^{{\rm A}({\rm B})}({\boldsymbol k})$ is calculated by summing both degenerate bands. As shown in Fig. 4(c), we find large and opposite spin polarizations of the two LuI sectors for the lowest two conduction bands. Remarkably, we observe a strong Rashba-type HSP pattern around $M$ forming a clear domain that spans nearly 80% of the whole BZ. In addition, the spin texture around the $X$ point is Dresselhaus-type, while around $\varGamma$, there is a small region with vanishing HSP due to the strong hopping effect between the two sectors. As discussed above, the intersector hopping term totally vanishes only along the BZ boundary $X$–$M$, and the size of large HSP domain in the whole BZ depends on the ratio between the SOC strength and hopping parameter between two sectors $\lambda_{\rm R}/t_{1}$, rendering ideal HSP effects in the materials with relatively strong SOC. Discussion. We note that for the energy bands with multi-orbital features, the central physics, i.e., sector polarization and the resultant strong HSP protected by nonsymmorphic symmetry, still persists along the BZ boundary. On the other hand, the spin configuration for the other parts of the BZ could form a more complicated pattern. This is because different atomic orbitals could couple different spin textures, while the total sector-projected spin texture is the superposition of the contributions from all considered orbitals.[41] To illustrate this, we further derive a four-orbital model Hamiltonian ($s$, $p_{x}$, $p_{y}$, and $p_{z}$ orbitals) containing 16 bands and make correspondence to the representative materials from DFT calculations. One of the main features induced by the multi-orbital nature is the retention of HSP around the $\varGamma$ point for the bands with considerable $p_{x}$ and $p_{y}$ components, as shown in Section D of Supplementary Material. Recently, a theoretical proposal using a 1D chain model claimed that HSP vanishes in the vicinity of TRI momenta due to the coupling between sectors.[42] In contrast, our results indicate that by simply adding either nonsymmorphic symmetry or multi-orbital features,[11] an HSP system would manifest a nontrivial spin pattern for each inversion-asymmetric sector around TRI momenta. Thus, it naturally closes such debate around the practical significance of HSP, invoking further motivations to look for materials with remarkable hidden spin textures and technologically relevant properties. Since the probing beam of ARPES distinguishes different sectors by penetrating depth, the measurement of momentum-resolved HSP is highly accessible by counting the difference between the numbers of spin-up and spin-down photoelectrons. Hence, the experimental validation of our predictions on the physical effects and material candidates is strongly needed. HSP protected by nonsymmorphic symmetry manifests two sets of spin-splitting bands that are degenerate in energy but localized at different inversion-asymmetric sectors. It would be desirable to exploit such properties for future spintronic applications, such as the modified spin field effect transistor model[5] and spin-dependent photogalvanic devices.[43] Finally, we propose that HSP might further provide a platform for tuning the hybridization between sectors to realize more exotic physics, such as topological superconductivity. In a superconductor with time-reversal symmetry and inversion symmetry, an odd-parity topological superconductor can be realized if the Fermi surface of Bloch bands encloses an odd number of TRI momenta.[25] In particular, such a kind of time-reversal-invariant topological superconductivity is proposed to exist in an interacting bilayer Rashba system,[14] of which the band model is nothing but an HSP system with a hybridization gap at the Rashba band-crossing point. The hybridization effect at the $\varGamma$ point requires an appropriate buffer layer between the two Rashba layers, posing challenges to sample growing and integration.[44] In comparison, HSP systems offer us an alternative way to achieve the required band structure for experimental realization. As shown in Fig. 4(d), a tensile strain along [110] direction of LuIO breaks the screw axis symmetry and thus opens a hybridization gap ${\mathit \Delta}$ at the $M$ point. The gap is 13 meV with a monoclinic distortion of the lattice $\gamma = 89^{\circ}$ and increases monotonically with larger distortion. Our approach to achieve interacting Rashba bilayer bands could be used to combine with other design principles such as odd-parity pairing symmetry for the future screening of topological superconductivity.[25] We thank Prof. Alex Zunger, Jun-Wei Luo, and Xiuwen Zhang, Dr. Wen Huang, Quansheng Wu, and Zhongjia Chen for helpful discussions. 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