The Rashba spin–orbit coupling effect, primarily arising from structural-inversion asymmetry in periodic crystals, has garnered considerable attention due to its tunability and potential applications in spintronics. Its capability to manipulate electron spin without an external magnetic field opens new avenues for spintronic device design, particularly in semiconductor technology. Within this framework, 2D Rashba materials hold special interest due to their inherent characteristics, which facilitate miniaturization and engineering capabilities. In this Perspective article, we provide an overview of recent advancements in the research of 2D Rashba materials, aiming to offer a comprehensive understanding of the diverse manifestations and multifaceted implications of the Rashba effect in material science. Rather than merely presenting a list of materials, our approach involves synthesizing various viewpoints, assessing current trends, and addressing challenges within the field. Our objective is to bridge the gap between fundamental research and practical applications by correlating each material with the necessary advancements required to translate theoretical concepts into tangible technologies. Furthermore, we highlight promising avenues for future research and development, drawing from insights gleaned from the current state of the field.

Spintronics is a rapidly burgeoning research field with the potential to supersede conventional electronics in seamlessly integrating memory, processing, communication, and sensing by utilizing electrons’ spin degree of freedom as a logical unit.1–9 The field of spintronics represents a significant shift in the approach to electronic devices, focusing on the transport of electron spin rather than on the conventional flow of charge current. Various methods, including thermal, optical, electric, and magnetic techniques, have facilitated this paradigm shift, contributing to the versatility and potential of spintronic applications.10,11 However, the design of most spintronic devices relies heavily on the principles of giant magnetoresistance (GMR)12,13 and tunneling magnetoresistance (TMR).14 These principles, first identified in the late 1980s and the early 1990s, have been foundational to developing spintronics.15–20 GMR involves significant changes in the electrical resistance in response to an external magnetic field in layered ferromagnetic materials. Conversely, TMR is a phenomenon in which the resistance of a magnetic tunnel junction changes depending on the relative alignment of the magnetization in ferromagnetic layers separated by an insulating barrier.

A common characteristic of most of the spintronic devices is ferromagnetic materials. While ferromagnetic materials are integral to the functioning of GMR- and TMR-based devices, they pose several challenges, particularly regarding device integration.20–24 Integration issues stem from factors such as the incompatibility of ferromagnetic materials with standard semiconductor processes, difficulties in controlling their magnetic properties at the nanoscale, and challenges in maintaining consistent performance under varying operational conditions. In response to these challenges, semiconductor spintronics has emerged as a promising alternative. This approach leverages the properties of semiconductors combined with spin–orbit coupling (SOC) to generate and manipulate spin currents. SOC is a relativistic effect that arises from the interaction between an electron’s spin and its orbital motion.

Among the various SOC effects explored in semiconductor spintronics, Rashba spin–orbit coupling (RSOC) has gained considerable attention.25–28 RSOC is a phenomenon in which an external electric field can tune the SOC-induced effects, making it highly adaptable for spintronic device applications. This tunability is a crucial advantage of RSOC over other SOC effects because it allows the manipulation of spin currents without requiring an external magnetic field. This independence from magnetic fields is particularly beneficial in semiconductor spintronics because it enables the design of spintronic devices using nonmagnetic materials. This approach circumvents the integration challenges of ferromagnetic materials and opens new avenues for device functionality and miniaturization.29–32 

The concept of RSOC was initially proposed by Emmanuel Rashba in 1959.25 However, it gained significant prominence in the scientific community following two publications in 1984 by Bychkov and Rashba27,28 in response to the experimental findings related to the quantum Hall effect (QHE).33,34 These foundational papers, published in the early 1980s, provide crucial insights into the quantum Hall effect, a phenomenon observed in two-dimensional (2D) electron systems under strong magnetic fields, which later played a pivotal role in understanding various quantum phenomena.

Originally, RSOC was believed to be exclusive to 2D surfaces and interfaces. This assumption was based on the understanding that RSOC arises from structural inversion asymmetry (SIA) in these confined systems.21 The relativistic motion of electrons in an asymmetric crystal potential induced by the SIA leads to the generation of an effective magnetic field, denoted as B e f f. This field acts analogously to an external magnetic field but is intrinsic to material’s crystal structure. However, subsequent research has significantly expanded our understanding of RSOC, revealing its presence even in bulk three-dimensional crystal structures28,35–39 and in centrosymmetric crystals (hidden Rashba effect).40–47 This revelation, supported by various studies conducted in the 21st century, underscores the ubiquity and versatility of RSOC in different material systems.48 

The effective magnetic field B e f f resulting from the SIA has profound implications for the behavior of electron spins in nonmagnetic materials. When electrons are subjected to this field, their spins experience torque, causing them to precess. This precession is the physical manifestation of the RSOC and is a critical factor in manipulating spin currents in spintronic devices.22 

Spin field-effect transistors (s-FETs), first proposed by Datta and Das in 1990, mark one of the significant advancements toward applying RSOC in practical spintronic devices.49 In s-FETs, spins injected from a ferromagnetic (FM) source are transported to the ferromagnetic drain by their precessional motion around the magnetic field governed by the RSOC in a two-dimensional electron gas (2DEG) hosted by a semiconducting channel, as schematically shown in Fig. 1. The spin-precession length and, hence, the spin orientation of the carriers can be modulated by an external gate voltage that tunes the Rashba spin-splitting parameter.

FIG. 1.

(a)Schematic representation of electrons traversing from a ferromagnetic source to a ferromagnetic drain within a 2DEG hosted by a semiconducting channel, in the presence of a perpendicular electric field E z that induces a Rashba magnetic field B R y. (b) Precessional spin movement of electrons around the Rashba magnetic field while traveling from the source to the drain, modulated by the gate voltage.

FIG. 1.

(a)Schematic representation of electrons traversing from a ferromagnetic source to a ferromagnetic drain within a 2DEG hosted by a semiconducting channel, in the presence of a perpendicular electric field E z that induces a Rashba magnetic field B R y. (b) Precessional spin movement of electrons around the Rashba magnetic field while traveling from the source to the drain, modulated by the gate voltage.

Close modal

Apart from this, the interplay of RSOC with superconductivity brings up the possibility of realizing various exotic quantum states, which might have a revolutionary impact in the field of quantum information processing.50–52 The combination of 2D Rashba materials with s-wave superconductors under broken time-reversal symmetry is predicted to host topologically protected Majorana edge modes.53,54 These Majorana states are unique and robust against certain types of perturbations, making them promising candidates for fault-tolerant topological quantum computation if realized experimentally.55–57 

In the context of spintronic device applications, 2D Rashba materials are particularly attractive due to their inherent characteristics, facilitating miniaturization and engineering capabilities. In this Perspective article, we aim to offer an overview of recent progress in researching 2D Rashba materials. Starting with Sec. II, which delves into the fundamental theoretical concepts essential for understanding the Rashba effect, Sec. III provides a comprehensive summary of the recent advancements in the study of various 2D Rashba materials. These include AB binary monolayers, transition metal dichalcogenides (TMDs), Janus TMDs, and other Janus monolayers, as well as van der Waals (vdW) heterostructures. Additionally, Sec. III explores diverse strategies for manipulating Rashba spin splitting in materials. This includes applying external electric fields, strain engineering, and the influence of substrate proximity—critical factors in identifying materials suitable for prospective spintronic device design. Furthermore, the role of RSOC in governing the nontrivial-topological electronic phase transition in specific 2D materials is discussed in SubSec. III F. The optical manipulation of RSOC is included in the subsequent Subsec. III G. Section IV presents an overview of the practical applications of RSOC in spintronic technology, followed by a Summary and Outlook in Sec. V.

SOC is a relativistic phenomenon that lifts the spin degeneracy of electronic bands provided both space-inversion (I) and time-reversal (TR) symmetries are not conserved simultaneously.58–61 In the case of free electron gas, the presence of I and TR symmetries mandates the parabolic nature of electronic bands to retain their spin degeneracy, as illustrated in Fig. 2(a). The TR symmetry dictates E(k,  ) = E(-k,  ), while the I symmetry mandates E(k,  ) = E(-k,  ), where E and k denote the energy eigenvalues and crystal momentum, respectively. The up and down arrows represent the spin-up and spin-down states, respectively. The breaking of inversion symmetry (while preserving TR symmetry) induces spin splitting of energy bands at all generic k points, except at the time-reversal-invariant momenta (TRIM) points, where Kramer’s degeneracy holds.62 

FIG. 2.

(a)Schematic representation of the spin degenerate electronic band structure for 2DEG without SOC. (b) Rashba spin splitting resulting in lifted spin degeneracy at all k points, except at the TRIM point. E R and k R denote the Rashba energy and Rashba momentum offset, respectively. (c) Schematic 3D representation of the electronic band structure in the presence of the Rashba effect. (d) Helical spin texture at a constant energy surface for the case of pure Rashba spin splitting.

FIG. 2.

(a)Schematic representation of the spin degenerate electronic band structure for 2DEG without SOC. (b) Rashba spin splitting resulting in lifted spin degeneracy at all k points, except at the TRIM point. E R and k R denote the Rashba energy and Rashba momentum offset, respectively. (c) Schematic 3D representation of the electronic band structure in the presence of the Rashba effect. (d) Helical spin texture at a constant energy surface for the case of pure Rashba spin splitting.

Close modal
In nonmagnetic polar materials, i.e., preserved TR but broken I symmetry, the RSOC can be described as a linear coupling between electron’s spin σ and crystal momentum k. A naive concept of RSOC can be illustrated by considering the case of a free 2DEG.29 A relativistic electron of mass m with momentum k moving in an electric field ( E = E z ^), acting in the direction of broken inversion symmetry experiences a magnetic field B e f f = m c 2 ( k × E ), where is the reduced Planck’s constant and c is the speed of light. The Zeeman interaction of electron’s spin moment with the effective magnetic field mimics the form of RSOC. Thus, the Rashba Hamiltonian ( H ^ R) can be expressed as
H ^ R = e 2 m ( σ B ) = α R ( σ × k ) z ^ ,
(1)
where α R is the Rashba parameter, which represents the strength of the SOC, and it is a crucial design parameter on Rashba materials, as it represents the strength of the spin splitting. The Hamiltonian for a 2DEG, including the Rashba term, reads
H ^ = 2 k 2 2 m + α R ( σ × k ) z ^ .
(2)
Solving the Schrödinger equation H ^ | ψ = E | ψ yields the following energy eigenvalues and eigenstates:
E ± = 2 k 2 2 m ± α R k = 2 2 m ( k ± k R ) 2 + E R a n d
(3)
| ψ ± ( k ) = 1 2 ( ± e i θ 1 ) .
(4)

Figure 2(b) represents a prototypical electronic band structure illustrating the lifting of spin degeneracy due to RSOC. The offset Rashba momentum k R ( = m α R ) and Rashba energy E R ( = m α R 2 / 2 ) can be determined computationally and the Rashba parameter α R can be calculated using the expression α R = 2 E R / k R. Furthermore, angle θ in Eq. (4) can be represented as tan 1(k y/k x). On the other hand, the experimental observation of RSOC is possible using either electron photoemission or transport experiments.

The spin texture is obtained by calculating the expectation value of the spin operator σ, expressed as
ψ ± ( k ) | σ | ψ ± ( k ) = ± ( cos θ sin θ 0 ) .
(5)

Hence, the spin texture is independent of the magnitude of k; rather, it depends on its direction within the plane of the 2DEG. As k k, the angle θ varies from 0 to π, reversing the spin orientation. This ensures the conservation of TR symmetry in pure Rashba materials.

Often, materials possessing high-symmetry surfaces, such as those characterized by point groups C 3 v and C 4 v, exhibit anisotropic behavior in their Rashba spin splitting, which may not be captured within the conventional Rashba model having linear-momentum dependence. In such cases, the Hamiltonian must incorporate terms up to the third-order in k.63,64

In this Perspective article, our primary focus is on conventional 2D Rashba materials. Our discussion aims to provide a holistic view of the diversity of Rashba effect and its multifaceted implications in materials science (see Table I). Through this approach, we aim to enrich the discourse on Rashba materials, offering insights into the conventional Rashba materials and thereby contributing to the broader understanding and application of Rashba physics in advanced materials development.

TABLE I.

List of Rashba parameters (ER, kR, and αR) of some 2D Rashba materials. Additionally, here are displayed several tentative functionalities depending on the family and particular compound. Those could cover, but are not limited to, spin field-effect transistors (s-fet), 2D-ferroelectric Rashba semiconductors (2D-fersc), and optical-Rashba devices (ord).

MaterialsER (meV)kR−1)αR (eVÅ)ReferenceFunctionality
AB binary monolayer     
h-TaN 74 … 4.23 84  
h-NbN 52 … 2.9 84  
AlBi 22 0.016 2.8 83,145  s-fet, 2D-fersc 
PbSi 0.007 2.7 83,145  s-fet, 2D-fersc 
BiSb 13 0.011 2.3 66  
BiAs … … 1.95 83  s-fet 
TlP … … 1.79 83,145  s-fet, 2D-fersc 
BiP … … 1.6 83   
PbBi 20 0.025 1.6 82  
PBi … … 1.56 82  
GaSb … … 1.45 145  s-fet, 2D-fersc 
BiN … … 1.31 83   
WSe … … 1.26 88  
WS … … 1.2 88  
BiB … … 1.16 83,145  s-fet 
ZnTe … … 1.06 86  
PbTe bilayer … … 1.05 87  
SnTe bilayer … … 1.02 87  
WC … … 1.02 88  
GeTe bilayer … … 1.0 87  s-fet, 2D-fersc 
GaTe 15 0.029 1.0 89  
MgTe … … 0.63 85  
GeTe monolayer … 0.6 87  
PbTe monolayer … 0.6 87  
SnTe Monolayer … … 0.6 87  
PSb … 0.006 0.4 82  
MoC … … 0.14 88  
PAs 0.1 0.002 0.1 82  
2D Janus Monolayers      
BiTeBr … … 9.15 98,155  s-fet 
BiTeCl … … 7.48 98  
Mo2COI (AA) 0.1129 0.0571 3.9491 68  
Mo2COTe (BB) 0.0768 0.0640 2.3967 68  
Mo2COSe (BB) 0.1789 0.1470 2.3247 68  
BiTeI 40 0.043 1.97 99,155  s-fet 
Mo2COS (BB) 0.1789 0.1879 1.9045 68  
Janus Sb2Se2Te armchair (zigzag) … … 1.53 (1.52) 156  
Mo2COCl (BB) 0.1022 0.1376 1.4854 68  
SbTeI 17 0.024 1.39 99  
CrSeTe … … 1.23 94,145  
TiS2Se 40 0.074 1.081 101   
InTeF monolayer … … 1.08 100  
Sb2SeTe2 monolayers: armchair (zigzag) … … 1.00 (1.12) 156  
WSeTe 52 0.17 0.92 145,157,158  s-fet 
Mo2COBr (AA) 0.0072 0.0176 0.8185 68  
SnSTe 6.95 0.0184 0.755 97  
WSSe 3.6 0.01 0.72 145,157   
ZrS2Se 19 0.053 0.717 101   
HfS2Se 15 0.053 0.566 101  
MoSSe 1.4 0.005 0.53 145,157  s-fet 
CrSTe … … 0.31 94  
RbKNaBi 1.3 … 0.274 135   
SnSeTe 2.46 0.018 0.273 97  
CrSSe … … 0.26 94,145  
WSTe 7.78 0.0631 0.247 97,145  s-fet 
WSiGeN4 4.2 0.076 0.111 103   
SnSSe (Γ-K1.03 0.0189 0.109 97  
MoSiGeN4 0.8 0.048 0.033 103   
MoSeTe … … 0.012 145,157  s-fet 
LaOBiS2 (R-2) 119 … 4.78 40  s-fet 
LuIO (R-2) 70 0.08 1.75 44  s-fet 
2D van der Waals Heetrostructures     
1QL(2QL) Bi2Se3/1L PtSe2 4.8 (4.0) 0.002 (0.002) 4.8 (4.0) 111  
BiSb/AlN … … 1.5 66  
PtSe2/MoSe2 … … 1.3 108  s-fet, ord 
AlN/InTeF 11 … 1.13 100  
BN/InTeF 10 … 1.08 100  
J-SnSSe/WSSe 42.91 0.126 0.681 97  
Bi(111) surface 14 0.05 0.55 159  
InSe/GaTe … … 0.5 106  
GaSe/MoSe2 31 0.13 0.49 107  s-fet 
InSe/InTe … … 0.44 106  
Au (111) surface 2.1 0.012 0.33 160  
J-SnSTe/WSTe 2.47 0.0366 0.135 97  
InGaAs/InAlAs surface <1.0 0.028 0.07 161,162  s-fet 
LaAlO3/SrTiO3 interface ∼1.0 … 0.02 163,164  ord 
GaS/MoS2 0.05 … 107  s-fet 
GaS/WS2 0.03 … 107  s-fet, ord 
GaSe/WSe2 22 0.11 … 107  s-fet 
GaTe/MoTe2 48 0.12 … 107  s-fet 
GaTe/WTe2 47 0.11 … 107  s-fet 
KTaO3/K(Zn,Ni)F3 interface 64 0.044 … 164,165  ord 
MaterialsER (meV)kR−1)αR (eVÅ)ReferenceFunctionality
AB binary monolayer     
h-TaN 74 … 4.23 84  
h-NbN 52 … 2.9 84  
AlBi 22 0.016 2.8 83,145  s-fet, 2D-fersc 
PbSi 0.007 2.7 83,145  s-fet, 2D-fersc 
BiSb 13 0.011 2.3 66  
BiAs … … 1.95 83  s-fet 
TlP … … 1.79 83,145  s-fet, 2D-fersc 
BiP … … 1.6 83   
PbBi 20 0.025 1.6 82  
PBi … … 1.56 82  
GaSb … … 1.45 145  s-fet, 2D-fersc 
BiN … … 1.31 83   
WSe … … 1.26 88  
WS … … 1.2 88  
BiB … … 1.16 83,145  s-fet 
ZnTe … … 1.06 86  
PbTe bilayer … … 1.05 87  
SnTe bilayer … … 1.02 87  
WC … … 1.02 88  
GeTe bilayer … … 1.0 87  s-fet, 2D-fersc 
GaTe 15 0.029 1.0 89  
MgTe … … 0.63 85  
GeTe monolayer … 0.6 87  
PbTe monolayer … 0.6 87  
SnTe Monolayer … … 0.6 87  
PSb … 0.006 0.4 82  
MoC … … 0.14 88  
PAs 0.1 0.002 0.1 82  
2D Janus Monolayers      
BiTeBr … … 9.15 98,155  s-fet 
BiTeCl … … 7.48 98  
Mo2COI (AA) 0.1129 0.0571 3.9491 68  
Mo2COTe (BB) 0.0768 0.0640 2.3967 68  
Mo2COSe (BB) 0.1789 0.1470 2.3247 68  
BiTeI 40 0.043 1.97 99,155  s-fet 
Mo2COS (BB) 0.1789 0.1879 1.9045 68  
Janus Sb2Se2Te armchair (zigzag) … … 1.53 (1.52) 156  
Mo2COCl (BB) 0.1022 0.1376 1.4854 68  
SbTeI 17 0.024 1.39 99  
CrSeTe … … 1.23 94,145  
TiS2Se 40 0.074 1.081 101   
InTeF monolayer … … 1.08 100  
Sb2SeTe2 monolayers: armchair (zigzag) … … 1.00 (1.12) 156  
WSeTe 52 0.17 0.92 145,157,158  s-fet 
Mo2COBr (AA) 0.0072 0.0176 0.8185 68  
SnSTe 6.95 0.0184 0.755 97  
WSSe 3.6 0.01 0.72 145,157   
ZrS2Se 19 0.053 0.717 101   
HfS2Se 15 0.053 0.566 101  
MoSSe 1.4 0.005 0.53 145,157  s-fet 
CrSTe … … 0.31 94  
RbKNaBi 1.3 … 0.274 135   
SnSeTe 2.46 0.018 0.273 97  
CrSSe … … 0.26 94,145  
WSTe 7.78 0.0631 0.247 97,145  s-fet 
WSiGeN4 4.2 0.076 0.111 103   
SnSSe (Γ-K1.03 0.0189 0.109 97  
MoSiGeN4 0.8 0.048 0.033 103   
MoSeTe … … 0.012 145,157  s-fet 
LaOBiS2 (R-2) 119 … 4.78 40  s-fet 
LuIO (R-2) 70 0.08 1.75 44  s-fet 
2D van der Waals Heetrostructures     
1QL(2QL) Bi2Se3/1L PtSe2 4.8 (4.0) 0.002 (0.002) 4.8 (4.0) 111  
BiSb/AlN … … 1.5 66  
PtSe2/MoSe2 … … 1.3 108  s-fet, ord 
AlN/InTeF 11 … 1.13 100  
BN/InTeF 10 … 1.08 100  
J-SnSSe/WSSe 42.91 0.126 0.681 97  
Bi(111) surface 14 0.05 0.55 159  
InSe/GaTe … … 0.5 106  
GaSe/MoSe2 31 0.13 0.49 107  s-fet 
InSe/InTe … … 0.44 106  
Au (111) surface 2.1 0.012 0.33 160  
J-SnSTe/WSTe 2.47 0.0366 0.135 97  
InGaAs/InAlAs surface <1.0 0.028 0.07 161,162  s-fet 
LaAlO3/SrTiO3 interface ∼1.0 … 0.02 163,164  ord 
GaS/MoS2 0.05 … 107  s-fet 
GaS/WS2 0.03 … 107  s-fet, ord 
GaSe/WSe2 22 0.11 … 107  s-fet 
GaTe/MoTe2 48 0.12 … 107  s-fet 
GaTe/WTe2 47 0.11 … 107  s-fet 
KTaO3/K(Zn,Ni)F3 interface 64 0.044 … 164,165  ord 

Beyond transition metal dichalcogenides (TMDs) and group-IV monolayers such as silicene, germanene, and stanene (as discussed below in this section), various emerging classes of 2D materials like halide perovskites70 and topological insulators also showcase Rashba-induced phenomena.30 These materials, characterized by high carrier mobility, adjustable bandgaps, and robust light–matter interactions, are further empowered by the Rashba effect to support innovative device functionalities. For example, the manipulation of spin texture and spin-momentum locking in topological insulators, facilitated by the Rashba effect, enables the development of new quantum devices that leverage the topological protection of edge states for efficient, low-dissipation transport.71–74 Exploring the Rashba effect in 2D materials extends beyond practical applications to deepen our understanding of fundamental physics, such as the dynamics between spin–orbit coupling and electron–electron interactions in reduced dimensions.75–80 This foundational knowledge is vital for advancing quantum materials and devices grounded in topological physics and spintronics principles. The Rashba effect in 2D materials expands the functional material library for cutting-edge applications. It pushes the boundaries of our theoretical and experimental grasp of spin–orbit phenomena in condensed matter physics.81 

Below, we provide a comprehensive overview of various 2D materials exhibiting Rashba effect, as detailed in Table I. Our discussion is not just a catalog of materials; it represents a synthesis of our viewpoints with the current trends and challenges within the field. We aim to bridge the gap between fundamental research and practical applications by correlating each material with the necessary advancements to transition theoretical concepts into tangible technologies. This analysis encompasses an assessment of the current state of the field, identifying promising avenues for future research and development.

Centrosymmetric 2D phosphorenes do not exhibit the Rashba effect. However, RSOC can be induced by alloying phosphorene with heavier elements. The resulting buckled hexagonal P X ( X = As, Bi, and Sb) monolayers possess significant Rashba splitting near the Γ point of conduction band minimum (CBM) and α R shows an increasing trend from 0.13 to 1.56 eV Å with increasing atomic number of X in agreement with the expected increasing of the SOC strength.82 Moreover, the stable free-standing buckled honeycomb monolayers of BiSb and AlBi exhibit a large and tunable RSOC with α R of 2.366 and 2.77 eV Å,83 respectively. In the AlBi monolayer, RSOC is sensitive to strain, while in the BiSb monolayer, it is sensitive to both strain and electric field.83 Wu et al.83 designed a spin field-effect transistor (s-FET) based on BiSb monolayer and reported a short spin channel length (42 nm, tunable with strain) compared to conventional s-FETs (about 2–5  μm).

Isostructural monolayers of h-NbN and h-TaN are reported to exhibit a substantial Rashba spin splitting, with α R values of 2.9 and 4.23 eVÅ, respectively.84 The h-TaN monolayer has a higher value of α R compared to h-NbN due to the larger SOC induced by the heavier Ta atoms.84 Among the monolayer Mg X (X = S, Se, Te) family, MgTe exhibits the highest Rashba spin splitting with α R of 0.63 eVÅ.85 Likewise, ZnTe and CdTe exhibit moderate Rashba spin splitting with α R of 1.06 and 1.27 eVÅ, respectively.86 Based on the first-principles calculations, Liu et al.87 report moderate α R values of 0.60, 0.62, and 0.60 eVÅ for GeTe, SnTe, and PbTe monolayers, respectively. Rehman et al.,88 on the other hand, investigated the Rashba properties of M X monolayers ( M = Mo, W; and X = C, S, Se) and determined the Rashba parameter near the Γ point of valence band maximum (VBM) to be 0.14, 1.02, 1.2, and 1.26 eVÅ, for MoC, WC, WS, and WSe monolayers, respectively. Remarkably, these monolayers demonstrated a quantum valley Hall effect due to their distinctive Berry curvatures.88  Figure 3(a) shows the crystal structure of the binary buckled square monolayers of Pb X (where X = S, Se, and Te), which inherently exhibit RSOC. Specifically, the buckled square PbS monolayers exhibit α R values of 1.03 and 5.10 eV Å near the Γ- and M-points of Brillouin zone for the conduction band minimum (CBM), respectively. However, the Rashba effect is absent in the planar square PbS monolayer due to the presence of inversion symmetry.65 

FIG. 3.

Some of the typical 2D Rashba materials. (a) Structure of 2D square buckled PbX (X = S, Se, Te) monolayer from top and side views, employing distinct colors for different atom types.65 (b) 2D hexagonal buckled configuration of BiSb, where purple and pink balls symbolize Bi and Sb atoms, respectively.66 (c) Structure of Janus MoSSe, a representative structure of the family of 2D Janus TMDs.67 (d) Left: AA and BB configuration of symmetrically passivated M o 2CO 2 from the side view. Right: Non-centrosymmetric AA and BB terminating configurations of Janus M o 2COX (X = S, Se, Te; F, Cl, Br, I).68 (e) Side view of the Janus bilayer of WSeTe/MoSeTe for five stacking orders.69 Reproduced with permission from Hanakata et al., Phys. Rev. B 96, 161401 (2017). Copyright 2017 American Physical Society. Reproduced with permission from Singh et al., Phys. Rev. B 95, 165444 (2017). Copyright 2017 American Physical Society. Reproduced with permission from Yu et al., Phys. Rev. B 104, 075435 (2021). Copyright 2021 American Physical Society. Reproduced with permission from Karmakar et al., Phys. Rev. B 107, 075403 (2023). Copyright 2023 American Physical Society. Reproduced with permission from Rezavand et al., J. Magn. Magn. Mater. 544, 168721 (2022). Copyright 2021 Elsevier B.V.

FIG. 3.

Some of the typical 2D Rashba materials. (a) Structure of 2D square buckled PbX (X = S, Se, Te) monolayer from top and side views, employing distinct colors for different atom types.65 (b) 2D hexagonal buckled configuration of BiSb, where purple and pink balls symbolize Bi and Sb atoms, respectively.66 (c) Structure of Janus MoSSe, a representative structure of the family of 2D Janus TMDs.67 (d) Left: AA and BB configuration of symmetrically passivated M o 2CO 2 from the side view. Right: Non-centrosymmetric AA and BB terminating configurations of Janus M o 2COX (X = S, Se, Te; F, Cl, Br, I).68 (e) Side view of the Janus bilayer of WSeTe/MoSeTe for five stacking orders.69 Reproduced with permission from Hanakata et al., Phys. Rev. B 96, 161401 (2017). Copyright 2017 American Physical Society. Reproduced with permission from Singh et al., Phys. Rev. B 95, 165444 (2017). Copyright 2017 American Physical Society. Reproduced with permission from Yu et al., Phys. Rev. B 104, 075435 (2021). Copyright 2021 American Physical Society. Reproduced with permission from Karmakar et al., Phys. Rev. B 107, 075403 (2023). Copyright 2023 American Physical Society. Reproduced with permission from Rezavand et al., J. Magn. Magn. Mater. 544, 168721 (2022). Copyright 2021 Elsevier B.V.

Close modal

Generally, the intrinsic RSOC present in 2D materials can be effectively tuned by applying external perturbative methods, including electric fields and strain. This tunability makes these materials suitable candidates for spintronic devices. PBi monolayer, which exhibits the highest Rashba splitting among all P X monolayers, shows a variation of α R from 1.56 to 4.41 eVÅ under 10% biaxial tensile strain.82 In BiSb, α R increases from 2.3 to 3.56 eVÅ at 6% biaxial tensile strain and reduces to 1.77 eVÅ at 4% compressive strain. Furthermore, when subjected to strain, BiSb monolayer, crystal structure shown in Fig. 3(b), changes from a direct to an indirect bandgap semiconductor.66 On the contrary, RSOC in the GaTe monolayer is nearly insensitive to biaxial strain.89, α R in the MgTe monolayer can be tuned up to ± 0.2 eVÅ under applied biaxial strain.85 Similarly, applying a positive electric field or compressive strain can significantly enhance α R in M X monolayers. Conversely, applying a negative electric field or tensile strain weakens the Rashba effect in these materials.88 

The hexagonal 2D transition-metal dichalcogenides (TMDs) ( M X 2, M = Mo, W, and X = S, Se, Te) do not exhibit intrinsic RSOC due to their out-of-plane mirror symmetry. However, this symmetry can be broken and RSOC can be induced in these materials by replacing one of the chalcogen atoms, X, with another chalcogen element, Y, resulting in the formation of Janus TMDs ( M = Mo, W; and { X, Y} = S, Se, Te; where X Y) that stabilize in a hexagonal crystal structure as shown in Fig. 3(c).90 Typical Janus M X Y semiconductors have a stable structure at ambient conditions similar to the conventional 2D TMDs and possess intrinsic RSOC due to the built-in electric field present, perpendicular to the monolayer plane from the chalcogen atom with a larger atomic number (lower electronegativity) to the one with smaller atomic no (higher electronegativity).91–93 With the same chalcogenides, W X Y has a higher value of α R than that of Mo X Y since W has a higher SOC than Mo. The value of α R in these materials can be increased further with the application of an external electric field ( E e x t) parallel to the intrinsic field as it increases the charge accumulation in the chalcogen atom with a smaller atomic number.93 Conversely, E e x t applied opposite to the intrinsic field weakens the Rashba effect. As presented in Fig. 4(a), among all other M X Y monolayers, WSeTe exhibits the most significant change in α R with an increase of 0.031 eV Å near the Γ point under an electric field of 0.5 V/Å when compared to its intrinsic value. On the contrary, MoSTe and WSTe are nearly insensitive to the applied field.93 

FIG. 4.

Modulation of the Rashba parameter α R in 2D Janus TMDs under the influence of (a) an external electric field and (b) biaxial strain. Figure adopted from Ref. 93. Reproduced with permission from Hu et al., Phys. Rev. B 97, 235404 (2018). Copyright 2018 American Physical Society.

FIG. 4.

Modulation of the Rashba parameter α R in 2D Janus TMDs under the influence of (a) an external electric field and (b) biaxial strain. Figure adopted from Ref. 93. Reproduced with permission from Hu et al., Phys. Rev. B 97, 235404 (2018). Copyright 2018 American Physical Society.

Close modal

Janus TMDs exhibit a nonlinear change in α R when subjected to an in-plane biaxial strain. Figure 4(b) depicts the change in α R of the Janus MXY monolayers with applied biaxial strain. Among all others, MoSeTe shows the highest increase in α R from 0.5 to 1.1 eV Å under a compressive strain of 3 %. Hu et al.93 also reported a significant increase in the anisotropic Rashba spin splitting of these materials with the application of a compressive strain.

On the other hand, compared to other Janus TMDs such as MoSSe and WSSe, strain-free monolayers of CrSSe, CrSTe, and CrSeTe show relatively higher values of the intrinsic Rashba parameter of 0.26, 0.31, and 1.23 eV Å, respectively. With an applied compressive strain of 2 %, α R of CrSSe, CrSTe, and CrSeTe monolayers increases up to 0.66, 0.50, and 2.11 eV Å, respectively.94 Anisotropic Rashba spin splitting arising from RSOC is observed around the M-point in Pt X Y ( X = Y = S, Se, Te; X Y) monolayers. The calculated values of α R from the M- to the Γ-point are 1.654, 1.103, and 0.435 eV Å, while the values from the M- to K-point are 1.333, 1.244, and 0.746 eV Å for PtSSe, PtSTe, and PtSeTe, respectively.95 

RSOC can also be induced in centrosymmetric TMDs by applying an external electric field. E e x t breaks the inversion symmetry in these crystals and induces Rashba spin splitting. Yao et al.96 reported a significant linear increase of α R with increasing E e x t in six M X 2 monolayers. Figure 5 illustrates the variation in the Rashba parameter of 2D TMDs in response to an applied external electric field. Notably, the anions play a significant role in the electric-field dependence of RSOC in these materials. Conversely, the cations make a minimal contribution to the field dependence of RSOC as they are strongly shielded from the external electric field by the anions.96 

FIG. 5.

Variation in the Rashba parameter of 2D TMDs in response to an applied external electric field. Figure adopted from Ref. 96. Reproduced with permission from Yao et al., Phys. Rev. B 95, 165401 (2017). Copyright 2017 American Physical Society.

FIG. 5.

Variation in the Rashba parameter of 2D TMDs in response to an applied external electric field. Figure adopted from Ref. 96. Reproduced with permission from Yao et al., Phys. Rev. B 95, 165401 (2017). Copyright 2017 American Physical Society.

Close modal

Apart from the Janus TMDs, various other Janus monolayers have also gained significant attention due to their large Rashba spin splitting and potential applications in spintronic devices. Bhat et al.97 performed ab initio calculations on Janus Sn X Y monolayers ({ X, Y} = S, Se, Te; with X Y) to study how RSOC is affected by the replacement of the transition metal in Janus TMDs with a metal. The resulting Janus SnSSe, SnSeTe, and SnSTe monolayers exhibit anisotropic Rashba spin splitting having α R of 0.109, 0.273, and 0.755 eV Å, respectively.97 Notably, Bafekry et al.98 reported a huge Rashba spin splitting in BiTeCl and BiTeBr monolayers with α R as 7.48 and 9.15 eV Å, respectively. Conversely, BiTeI and SbTeI monolayers are reported to exhibit a relatively smaller α R of 1.97 and 1.39 eV Å  respectively.99 The value of α R for InTeF was determined to be 1.08 eV Å.100 

Another interesting class of 2D Rashba materials is the Janus transitional-metal trichalcogenide (TMTC) monolayers with the chemical formula of M X 2 Y ( M = Ti, Zr, Hf; X Y = S, Se). These materials show RSOC due to the cumulative effect of SOC and the lack of in-plane mirror symmetry. In the case of MS 3-based TMTC monolayers, the Rashba spin splitting occurring near the Γ point of VBM is mainly contributed by the d x y orbitals of the transition-metal atoms. The Rashba parameter in these systems varies as TiS 2 Se > ZrS 2 Se > HfS 2 Se due to a decrease in the built-in electric field. On the contrary, MSe 3-based TMTC monolayers do not exhibit any RSOC, owing to the lack of any contribution from the d x y orbitals of the transition metal near the Fermi level. The calculated value of Δ E R, Δ k R and α R for the Janus TiS 2Se monolayer is 40 meV, 0.074 Å 1, and 1.081 eV Å, respectively.101 

Indirect bandgap semiconductors of 2D Janus tellurite (Te 2Se) monolayers show RSOC near the Γ point of CBM, the significant contribution of which arises from the p z orbitals of Te atoms.102 Guo et al.103 recently predicted a new class of Janus materials M A 2 Z 4 ( M = transition metal such as W, V, Nb, Ta, Ti, Zr, Hf, or Cr; A = Si or Ge, and Z = N, P, or As) with intrinsic RSOC. The predicted monolayers of MSiGeN 4 ( M = Mo and W) are dynamically and thermally stable and exhibit RSOC with a Rashba energy of 0.8 and 4.2 meV for MoSiGeN 4 and WSiGeN 4, respectively.103,104

An exciting class of 2D Janus Rashba materials with a substantially large value of α R has been recently proposed by Karmakar et al.68 These materials belong to the Janus Mo 2CO X structures, which are derived from the parent compound Mo 2CO 2, a popular class of 2D MXenes in the family of 2D transition metal carbides/nitrides/carbonitrides with the generic formula of M n + 1 X n T 2 ( M = 3 d or 4 d transition metals, X = C or N, T = surface termination unit, n = 1 3). Here, RSOC is induced by breaking the inversion symmetry present in the parent compound Mo 2CO 2, with the replacement of one of the terminating O layers either with a halogen (F, Cl, Br, and I) or a chalcogen (S, Se, and Te) layer [see Fig. 3(d)].

Constructing vdW heterostructures is a promising approach for manipulating the properties of the constituent monolayers. The structural asymmetry within these heterostructures creates a net out-of-plane electric field which impacts the system’s RSOC. Consequently, electronic properties undergo significant changes compared to the individual monolayers. For instance, with the formation of a bilayer, Liu et al.87 observed an almost twofold increase in the Rashba spin splitting at the CBM of MTe ( M = Ge, Sn, and Pb) compared to its monolayer. This increase is attributed to the doubling of the dipole moment. The calculated values of α R are 1.10, 1.02, and 1.05 eV Å for GeTe, SnTe, and PbTe bilayers, respectively.87 

The 2D III-VI chalcogenide N X ( N = Ga, In; X = S, Se, Te) monolayers do not exhibit any intrinsic RSOC owing to the out-of-plane mirror symmetry. Nevertheless, this limitation is overcome by constructing vdW heterostructures of InSe with GaTe and InTe. These heterostructures, namely, InSe/GaTe and InSe/InTe, exhibit significant RSOC at the CBM with α R of 0.50 and 0.44 eV Å, respectively, where RSOC mainly originates from the InSe layer.106 First principles studies on the Ga X/ M X 2 ( M = Mo, W; X = S, Se, Te) heterostructures reveal an increase in Rashba spin splitting with increased SOC in the p-orbitals of chalcogen atoms. Strikingly, the replacement of Mo with W decreases the strength of RSOC due to the increase in d-orbital contribution near the valance band edge at the expense of the p-orbital of the chalcogen atoms confirming the major contribution of the p-orbitals to the Rashba spin splitting. Similar observations hold true for the MoS 2/Bi(111) system.107 

Furthermore, InTeF/AlN and InTeF/BN heterostructures exhibit α R of 1.13 and 1.08 eV Å, respectively.100 A significant Rashba spin splitting with an energy of 110 meV was observed near the Γ point of PtSe 2/MoSe 2 heterostructures, which is mainly attributed to the strong interfacial SOC arising from the hybridization between the two constituent monolayers.108 Moreover, by applying strain and electric fields, the RSOC in these materials can be tuned effectively. Particularly, with the application of 6% in-plane strain, the Rashba coefficient increases up to 1.33 and 1.26 eV Å for InSe/GaTe and InSe/InTe, respectively.106 First-principles calculations on InTe/PtSe 2 heterostructures reveal Rashba-type spin splitting near the Γ point of VBM, which can be manipulated by strain engineering. While tensile strain increases α R, compressive strain weakens the RSOC. On the other hand, strong interlayer coupling promotes RSOC resulting in an enhanced α R with decreased interlayer distances.109 

Peng et al.110 report a significant Rashba spin splitting near the Γ point of MoS 2/Bi 2Te 3 that mainly arises from the Bi 2Te 3 layer, while the MoS 2 layer plays an inductive role. α R in these systems are comparable to those observed in {Bi 2Se 3} 2/InP(111) heterostructures as observed from the angle-resolved photoemission spectroscopy (ARPES) measurements. The spin textures measured experimentally for different thicknesses of {Bi 2Se 3} 2/InP(111) heterostructures reveal Rashba-like splitting of the massive Dirac cones in the surface states as a result of the substrate-induced inversion asymmetry.102 

In a recent study, Sattar and Larsson111 observed that α R varies significantly with changing thickness of the constituting layers in Bi 2Se 3/PtSe 2 heterostructures. Specifically, the heterostructure consisting of 2-quintuple layers (QL) Bi 2Se 3/ 2L PtSe 2 exhibits α R of 16.84 and 4.4 eV Å at VBM and CBM, respectively, which are among the highest reported values for known 2D Rashba materials.111 Intrinsic RSOC is also present in Janus SnSSe/WSSe semiconductor heterostructures, which exhibit a Rashba spin splitting near the Fermi level with α R of 0.7 eV Å. With the application of 12.8% of compressive strain, this α R increases up to 1.07 eV Å. On the contrary, metallic heterostructures of Janus layers SnSTe/WSTe and SnSeTe/WSeTe do not exhibit any Rashba-type splitting.97 

Further, Kunihashi et al.112 studied the effect of incorporating heavier Bi elements into GaAs heterostructures by employing time and spatially resolved Kerr rotation measurements on the 70 nm thick epitaxially grown layer of GaAs 0.961Bi 0.039. Their results revealed that RSOC prevailed over the Dresselhaus effect, with the measured value of α R as 2.5 meVÅ. Interestingly, a decrease in α R was observed with an increase in the pump laser intensity due to the light-induced screening effect on the built-in potential gradient.112 GaSe/MoSe 2 heterostructures also exhibit a Rashba splitting of 0.49 eV Å. Notably, these heterostructures exhibit the coexistence of Rashba splitting and band splitting at the K and K valleys, highlighting their potential for applications in spintronics and valleytronics.107 

Proximity-induced SOC primarily occurs within a few atomic layers of the low-SOC material near its interface when a heterostructure is formed with a material exhibiting stronger SOC. This phenomenon may result in the charge transfer between the material and the substrate or the formation of a thin interfacial layer with slightly altered electronic states.114–117 Graphene shows a substantial increase in RSOC with the proximity effect of substrates. In its free-standing form, graphene possesses the intrinsic spin–orbit splitting of around 50  μeV. However, the Rashba spin splitting of the Dirac cones can be enhanced up to 13–100 meV when synthesized epitaxially with high SOC elements including Ag, Au, and Pb intercalated, or in direct contact with different substrates such as Ni, Ir, and Co.105,118–123 The anomalous increase in the spin splitting can be attributed to a strong π-d hybridization between graphene and the substrate. Figure 6(b) shows the variation in RSOC of graphene in the graphene/Au/Ni(111) heterostructure on account of the proximity-induced effect of the substrate.

FIG. 6.

(a) ARPES spectra around the K point in the Brillouin zone of graphene in graphene/Au/Ni(111). Cross symbols indicate the positions in the Dirac cone of the p-states in graphene probed by spin- and angle-resolved photoemission spectra as shown in (b). Blue (red) lines in (b) represent spin-up (down) spectra. k is the in-plane component of the wave vector. Figure reproduced with permission from Marchenko et al., Nat. Commun. 3, 1232 (2012). Copyright 2012 Macmillan Publishers Limited.

FIG. 6.

(a) ARPES spectra around the K point in the Brillouin zone of graphene in graphene/Au/Ni(111). Cross symbols indicate the positions in the Dirac cone of the p-states in graphene probed by spin- and angle-resolved photoemission spectra as shown in (b). Blue (red) lines in (b) represent spin-up (down) spectra. k is the in-plane component of the wave vector. Figure reproduced with permission from Marchenko et al., Nat. Commun. 3, 1232 (2012). Copyright 2012 Macmillan Publishers Limited.

Close modal

The use of thin layers of 3D topological insulators like Bi 2Se 3 or Bi 2Te 3 as a substrate can further increase RSOC in graphene.124,125 On the other hand, TMDs, as mentioned above, do not exhibit any intrinsic RSOC. However, MoTe 2 when placed on top of EuO substrate shows a relatively large Rashba spin splitting owing to the proximity-induced interactions.126 Furthermore, the Janus WSSe semiconductor exhibits an increase in α R from 0.17 to 0.95 eVÅ under the proximity effect of MnO (111).127 

Recent theoretical and experimental studies on topological insulators (TIs) have developed a profound understanding of how electronic band structure in certain materials can be altered significantly by SOC effects. SOC often leads to band inversion near the Fermi level, driving the topological phase transition in materials.71–74 In the case of 2D (3D) TIs, this type of transformation leads to the formation of Dirac-cone states, characterized by a distinctive spin-momentum interdependence on the metallic edge (surface) states. This exciting feature forms the basis of the quantum spin Hall effect, with profound implications for condensed matter physics and potential applications in novel spintronic devices. Due to the conservation of TR-symmetry, the opposite edge states of the 2D nontrivial TIs possess opposite spin chirality and are charge neutral.128–131 

BiTeI well exemplifies this phenomenon. Under normal conditions, BiTeI exhibits a substantial Rashba spin splitting of 3.85 eV Å. Strikingly, BiTeI can transform a topologically trivial state to a nontrivial TI by applying hydrostatic pressure, as shown in Fig. 7. Under ambient conditions, the system possesses a conventional bandgap of 286 meV and a Rashba energy of 110 meV. However, with hydrostatic compression, beyond a critical point an inverted bandgap emerges between the top valence band and the bottom conduction band leading to a topologically nontrivial phase.113 Similarly, Jozwiak et al.,132 based on their DFT calculations on a 7-quintuple layer thick slab of Bi 2Se 3, demonstrated that the emergence of surface band inversion in the surface electronic configuration of the topological insulator Bi 2Se 3 is mainly caused by RSOC.132–134 

FIG. 7.

Computed electronic band dispersions in proximity to the Fermi level (EF) for both I-terminated (top panels) and Te-terminated (bottom panels) surfaces of BiTeI, subjected to hydrostatic compression by (a) and (b) V/V0 = 1, (c) and (d) V/V0 = 0.89, and (e) and (f) V/V0 = 0.86. The corresponding Fermi surfaces for cases (e) and (f) are depicted in (g) and (h), respectively. Reproduced with permission from Bahramy et al., Nat. Commun. 3, 679 (2012). Copyright 2012 Macmillan Publishers Limited.

FIG. 7.

Computed electronic band dispersions in proximity to the Fermi level (EF) for both I-terminated (top panels) and Te-terminated (bottom panels) surfaces of BiTeI, subjected to hydrostatic compression by (a) and (b) V/V0 = 1, (c) and (d) V/V0 = 0.89, and (e) and (f) V/V0 = 0.86. The corresponding Fermi surfaces for cases (e) and (f) are depicted in (g) and (h), respectively. Reproduced with permission from Bahramy et al., Nat. Commun. 3, 679 (2012). Copyright 2012 Macmillan Publishers Limited.

Close modal

In 2D materials, one of the intriguing cases featuring the coexistence of both RSOC and topologically nontrivial edge states is the Janus RbKNaBi monolayer.135 It is a quantum spin Hall insulator with a relatively large bandgap and is dynamically and thermally stable. RbKNaBi possesses intrinsic RSOC owing to a built-in electric field because of the difference in electronegativities between the top and bottom atomic layers. Figures 8(a) and 8(b) show the electronic band structure of RbKNaBi with the orbital projection of s, p x + i p y, and p z orbitals of Bi, calculated using GGA and GGA + SOC approximations, respectively. A SOC-induced band inversion can be noticed at the Fermi level near the Γ point. Interestingly, RbKNaBi shows a topologically nontrivial behavior, as shown in Figs. 8(c) and 8(d). A single pair of helical edge states are present within the bandgap of nearly 229 meV. The bandgap is sufficiently large to protect the helical edge states against thermally activated carriers, enabling the realization of the room temperature quantum spin Hall effect.

FIG. 8.

Electronic band structure of RbKNaBi with the orbital projection of s, p x + p y, and p z orbitals of Bi calculated (a) without and (b) with SOC. (c) Evolution of WCC computed using GGA + SOC, indicating a nontrivial Z 2 topological invariant ( Z 2 = 1). (c) Projected edge spectrum revealing a nontrivial metallic edge state within the bandgap. Reproduced with permission from Guo et al., Phys. Rev. Mater. 7, 044604 (2023). Copyright 2023 American Physical Society.

FIG. 8.

Electronic band structure of RbKNaBi with the orbital projection of s, p x + p y, and p z orbitals of Bi calculated (a) without and (b) with SOC. (c) Evolution of WCC computed using GGA + SOC, indicating a nontrivial Z 2 topological invariant ( Z 2 = 1). (c) Projected edge spectrum revealing a nontrivial metallic edge state within the bandgap. Reproduced with permission from Guo et al., Phys. Rev. Mater. 7, 044604 (2023). Copyright 2023 American Physical Society.

Close modal

An intriguing area of research has emerged recently, focusing on proposing engineering of TIs through strategic layering of topologically trivial Rashba monolayers, utilizing first-principles calculations. In 2013, Das et al.136 introduced a novel approach of designing a 3D TI by stacking bilayers composed of two-dimensional Fermi gases, each exhibiting opposite RSOC on adjacent layers. They observed that while a single bilayer consistently demonstrated topologically trivial behavior, topologically nontrivial insulating states emerged only in the bulk after crossing a critical number of bilayers. On the other hand, Nechaev et al.,137 through theoretical investigations, found that a centrosymmetric sextuple layer formed by combining two BiTeI trilayers with opposite RSOC exhibits an inverted bandgap of sufficient magnitude for practical applications. However, they observed that the sextuple layer transitioned to a topologically trivial state with just a 5% increase in vdW spacing. This strategic approach presents new avenues for designing intriguing topological materials by leveraging the inherent RSOC in 2D materials.

RSOC plays a crucial role in spin field-effect transistors, enabling information processing and storage without reliance on external magnetic fields. However, the spin relaxation mechanism in the 2D semiconducting channels limits the precision and accuracy of these devices. Achieving a persistent spin helix (PSH) condition in the 2DEG is an effective solution to tackle this challenge. Hence, efficient engineering of RSOC and Dresselhaus effect in 2D materials is of subsequent importance.138 While the Rashba parameter can be manipulated by an external electric field and/or strain, and the Dresselhaus effect can be tuned by controlling the width of the quantum wells,139 these adjustments often require complex fabrication processes. However, replacing the gate in state-of-the-art s-FETs with an optical field presents a promising avenue for developing faster and more energy-efficient devices. This approach offers a flexible and efficient means to control the RSOC.140 Notably, optical tuning is a non-destructive as well as reversible technique that can accurately alter the electron density and effectively screen the intrinsic electric field in the system without relying on the excitation beam.141 

Ma et al.142 investigated the optical tuning of RSOC and Dresselhaus effect in the 2DEG of GaAs/Al 0.3Ga 0.7As by measuring the spin-galvanic effect (SGE). They introduced an additional control light above the barrier’s bandgap to tune the SGE excited by a circularly polarized light below the bandgap of GaAs. Their observations reveal an efficient optical tunability of RSOC compared to Dresselhaus SOC in GaAs/Al 0.3Ga 0.7As. Specifically, they demonstrate that the ratio of Rashba- and Dresselhaus-related SGE currents varies systematically with the increasing power of the control light, as shown in Fig. 9(a). Above a critical point, inverse PSH emerges resulting in an extended spin lifetime. This emphasizes the potential of optical tuning as an effective technique for modulating SOC, offering implications for the design of spintronic devices with prolonged spin coherence time.142 

FIG. 9.

(a) Optical manipulation of the Rashba-to-Dresselhaus coefficient ratio ( α / β) in GaAs/Al 0.3Ga 0.7As with increasing power of the control light, displayed in a semi-logarithmic plot. The solid line illustrates the baseline ratio in the absence of control light. (b) Orientation of the effective magnetic field vector, B e f f, within the momentum space when the inverse persistent spin helix ( α = β) condition is satisfied. While the arrows indicate the direction of B e f f, their lengths symbolize the corresponding field strength. B e f f is the resultant of the Dresselhaus and Rashba effective magnetic fields. An unidirectional alignment of B e f f is ensured by the inverse persistent spin helix condition. Reproduced with permission from Ma et al., Opt. Express 31, 14473 (2023). Copyright 2023 Optica Publishing Group.

FIG. 9.

(a) Optical manipulation of the Rashba-to-Dresselhaus coefficient ratio ( α / β) in GaAs/Al 0.3Ga 0.7As with increasing power of the control light, displayed in a semi-logarithmic plot. The solid line illustrates the baseline ratio in the absence of control light. (b) Orientation of the effective magnetic field vector, B e f f, within the momentum space when the inverse persistent spin helix ( α = β) condition is satisfied. While the arrows indicate the direction of B e f f, their lengths symbolize the corresponding field strength. B e f f is the resultant of the Dresselhaus and Rashba effective magnetic fields. An unidirectional alignment of B e f f is ensured by the inverse persistent spin helix condition. Reproduced with permission from Ma et al., Opt. Express 31, 14473 (2023). Copyright 2023 Optica Publishing Group.

Close modal

On the other hand, Michiardi et al.,143 as part of their study on topological insulator Bi 2Se 3, demonstrated efficient tuning of RSOC using optical pulses with a picosecond timescale. According to their proposed mechanism, optical excitation above the energy gap induces charge redistribution perpendicular to the surface in the presence of a band-bending surface potential. This generates an ultrafast photovoltage that modulates the α R within a sub-picosecond timeframe. The measured change in Rashba momentum Δ k R within the first quantum well state (QWS1), approximately 3.5 × 10 3 Å 1, corresponds to an alteration in the spin precession angle of π over a distance of less than 100 nm.143 This implies that the effect becomes noticeable in devices of similar length under ballistic transport conditions. The use of optical pulses to manipulate Rashba splitting in 2DEGs with a picosecond-level timescale represents a significant advancement in optically controlled spintronic devices.

Another study on Ge/Si 0.15Ge 0.85 multiple quantum wells further supports the viability of contactless optical excitation as an effective method for tuning SOC, thus paving the way for electro-optic modulation of spin-based quantum devices consisting of group IV heterostructures.144 

More recently, the realization of 2D ferroelectricity has been demonstrated in various compounds.145–152 Some of which, profit from the spontaneous polarization coupling with the strong SOC leading to a reversible Rashba 2D compound. In striking difference to bulk ferroelectrics that aim to be growth in ultra-thin films, the 2D ferroelectric materials do not suffer from the polarization cancelation induced by the depolarizing fields. The latter presented as a result of the ultra thin-films dimensionality. Additionally, due to the weak vdW interaction between the layers, the 2D ferroelectric Rashba compounds are ideal for the experimental processing showing a large CMOS growth compatibility. Finally, and as an additional advantage of the 2D ferroelectric Rashba compounds is that, as demonstrated in the WTe 2 compounds,153,154 the covalent bonding in the layer favors the charge screening resulting in a metallic ferroelectricity behavior ideal for controlling the spin-texture by electric field in 2D devices. For more details on 2D-ferroelectric Rashba semiconductors (2D-fersc), we refer the reader to Refs. 145, 146, and 151.

Spintronic presents a promising next-generation platform, surpassing traditional electronics by leveraging electrons’ spin degree of freedom.7–9,166,167 Furthermore, manipulating spin as a logical unit in spintronic opens up novel avenues for neuromorphic168,169 and probabilistic computing.170 Spintronic improves scaling, processing speed, and energy efficiency compared to electronics and establishes a direct interface with existing technologies. For instance, spin valve and magnetic tunnel junction (MTJ) devices found swift applications in disk read heads, proximity sensors for automobiles, automated industrial tools, and biomedical devices following the discovery of giant magnetoresistance.12,13 However, reliance on external magnetic field limits the energy efficiency of MTJ- and GMR-based devices.

The discovery of spin-transfer torques (STT)171,172 enables all-electric control over spin states and resistance in GMR devices, enhancing scalability. This has led to the development of scalable nonvolatile magnetic random access memory (RAM) using STT, replacing static RAM and showing potential applications in dynamic RAM technology.173–175 While GMR devices are currently integrated into conventional electronic platforms, an all-spintronic platform requires further innovation in materials design and the fabrication of high-density and low-power components. Notably, STT devices face challenges such as dependence on high-performance magnets, spin filtration, low spin carrier lifetimes and diffusion lengths, Joule heating, and voltage breakdown.7–9,166

Recent phenomena include the spin Hall effect (SHE)176–180 and Rashba-Edelstein effect (REE),177 along with their optical181 and thermal182 equivalents, introduce new possibilities for efficient spin manipulation. Focusing on SHE and REE, an applied voltage generates a spin-polarized current and interfacial spin accumulation, similar to STT but without charge flowing through the magnetic layers. This process reduces the impact of Joule heating and minimizes the risk of voltage breakdown compared to STT. Additionally, these spin–orbit torques (SOT) can excite various magnetic materials,183 enabling the switching of single-layer magnets and efficient manipulation and excitation of domain walls, skyrmions, and spin waves. However, progress with SOT is hindered by low charge-spin conversion efficiency7 and reliance on heavy elements.

A diverse range of spin transistors, leveraging various operational principles, has been proposed in the literature.167,184 One notable example demonstrating practical applications of RSOC is the Datta-Das spin transistor, also referred to as the spin FET.49 In this transistor, an external gate voltage controls the flow of spin-polarized electrons. The device typically consists of a Rashba semiconductor (channel) with two ferromagnetic contacts, which act as a source and drain (see Fig. 1). The gate voltage is used to manipulate the spin of electrons, allowing for the modulation of the spin current between the ferromagnetic contacts.

Furthermore, the concept of a bipolar spin switch, introduced by Johnson185 in 1993, outlines a spin injection technique employing a thin ferromagnetic film to polarize the spin axes of electrons transporting an electric current in a ferromagnetic-nonmagnetic-ferromagnetic metal trilayer structure. This configuration yields a three-terminal, current-biased device with a bipolar voltage (or current) output dependent on the magnetization orientations of the two ferromagnets.

As observed in layered bulk compounds,186–190 the intrinsic coupling between the broken inversion symmetry and SOC opens the possibility of an electrically-controlled Rashba device. In this potential device design, realization relies on the feasibility of polarization switching in the 2D ferroelectric layer, unlocking spin-texture reversal and spin control. Potential candidates include the {Mo,W}Te 2,153,154,191–194 AgBiP 2Se 6,145 and A B monolayers145 in which, the out-of-plane polarization can be switched. Some efforts toward such direction have shown the successful design of the electric field control of valleytronics.145,195,196 Moreover, an additional advantage of the 2D vdW ferroelectrics is the feasible functionalization and growth on substrates, offering a doable engineering of the potential device.

Recently, globally centrosymmetric but locally polar 2D layers, termed R-2 Rashba systems, have been proposed as a novel platform for spin FETs, leveraging the spin-layer-locking mechanism in vertically-stacked distinct atomic layers having opposite spin textures.40–43,45,46 This involves operating the spin FET through electrical gates by tuning electrostatic doping and spin channel band splitting.44 

Other proposed devices benefit from the Aharonov–Casher effect197 in which charge-neutral magnetic moments experience quantum oscillations in the presence of an external electric field.198 Such a proposal considers Rashba active materials ring shaped, by lithography for examples, in which, the spin momenta, associated to the electons’s flow, present a difference at the end of the loop.

In this Perspective, we highlight the crucial significance of Rashba effect in broadening the research horizons within the domain of 2D materials, surpassing the confines of graphene. This investigation encompasses transition-metal dichalcogenides, Janus monolayers, silicene, germanene, stanene, and 2D van der Waals heterostructures. The Rashba effect enhances the comprehension of these materials and establishes the groundwork for actualizing diverse unprecedented physical occurrences and technological advancements. Key areas influenced by the Rashba effect include:

Spin-Orbitronics: This emerging field represents a synergistic blend of spintronics and orbitronics, where both spin and orbital degrees of freedom are manipulated.199,200 The Rashba effect plays a crucial role in this integration, enabling the simultaneous control of spin and orbital characteristics, which could revolutionize the design and functionality of electronic devices.201 

Nonlinear spintronics: In strong electron–electron interactions, the Rashba effect contributes to the emergence of nonlinear spintronic phenomena.202,203 This interaction can lead to higher-order harmonics in spin currents, opening up new possibilities for advanced information processing and storage technologies that leverage these complex spin dynamics. Furthermore, strong electron–electron interactions in Rashba materials can yield non-conventional correlated states, unusual collective modes, and bound electron pairs with non-trivial orbital and spin structures.75–80 

Spin-photovoltaics and 0ptospintronics: This area explores the intersection between photonics and spintronics, where the Rashba effect facilitates the coupling between light and spin-polarized currents.204,205 This coupling could lead to novel spin-photovoltaic devices, which harness light to generate spin currents, and optospintronics, which combines optical and spintronic functionalities for innovative applications.

Thermal spintronics: The Rashba effect is instrumental in controlling spin currents induced by thermal gradients, an essential aspect of thermal spintronics.206,207 This control is crucial for developing spin-based thermoelectric devices, which can convert waste heat into sound energy, offering a novel approach to energy efficiency and sustainability.

Ultrafast spin dynamics: The Rashba effect provides an ideal platform for the ultrafast manipulation of spin states.208 This capability is essential for creating ultrafast memory devices, where rapid and precise control over spin dynamics is paramount and can use materials with a significant Rashba effect.209 The ability to manipulate spin states on extremely short timescales could lead to a new generation of high-speed, high-efficiency memory and processing devices, significantly outperforming current technologies in speed and energy consumption.

Each of the aforementioned domains signifies a notable progression in materials science and technology, steered by the distinctive characteristics of the Rashba effect in 2D materials. The exploration of Rashba materials, particularly in the context of 2D vdW materials, has made significant strides. However, despite the extensive list of identified Rashba materials, there remains a gap in developing a comprehensive descriptor that can effectively pinpoint optimal systems with isolated spin states and large tunable splitting.64 The existence of heavy atoms in these materials stands as a pivotal element owing to their robust SOC, a fundamental aspect in the emergence of the Rashba effect. Furthermore, 2D layers with pronounced crystal-potential gradients are imperative. Nevertheless, the obstacle resides in pinpointing explicit, measurable factors that can be methodically utilized to foresee and enhance Rashba attributes in the materials’ design phase. This absence of a conclusive array of parameters or a descriptor constrains the capacity to effectively engineer materials with targeted Rashba characteristics.

From a theoretical perspective, current methodologies primarily involve analyzing the spin texture to confirm the existence of Rashba effects. While effective, this approach often requires a comprehensive analysis of the electronic band structure, including SOC effects, which can be complex and resource-intensive. A more streamlined method to calculate the Rashba parameter would be highly beneficial. Ideally, such a methodology would allow for estimating α R without necessitating a complete band structure analysis, thereby simplifying the identification and characterization of Rashba materials in high-throughput calculations.

To address all these challenges, future research could focus on:

  • Developing predictive models: Machine learning and data-driven approaches could be employed to develop predictive models that identify potential Rashba materials based on their atomic and electronic properties.64 Even if regression models are trained, obtaining a reasonable estimation of the Rashba parameter would be possible.

  • High-throughput screening: Leveraging computational tools for high-throughput screening of materials could expedite the discovery of new Rashba systems using established criteria and theoretical models.145,210

  • Advanced computational methods: Improving computational methods to more efficiently calculate α R (beyond the linear- k Rashba model) and other relevant parameters, possibly through developing new algorithms or adapting existing ones to target Rashba-related properties specifically.

  • Experimental validation: Complement theoretical advancements with experimental techniques to validate predictions and refine models, ensuring the theoretical descriptors are grounded in practical, observable phenomena.

In essence, although notable advancements have been achieved in pinpointing 2D Rashba materials, there exists a distinct requirement for more sophisticated descriptors and computational approaches to scrutinize both traditional and unconventional (hidden) Rashba systems. These progressions would substantially augment the capacity to devise and leverage materials with precise Rashba attributes, thereby facilitating a more effective and focused progression in spintronics development.

A.B. and S.S. acknowledge support from the University Research Awards at the University of Rochester. S.S. is supported by the U.S. Department of Energy, Office of Science, Office of Fusion Energy Sciences, Quantum Information Science program under Award No. DE-SC-0020340. A.H.R. thanks the Pittsburgh Supercomputer Center (Bridges2) and San Diego Supercomputer Center (Expanse) through allocation DMR140031 from the Advanced Cyberinfrastructure Coordination Ecosystem: Services & Support (ACCESS) program, which is supported by National Science Foundation Grant Nos. #2138259, #2138286, #2138307, #2137603, and #2138296. A.H.R. also recognizes support of West Virginia Research under the call Research Challenge Grand Program 2022 and NASA EPSCoR Award No. 80NSSC22M0173. A.C.G.-C. acknowledges support from the GridUIS-2 experimental testbed. The latter was developed under the Universidad Industrial de Santander (SC3-UIS) High Performance and Scientific Computing Centre with support from UIS Vicerrectoría de Investigación y Extensión (VIE-UIS) and several UIS research groups. A.C.G.-C. also acknowledge Grant No. 202303059C entitled “Optimización de las Propiedades Termoeléctricas Mediante Tensión Biaxial en la Familia de Materiales Bi 4 O 4 Se X 2 ( X = Cl, Br, I) Desde Primeros Principios” supported by the LNS-BUAP.

The authors have no conflicts to disclose.

Arjyama Bordoloi: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (lead); Methodology (equal); Validation (equal); Writing – original draft (lead); Writing – review & editing (equal). A. C. Garcia-Castro: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Validation (equal); Writing – review & editing (equal). Zachary Romestan: Formal analysis (supporting); Investigation (supporting); Methodology (supporting); Validation (equal); Writing – review & editing (supporting). Aldo H. Romero: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal). Sobhit Singh: Conceptualization (lead); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (lead); Methodology (equal); Supervision (lead); Validation (equal); Writing – original draft (lead); Writing – review & editing (equal).

The data that support the findings of this study are available within the article and the references therein.

1.
I.
Žutić
,
J.
Fabian
, and
S.
Das Sarma
, “
Spintronics: Fundamentals and applications
,”
Rev. Mod. Phys.
76
,
323
(
2004
).
2.
J.
Fabian
,
A.
Matos-Abiague
,
C.
Ertler
,
P.
Stano
, and
I.
Žutić
, “
Semiconductor spintronics
,”
Acta Phys. Slovaca. Reviews and Tutorials
57
,
565
907
(
2007
).
3.
S. A.
Wolf
,
D. D.
Awschalom
,
R. A.
Buhrman
,
J. M.
Daughton
,
S.
von Molnár
,
M. L.
Roukes
,
A. Y.
Chtchelkanova
, and
D. M.
Treger
, “
Spintronics: A spin-based electronics vision for the future
,”
Science
294
,
1488
(
2001
).
4.
C.
Chappert
,
A.
Fert
, and
F. N.
Van Dau
, “
The emergence of spin electronics in data storage
,”
Nat. Mater.
6
,
813
(
2007
).
5.
D. D.
Awschalom
and
M. E.
Flatté
, “
Challenges for semiconductor spintronics
,”
Nat. Phys.
3
,
153
(
2007
).
6.
H.
Zabel
, “
Progress in spintronics
,”
Superlattices Microstruct.
46
,
541
(
2009
).
7.
E. Y.
Vedmedenko
,
R. K.
Kawakami
,
D. D.
Sheka
,
P.
Gambardella
,
A.
Kirilyuk
,
A.
Hirohata
,
C.
Binek
,
O.
Chubykalo-Fesenko
,
S.
Sanvito
,
B. J.
Kirby
,
J.
Grollier
,
K.
Everschor-Sitte
,
T.
Kampfrath
,
C.-Y.
You
, and
A.
Berger
, “
The 2020 magnetism roadmap
,”
J. Phys. D: Appl. Phys.
53
,
453001
(
2020
).
8.
D.
Sander
,
S. O.
Valenzuela
,
D.
Makarov
,
C. H.
Marrows
,
E. E.
Fullerton
,
P.
Fischer
,
J.
McCord
,
P.
Vavassori
,
S.
Mangin
,
P.
Pirro
,
B.
Hillebrands
,
A. D.
Kent
,
T.
Jungwirth
,
O.
Gutfleisch
,
C. G.
Kim
, and
A.
Berger
, “
The 2017 magnetism roadmap
,”
J. Phys. D: Appl. Phys.
50
,
363001
(
2017
).
9.
R. L.
Stamps
,
S.
Breitkreutz
,
J.
Åkerman
,
A. V.
Chumak
,
Y.
Otani
,
G. E. W.
Bauer
,
J.-U.
Thiele
,
M.
Bowen
,
S. A.
Majetich
,
M.
Kläui
,
I. L.
Prejbeanu
,
B.
Dieny
,
N. M.
Dempsey
, and
B.
Hillebrands
, “
The 2014 magnetism roadmap
,”
J. Phys. D: Appl. Phys.
47
,
333001
(
2014
).
10.
A.
Hirohata
,
K.
Yamada
,
Y.
Nakatani
,
I.-L.
Prejbeanu
,
B.
Diény
,
P.
Pirro
, and
B.
Hillebrands
, “
Review on spintronics: Principles and device applications
,”
J. Magn. Magn. Mater.
509
,
166711
(
2020
).
11.
E. C.
Ahn
, “
2D materials for spintronic devices
,”
npj 2D Mater. Appl.
4
,
17
(
2020
).
12.
G.
Binasch
,
P.
Grünberg
,
F.
Saurenbach
, and
W.
Zinn
, “
Enhanced magnetoresistance in layered magnetic structures with antiferromagnetic interlayer exchange
,”
Phys. Rev. B
39
,
4828
(
1989
).
13.
M. N.
Baibich
,
J. M.
Broto
,
A.
Fert
,
F. N.
Van Dau
,
F.
Petroff
,
P.
Etienne
,
G.
Creuzet
,
A.
Friederich
, and
J.
Chazelas
, “
Giant magnetoresistance of (001)Fe/(001)Cr magnetic superlattices
,”
Phys. Rev. Lett.
61
,
2472
(
1988
).
14.
Y.
Yao
,
H.
Cheng
,
B.
Zhang
,
J.
Yin
,
D.
Zhu
,
W.
Cai
,
S.
Li
, and
W.
Zhao
, “
Tunneling magnetoresistance materials and devices for neuromorphic computing
,”
Mater. Futures
2
,
032302
(
2023
).
15.
S. S. P.
Parkin
, “
Giant magnetoresistance in magnetic nanostructures
,”
Annu. Rev. Mater. Res.
25
,
357
(
1995
).
16.
E. Y.
Tsymbal
and
D. G.
Pettifor
, “Perspectives of giant magnetoresistance,” in Solid State Physics, Vol. 56, edited by H. Ehrenreich and F. Spaepen (Academic Press, 2001), pp. 113–237.
17.
E.
Dagotto
, “Brief introduction to giant magnetoresistance (GMR),” in Nanoscale Phase Separation and Colossal Magnetoresistance: The Physics of Manganites and Related Compounds (Springer Berlin Heidelberg, Berlin, 2003), pp. 395–405.
18.
P. A.
Grünberg
, “
Nobel lecture: From spin waves to giant magnetoresistance and beyond
,”
Rev. Mod. Phys.
80
,
1531
(
2008
).
19.
R.
Weiss
,
R.
Mattheis
, and
G.
Reiss
, “
Advanced giant magnetoresistance technology for measurement applications
,”
Meas. Sci. Technol.
24
,
082001
(
2013
).
20.
A.
Fert
,
R.
Ramesh
,
V.
Garcia
,
F.
Casanova
, and
M.
Bibes
, “
Electrical control of magnetism by electric field and current-induced torques
,”
Rev. Mod. Phys.
96
,
015005
(
2024
).
21.
E. I.
Rashba
, “
Theory of electrical spin injection: Tunnel contacts as a solution of the conductivity mismatch problem
,”
Phys. Rev. B
62
,
R16267
(
2000
).
22.
A.
Fert
and
H.
Jaffrès
, “
Conditions for efficient spin injection from a ferromagnetic metal into a semiconductor
,”
Phys. Rev. B
64
,
184420
(
2001
).
23.
G.
Schmidt
and
L. W.
Molenkamp
, “
Spin injection into semiconductors, physics and experiments
,”
Semicond. Sci. Technol.
17
,
310
(
2002
).
24.
G.
Schmidt
,
D.
Ferrand
,
L. W.
Molenkamp
,
A. T.
Filip
, and
B. J.
van Wees
, “
Fundamental obstacle for electrical spin injection from a ferromagnetic metal into a diffusive semiconductor
,”
Phys. Rev. B
62
,
R4790
(
2000
).
25.
E. I.
Rashba
, “
Spin-orbit coupling in condensed matter physics
,”
Sov. Phys. Solid State
2
,
1109
(
1960
).
26.
E.
Rashba
, “
Properties of semiconductors with an extremum loop. I. Cyclotron and combinational resonance in a magnetic field perpendicular to the plane of the loop
,”
Sov. Phys.-Solid State
2
,
1109
(
1960
).
27.
Y. A.
Bychkov
and
E. I.
Rashba
, “
Oscillatory effects and the magnetic susceptibility of carriers in inversion layers
,”
J. Phys. C: Solid State Phys.
17
,
6039
(
1984
).
28.
Y. A.
Bychkov
and
É. I.
Rashba
, “
Properties of a 2D electron gas with lifted spectral degeneracy
,”
JETP Lett.
39
,
78
(
1984
).
29.
M.
Heide
,
G.
Bihlmayer
,
P.
Mavropoulos
,
A.
Bringer
, and
S.
Blügel
, Spin orbit driven physics at surfaces, Newsletter of the Psi-K Network 78 (2006).
30.
G.
Bihlmayer
,
O.
Rader
, and
R.
Winkler
, “
Focus on the Rashba effect
,”
New J. Phys.
17
,
050202
(
2015
).
31.
K.
Premasiri
and
X. P. A.
Gao
, “
Tuning spin–orbit coupling in 2D materials for spintronics: A topical review
,”
J. Phys.: Condens. Matter
31
,
193001
(
2019
).
32.
P.
Barla
,
V. K.
Joshi
, and
S.
Bhat
, “
Spintronic devices: A promising alternative to CMOS devices
,”
J. Comput. Electron.
20
,
805
(
2021
).
33.
D.
Stein
,
K. V.
Klitzing
, and
G.
Weimann
, “
Electron spin resonance on GaAs-Al xGa 1 xAs heterostructures
,”
Phys. Rev. Lett.
51
,
130
(
1983
).
34.
H. L.
Stormer
,
Z.
Schlesinger
,
A.
Chang
,
D. C.
Tsui
,
A. C.
Gossard
, and
W.
Wiegmann
, “
Energy structure and quantized Hall effect of two-dimensional holes
,”
Phys. Rev. Lett.
51
,
126
(
1983
).
35.
K.
Ishizaka
,
M. S.
Bahramy
,
H.
Murakawa
,
M.
Sakano
,
T.
Shimojima
,
T.
Sonobe
,
K.
Koizumi
,
S.
Shin
,
H.
Miyahara
,
A.
Kimura
,
K.
Miyamoto
,
T.
Okuda
,
H.
Namatame
,
M.
Taniguchi
,
R.
Arita
,
N.
Nagaosa
,
K.
Kobayashi
,
Y.
Murakami
,
R.
Kumai
,
Y.
Kaneko
,
Y.
Onose
, and
Y.
Tokura
, “
Giant Rashba-type spin splitting in bulk BiTeI
,”
Nat. Mater.
10
,
521
(
2011
).
36.
M. S.
Bahramy
,
R.
Arita
, and
N.
Nagaosa
, “
Origin of giant bulk Rashba splitting: Application to BiTeI
,”
Phys. Rev. B
84
,
041202
(
2011
).
37.
E.
Plekhanov
,
P.
Barone
,
D.
Di Sante
, and
S.
Picozzi
, “
Engineering relativistic effects in ferroelectric SnTe
,”
Phys. Rev. B
90
,
161108
(
2014
).
38.
F.-X.
Xiang
,
X.-L.
Wang
,
M.
Veldhorst
,
S.-X.
Dou
, and
M. S.
Fuhrer
, “
Observation of topological transition of Fermi surface from a spindle torus to a torus in bulk Rashba spin-split BiTeCl
,”
Phys. Rev. B
92
,
035123
(
2015
).
39.
A.
Narayan
, “
Class of Rashba ferroelectrics in hexagonal semiconductors
,”
Phys. Rev. B
92
,
220101
(
2015
).
40.
Q.
Liu
,
Y.
Guo
, and
A. J.
Freeman
, “
Tunable Rashba effect in two-dimensional LaOBiS 2 films: Ultrathin candidates for spin field effect transistors
,”
Nano Lett.
13
,
5264
(
2013
).
41.
X.
Zhang
,
Q.
Liu
,
J.-W.
Luo
,
A. J.
Freeman
, and
A.
Zunger
, “
Hidden spin polarization in inversion-symmetric bulk crystals
,”
Nat. Phys.
10
,
387
(
2014
).
42.
L.
Yuan
,
Q.
Liu
,
X.
Zhang
,
J.-W.
Luo
,
S.-S.
Li
, and
A.
Zunger
, “
Uncovering and tailoring hidden Rashba spin–orbit splitting in centrosymmetric crystals
,”
Nat. Commun.
10
,
906
(
2019
).
43.
S.
Lee
and
Y.-K.
Kwon
, “
Unveiling giant hidden Rashba effects in two-dimensional Si 2Bi 2
,”
npj 2D Mater. Appl.
4
,
45
(
2020
).
44.
R.
Zhang
,
A.
Marrazzo
,
M. J.
Verstraete
,
N.
Marzari
, and
T. D. P.
Sohier
, “
Gate control of spin-layer-locking FETs and application to monolayer LuIO
,”
Nano Lett.
21
,
7631
(
2021
).
45.
S.
Lee
,
M.
Kim
, and
Y.-K.
Kwon
, “
Unconventional hidden Rashba effects in two-dimensional InTe
,”
npj 2D Mater. Appl.
7
,
43
(
2023
).
46.
Z.
Lin
,
C.
Wang
,
Y.
Xu
, and
W.
Duan
, “
Hidden physical effects in noncentrosymmetric crystals
,”
Phys. Rev. B
102
,
165143
(
2020
).
47.
K.
Zhang
,
S.
Zhao
,
Z.
Hao
,
S.
Kumar
,
E. F.
Schwier
,
Y.
Zhang
,
H.
Sun
,
Y.
Wang
,
Y.
Hao
,
X.
Ma
,
C.
Liu
,
L.
Wang
,
X.
Wang
,
K.
Miyamoto
,
T.
Okuda
,
C.
Liu
,
J.
Mei
,
K.
Shimada
,
C.
Chen
, and
Q.
Liu
, “
Observation of spin-momentum-layer locking in a centrosymmetric crystal
,”
Phys. Rev. Lett.
127
,
126402
(
2021
).
48.
M.
Dykman
,
A.
Efros
,
B.
Halperin
,
L.
Levitov
, and
C.
Marcus
, “
Editorial: Collection in honor of E. I. Rashba and his fundamental contributions to solid-state physics
,”
Phys. Rev. B
106
,
210001
(
2022
).
49.
S.
Datta
and
B.
Das
, “
Electronic analog of the electro-optic modulator
,”
Appl. Phys. Lett.
56
,
665
(
1990
).
50.
S.
Yoshizawa
,
T.
Kobayashi
,
Y.
Nakata
,
K.
Yaji
,
K.
Yokota
,
F.
Komori
,
S.
Shin
,
K.
Sakamoto
, and
T.
Uchihashi
, “
Atomic-layer Rashba-type superconductor protected by dynamic spin-momentum locking
,”
Nat. Commun.
12
,
1462
(
2021
).
51.
B.
Chandrasekhar
, “
A note on the maximum critical field of high-field superconductors
,”
Appl. Phys. Lett.
1
,
7
(
1962
).
52.
A. M.
Clogston
, “
Upper limit for the critical field in hard superconductors
,”
Phys. Rev. Lett.
9
,
266
(
1962
).
53.
J. D.
Sau
,
R. M.
Lutchyn
,
S.
Tewari
, and
S.
Das Sarma
, “
Generic new platform for topological quantum computation using semiconductor heterostructures
,”
Phys. Rev. Lett.
104
,
040502
(
2010
).
54.
J. D.
Sau
,
S.
Tewari
,
R. M.
Lutchyn
,
T. D.
Stanescu
, and
S.
Das Sarma
, “
Non-abelian quantum order in spin-orbit-coupled semiconductors: Search for topological majorana particles in solid-state systems
,”
Phys. Rev. B
82
,
214509
(
2010
).
55.
A.
Das
,
Y.
Ronen
,
Y.
Most
,
Y.
Oreg
,
M.
Heiblum
, and
H.
Shtrikman
, “
Zero-bias peaks and splitting in an Al–InAs nanowire topological superconductor as a signature of Majorana fermions
,”
Nat. Phys.
8
,
887
(
2012
).
56.
J. C. Y.
Teo
and
C. L.
Kane
, “
Majorana fermions and non-abelian statistics in three dimensions
,”
Phys. Rev. Lett.
104
,
046401
(
2010
).
57.
J. D.
Sau
,
D. J.
Clarke
, and
S.
Tewari
, “
Controlling non-abelian statistics of Majorana fermions in semiconductor nanowires
,”
Phys. Rev. B
84
,
094505
(
2011
).
58.
R. J.
Elliott
, “
Spin-orbit coupling in band theory—Character tables for some ‘double’ space groups
,”
Phys. Rev.
96
,
280
(
1954
).
59.
G.
Dresselhaus
,
A. F.
Kip
, and
C.
Kittel
, “
Spin-orbit interaction and the effective masses of holes in germanium
,”
Phys. Rev.
95
,
568
(
1954
).
60.
G.
Dresselhaus
, “
Spin-orbit coupling effects in zinc blende structures
,”
Phys. Rev.
100
,
580
(
1955
).
61.
G.
Dresselhaus
,
A. F.
Kip
, and
C.
Kittel
, “
Cyclotron resonance of electrons and holes in silicon and germanium crystals
,”
Phys. Rev.
98
,
368
(
1955
).
62.
W.
Roland
, “Spin-orbit coupling effects in two-dimensional electron and hole systems,” in Springer Tracts in Modern Physics (Springer, Berlin, 2003), Vol. 191.
63.
S.
Vajna
,
E.
Simon
,
A.
Szilva
,
K.
Palotas
,
B.
Ujfalussy
, and
L.
Szunyogh
, “
Higher-order contributions to the Rashba-Bychkov effect with application to the Bi/Ag(111) surface alloy
,”
Phys. Rev. B
85
,
075404
(
2012
).
64.
S.
Gupta
and
B. I.
Yakobson
, “
What dictates Rashba splitting in 2D van der Waals heterobilayers
,”
J. Am. Chem. Soc.
143
,
3503
(
2021
).
65.
P. Z.
Hanakata
,
A. S.
Rodin
,
A.
Carvalho
,
H. S.
Park
,
D. K.
Campbell
, and
A. H.
Castro Neto
, “
Two-dimensional square buckled Rashba lead chalcogenides
,”
Phys. Rev. B
96
,
161401
(
2017
).
66.
S.
Singh
and
A. H.
Romero
, “
Giant tunable rashba spin splitting in a two-dimensional BiSb monolayer and in BiSb/AlN heterostructures
,”
Phys. Rev. B
95
,
165444
(
2017
).
67.
S.-B.
Yu
,
M.
Zhou
,
D.
Zhang
, and
K.
Chang
, “
Spin Hall effect in the monolayer Janus compound MoSSe enhanced by Rashba spin-orbit coupling
,”
Phys. Rev. B
104
,
075435
(
2021
).
68.
S.
Karmakar
,
R.
Biswas
, and
T.
Saha-Dasgupta
, “
Giant Rashba effect and nonlinear anomalous Hall conductivity in a two-dimensional molybdenum-based Janus structure
,”
Phys. Rev. B
107
,
075403
(
2023
).
69.
A.
Rezavand
and
N.
Ghobadi
, “
Tuning the Rashba spin splitting in Janus MoSeTe and WSeTe van der Waals heterostructures by vertical strain
,”
J. Magn. Magn. Mater.
544
,
168721
(
2022
).
70.
M.
Kim
,
J.
Im
,
A. J.
Freeman
,
J.
Ihm
, and
H.
Jin
, “
Switchable S = 1 / 2 and J = 1 / 2 Rashba bands in ferroelectric halide perovskites
,”
Proc. Natl. Acad. Sci.
111
,
6900
(
2014
).
71.
M. Z.
Hasan
and
C. L.
Kane
, “
Colloquium: Topological insulators
,”
Rev. Mod. Phys.
82
,
3045
(
2010
).
72.
X.-L.
Qi
and
S.-C.
Zhang
, “
Topological insulators and superconductors
,”
Rev. Mod. Phys.
83
,
1057
(
2011
).
73.
B. A.
Bernevig
,
Topological Insulators and Topological Superconductors
(
Princeton University Press
,
2013
).
74.
D.
Vanderbilt
,
Berry Phases in Electronic Structure Theory: Electric Polarization, Orbital Magnetization and Topological Insulators
(
Cambridge University Press
,
2018
).
75.
Y.
Gindikin
and
V. A.
Sablikov
, “
Spin-orbit-driven electron pairing in two dimensions
,”
Phys. Rev. B
98
,
115137
(
2018
).
76.
Y.
Gindikin
and
V. A.
Sablikov
, “
The spin–orbit mechanism of electron pairing in quantum wires
,”
Phys. Stat. Sol. (RRL)—Rapid Res. Lett.
12
,
1800209
(
2018
).
77.
Y.
Gindikin
and
V. A.
Sablikov
, “
Coulomb pairing of electrons in thin films with strong spin-orbit interaction
,”
Phys. E: Low-Dimens. Syst. Nanostruct.
108
,
187
(
2019
).
78.
Y.
Gindikin
and
V. A.
Sablikov
, “
Pair spin–orbit interaction in low-dimensional electron systems
,”
Eur. Phys. J. Spec. Top.
229
,
503
(
2020
).
79.
Y.
Gindikin
and
V. A.
Sablikov
, “
The Coulomb impurity in 2D materials with strong spin–orbit interaction
,”
Phys. Stat. Sol. (B)
258
,
2000501
(
2021
).
80.
Y.
Gindikin
and
V. A.
Sablikov
, “
Spin-dependent electron–electron interaction in Rashba materials
,”
J. Exp. Theor. Phys.
135
,
531
(
2022
).
81.
G.
Bihlmayer
,
P.
Noël
,
D. V.
Vyalikh
,
E. V.
Chulkov
, and
A.
Manchon
, “
Rashba-like physics in condensed matter
,”
Nat. Rev. Phys.
4
,
642
(
2022
).
82.
L.
Zhu
,
T.
Zhang
,
G.
Chen
, and
H.
Chen
, “
Huge Rashba-type spin–orbit coupling in binary hexagonal PX nanosheets (X = As, Sb, and Bi)
,”
Phys. Chem. Chem. Phys.
20
,
30133
(
2018
).
83.
K.
Wu
,
J.
Chen
,
H.
Ma
,
L.
Wan
,
W.
Hu
, and
J.
Yang
, “
Two-dimensional giant tunable Rashba semiconductors with two-atom-thick buckled honeycomb structure
,”
Nano Lett.
21
,
740
(
2021
).
84.
R.
Ahammed
and
A.
De Sarkar
, “
Valley spin polarization in two-dimensional h M N ( M = Nb,Ta) monolayers: Merger of valleytronics with spintronics
,”
Phys. Rev. B
105
,
045426
(
2022
).
85.
M. K.
Mohanta
,
A.
Arora
, and
A.
De Sarkar
, “
Conflux of tunable Rashba effect and piezoelectricity in flexible magnesium monochalcogenide monolayers for next-generation spintronic devices
,”
Nanoscale
13
,
8210
(
2021
).
86.
M. K.
Mohanta
,
F.
IS
,
A.
Kishore
, and
A.
De Sarkar
, “
Spin-current modulation in hexagonal buckled ZnTe and CdTe monolayers for self-powered flexible-piezo-spintronic devices
,”
ACS Appl. Mater. Interfaces
13
,
40872
(
2021
).
87.
C.
Liu
,
H.
Gao
,
Y.
Li
,
K.
Wang
,
L. A.
Burton
, and
W.
Ren
, “
Manipulation of the Rashba effect in layered tellurides MTe (M = Ge, Sn, Pb)
,”
J. Mater. Chem. C
8
,
5143
(
2020
).
88.
M. U.
Rehman
and
Z.
Qiao
, “
MX family: An efficient platform for topological spintronics based on Rashba and Zeeman-like spin splittings
,”
J. Phys.: Condens. Matter
35
,
015001
(
2022
).
89.
M.
Ariapour
and
S. B.
Touski
, “
Spin splitting and Rashba effect at mono-layer GaTe in the presence of strain
,”
Mater. Res. Express
6
,
076402
(
2019
).
90.
A.-Y.
Lu
,
H.
Zhu
,
J.
Xiao
,
C.-P.
Chuu
,
Y.
Han
,
M.-H.
Chiu
,
C.-C.
Cheng
,
C.-W.
Yang
,
K.-H.
Wei
,
Y.
Yang
,
Y.
Wang
,
D.
Sokaras
,
D.
Nordlund
,
P.
Yang
,
D. A.
Muller
,
M.-Y.
Chou
,
X.
Zhang
, and
L.-J.
Li
, “
Janus monolayers of transition metal dichalcogenides
,”
Nat. Nanotechnol.
12
,
744
(
2017
).
91.
J.
Zhang
,
S.
Jia
,
I.
Kholmanov
,
L.
Dong
,
D.
Er
,
W.
Chen
,
H.
Guo
,
Z.
Jin
,
V. B.
Shenoy
,
L.
Shi
, and
J.
Lou
, “
Janus monolayer transition-metal dichalcogenides
,”
ACS Nano
11
,
8192
(
2017
).
92.
C.
Xia
,
W.
Xiong
,
J.
Du
,
T.
Wang
,
Y.
Peng
, and
J.
Li
, “
Universality of electronic characteristics and photocatalyst applications in the two-dimensional Janus transition metal dichalcogenides
,”
Phys. Rev. B
98
,
165424
(
2018
).
93.
T.
Hu
,
F.
Jia
,
G.
Zhao
,
J.
Wu
,
A.
Stroppa
, and
W.
Ren
, “
Intrinsic and anisotropic Rashba spin splitting in Janus transition-metal dichalcogenide monolayers
,”
Phys. Rev. B
97
,
235404
(
2018
).
94.
S.
Chen
,
Z.
Zeng
,
B.
Lv
,
S.
Guo
,
X.
Chen
, and
H.
Geng
, “
Large tunable Rashba spin splitting and piezoelectric response in Janus chromium dichalcogenide monolayers
,”
Phys. Rev. B
106
,
115307
(
2022
).
95.
P. A. L.
Sino
,
L.-Y.
Feng
,
R. A. B.
Villaos
,
H. N.
Cruzado
,
Z.-Q.
Huang
,
C.-H.
Hsu
, and
F.-C.
Chuang
, “
Anisotropic Rashba splitting in Pt-based Janus monolayers PtXY (X,Y = S, Se, or Te)
,”
Nanoscale Adv.
3
,
6608
(
2021
).
96.
Q.-F.
Yao
,
J.
Cai
,
W.-Y.
Tong
,
S.-J.
Gong
,
J.-Q.
Wang
,
X.
Wan
,
C.-G.
Duan
, and
J. H.
Chu
, “
Manipulation of the large Rashba spin splitting in polar two-dimensional transition-metal dichalcogenides
,”
Phys. Rev. B
95
,
165401
(
2017
).
97.
B. D.
Bhat
, “
Rashba spin-splitting in Janus SnXY/WXY (X, Y = S, Se, Te; X Y) heterostructures
,”
J. Phys.: Condens. Matter
35
,
435301
(
2023
).
98.
A.
Bafekry
,
S.
Karbasizadeh
,
C.
Stampfl
,
M.
Faraji
,
D. M.
Hoat
,
I. A.
Sarsari
,
S. A. H.
Feghhi
, and
M.
Ghergherehchi
, “
Two-dimensional Janus semiconductor BiTeCl and BiTeBr monolayers: A first-principles study on their tunable electronic properties via an electric field and mechanical strain
,”
Phys. Chem. Chem. Phys.
23
,
15216
(
2021
).
99.
S.-D.
Guo
,
X.-S.
Guo
,
Z.-Y.
Liu
, and
Y.-N.
Quan
, “
Large piezoelectric coefficients combined with high electron mobilities in Janus monolayer XTeI (X=Sb and Bi): A first-principles study
,”
J. Appl. Phys.
127
,
064302
(
2020
).
100.
K.
Li
,
X.
Xian
,
J.
Wang
, and
N.
Yu
, “
First-principle study on honeycomb fluorated-InTe monolayer with large Rashba spin splitting and direct bandgap
,”
Appl. Surf. Sci.
471
,
18
(
2019
).
101.
R.
Ahammed
,
N.
Jena
,
A.
Rawat
,
M. K.
Mohanta
,
Dimple
, and
A.
De Sarkar
, “
Ultrahigh out-of-plane piezoelectricity meets giant Rashba effect in 2D Janus monolayers and bilayers of group IV transition-metal trichalcogenides
,”
J. Phys. Chem. C
124
,
21250
(
2020
).
102.
G.
Landolt
,
S.
Schreyeck
,
S. V.
Eremeev
,
B.
Slomski
,
S.
Muff
,
J.
Osterwalder
,
E. V.
Chulkov
,
C.
Gould
,
G.
Karczewski
,
K.
Brunner
,
H.
Buhmann
,
L. W.
Molenkamp
, and
J. H.
Dil
, “
Spin texture of Bi 2Se 3 thin films in the quantum tunneling limit
,”
Phys. Rev. Lett.
112
,
057601
(
2014
).
103.
S.-D.
Guo
,
W.-Q.
Mu
,
Y.-T.
Zhu
,
R.-Y.
Han
, and
W.-C.
Ren
, “
Predicted septuple-atomic-layer Janus MSiGeN 4 (M = Mo and W) monolayers with Rashba spin splitting and high electron carrier mobilities
,”
J. Mater. Chem. C
9
,
2464
(
2021
).
104.
Y.-L.
Hong
,
Z.
Liu
,
L.
Wang
,
T.
Zhou
,
W.
Ma
,
C.
Xu
,
S.
Feng
,
L.
Chen
,
M.-L.
Chen
,
D.-M.
Sun
,
X.-Q.
Chen
,
H.-M.
Cheng
, and
W.
Ren
, “
Chemical vapor deposition of layered two-dimensional MoSi 2N 4 materials
,”
Science
369
,
670
(
2020
).
105.
D.
Marchenko
,
A.
Varykhalov
,
M. R.
Scholz
,
G.
Bihlmayer
,
E. I.
Rashba
,
A.
Rybkin
,
A. M.
Shikin
, and
O.
Rader
, “
Giant Rashba splitting in graphene due to hybridization with gold
,”
Nat. Commun.
3
,
1232
(
2012
).
106.
W.
Ju
,
Y.
Xu
,
T.
Li
,
M.
Li
,
K.
Tian
,
J.
Chen
, and
H.
Li
, “
Rashba states localized to InSe layer in InSe/GaTe(InTe) heterostructure
,”
Appl. Surf. Sci.
595
,
153528
(
2022
).
107.
Q.
Zhang
and
U.
Schwingenschlögl
, “
Rashba effect and enriched spin-valley coupling in Ga X/ M X 2 ( M = Mo, W; X = S, Se, Te) heterostructures
,”
Phys. Rev. B
97
,
155415
(
2018
).
108.
L.
Xiang
,
Y.
Ke
, and
Q.
Zhang
, “
Tunable giant Rashba-type spin splitting in PtSe 2/MoSe 2 heterostructure
,”
Appl. Phys. Lett.
115
,
203501
(
2019
).
109.
W.
Ju
,
Y.
Zhang
,
Z.
Gao
,
Q.
Zhou
,
D.
Kang
,
T.
Li
,
M.
Li
,
G.
Hu
, and
H.
Li
, “
Rashba states situated inside the band gap of InTe/PtSe 2 heterostructure
,”
Results Phys.
28
,
104673
(
2021
).
110.
Q.
Peng
,
Y.
Lei
,
X.
Deng
,
J.
Deng
,
G.
Wu
,
J.
Li
,
C.
He
, and
J.
Zhong
, “
Giant and tunable Rashba spin splitting in MoS 2/Bi 2Te 3 heterostructures
,”
Phys. E: Low-Dimens. Syst. Nanostruct.
135
,
114944
(
2022
).
111.
S.
Sattar
and
J. A.
Larsson
, “
Tunable electronic properties and large Rashba splittings found in few-layer Bi 2Se 3/PtSe 2 van der Waals heterostructures
,”
ACS Appl. Electron. Mater.
2
,
3585
(
2020
).
112.
Y.
Kunihashi
,
Y.
Shinohara
,
S.
Hasegawa
,
H.
Nishinaka
,
M.
Yoshimoto
,
K.
Oguri
,
H.
Gotoh
,
M.
Kohda
,
J.
Nitta
, and
H.
Sanada
, “
Bismuth induced enhancement of Rashba spin–orbit interaction in GaAsBi/GaAs heterostructures
,”
Appl. Phys. Lett.
122
,
182402
(
2023
).
113.
M. S.
Bahramy
,
B.-J.
Yang
,
R.
Arita
, and
N.
Nagaosa
, “
Emergence of non-centrosymmetric topological insulating phase in BiTeI under pressure
,”
Nat. Commun.
3
,
679
(
2012
).
114.
A. C. N.
Nonnig
,
A.
da Cas Viegas
,
F. M.
da Rosa
,
P.
Pureur
, and
M. A.
Tumelero
, “
Spin–orbit proximity effect in Bi/Co multilayer: The role of interface scattering
,”
J. Magn. Magn. Mater.
567
,
170312
(
2023
).
115.
T.
Naimer
,
K.
Zollner
,
M.
Gmitra
, and
J.
Fabian
, “
Twist-angle dependent proximity induced spin-orbit coupling in graphene/transition metal dichalcogenide heterostructures
,”
Phys. Rev. B
104
,
195156
(
2021
).
116.
S.
Singh
,
A. M.
Alsharari
,
S. E.
Ulloa
, and
A. H.
Romero
, “Proximity-induced topological transition and strain-induced charge transfer in graphene/MoS 2 bilayer heterostructures,” in Handbook of Graphene Set (John Wiley & Sons, Ltd, 2019), Chap. 1, pp. 1–28.
117.
S.
Singh
,
C.
Espejo
, and
A. H.
Romero
, “
Structural, electronic, vibrational, and elastic properties of graphene/MoS 2 bilayer heterostructures
,”
Phys. Rev. B
98
,
155309
(
2018
).
118.
M.
Peralta
,
E.
Medina
, and
F.
Mireles
, “
Proximity-induced exchange and spin-orbit effects in graphene on Ni and Co
,”
Phys. Rev. B
99
,
195452
(
2019
).
119.
A.
Varykhalov
,
D.
Marchenko
,
M. R.
Scholz
,
E. D. L.
Rienks
,
T. K.
Kim
,
G.
Bihlmayer
,
J.
Sánchez-Barriga
, and
O.
Rader
, “
Ir(111) surface state with giant Rashba splitting persists under graphene in air
,”
Phys. Rev. Lett.
108
,
066804
(
2012
).
120.
E. I.
Rashba
, “
Graphene with structure-induced spin-orbit coupling: Spin-polarized states, spin zero modes, and quantum Hall effect
,”
Phys. Rev. B
79
,
161409
(
2009
).
121.
M.
Krivenkov
,
E.
Golias
,
D.
Marchenko
,
J.
Sánchez-Barriga
,
G.
Bihlmayer
,
O.
Rader
, and
A.
Varykhalov
, “
Nanostructural origin of giant Rashba effect in intercalated graphene
,”
2D Mater.
4
,
035010
(
2017
).
122.
M. M.
Otrokov
,
I. I.
Klimovskikh
,
F.
Calleja
,
A. M.
Shikin
,
O.
Vilkov
,
A. G.
Rybkin
,
D.
Estyunin
,
S.
Muff
,
J. H.
Dil
,
A. L. V.
de Parga
,
R.
Miranda
,
H.
Ochoa
,
F.
Guinea
,
J. I.
Cerdá
,
E. V.
Chulkov
, and
A.
Arnau
, “
Evidence of large spin-orbit coupling effects in quasi-free-standing graphene on Pb/Ir(111)
,”
2D Mater.
5
,
035029
(
2018
).
123.
E.
Zhizhin
,
A.
Varykhalov
,
A.
Rybkin
,
A.
Rybkina
,
D.
Pudikov
,
D.
Marchenko
,
J.
Sánchez-Barriga
,
I.
Klimovskikh
,
G.
Vladimirov
,
O.
Rader
, and
A.
Shikin
, “
Spin splitting of dirac fermions in graphene on Ni intercalated with alloy of Bi and Au
,”
Carbon
93
,
984
(
2015
).
124.
K.
Zollner
and
J.
Fabian
, “
Single and bilayer graphene on the topological insulator Bi 2Se 3: Electronic and spin-orbit properties from first principles
,”
Phys. Rev. B
100
,
165141
(
2019
).
125.
K.
Song
,
D.
Soriano
,
A. W.
Cummings
,
R.
Robles
,
P.
Ordejón
, and
S.
Roche
, “
Spin proximity effects in graphene/topological insulator heterostructures
,”
Nano Lett.
18
,
2033
(
2018
).
126.
J.
Qi
,
X.
Li
,
Q.
Niu
, and
J.
Feng
, “
Giant and tunable valley degeneracy splitting in MoTe 2
,”
Phys. Rev. B
92
,
121403
(
2015
).
127.
W.
Zhou
,
Z.
Yang
,
A.
Li
,
M.
Long
, and
F.
Ouyang
, “
Spin and valley splittings in Janus monolayer WSSe on a MnO(111) surface: Large effective Zeeman field and opening of a helical gap
,”
Phys. Rev. B
101
,
045113
(
2020
).
128.
L.
Fu
and
C. L.
Kane
, “
Topological insulators with inversion symmetry
,”
Phys. Rev. B
76
,
045302
(
2007
).
129.
J. E.
Moore
and
L.
Balents
, “
Topological invariants of time-reversal-invariant band structures
,”
Phys. Rev. B
75
,
121306
(
2007
).
130.
X.-L.
Qi
and
S.-C.
Zhang
, “
The quantum spin Hall effect and topological insulators
,”
Phys. Today
63
,
33
(
2010
).
131.
S.
Singh
,
Z.
Zanolli
,
M.
Amsler
,
B.
Belhadji
,
J. O.
Sofo
,
M. J.
Verstraete
, and
A. H.
Romero
, “
Low-energy phases of Bi monolayer predicted by structure search in two dimensions
,”
J. Phys. Chem. Lett.
10
,
7324
(
2019
).
132.
C.
Jozwiak
,
J. A.
Sobota
,
K.
Gotlieb
,
A. F.
Kemper
,
C. R.
Rotundu
,
R. J.
Birgeneau
,
Z.
Hussain
,
D.-H.
Lee
,
Z.-X.
Shen
, and
A.
Lanzara
, “
Spin-polarized surface resonances accompanying topological surface state formation
,”
Nat. Commun.
7
,
13143
(
2016
).
133.
C.-X.
Liu
,
X.-L.
Qi
,
H.
Zhang
,
X.
Dai
,
Z.
Fang
, and
S.-C.
Zhang
, “
Model Hamiltonian for topological insulators
,”
Phys. Rev. B
82
,
045122
(
2010
).
134.
H.
Zhang
,
C.-X.
Liu
,
X.-L.
Qi
,
X.
Dai
,
Z.
Fang
, and
S.-C.
Zhang
, “
Topological insulators in Bi 2Se 3, Bi 2Te 3, and Sb 2Se 3, with a single Dirac cone on the surface
,”
Nat. Phys.
5
,
438
(
2009
).
135.
S.-D.
Guo
,
J.-X.
Zhu
,
G.-Z.
Wang
,
H.-T.
Guo
,
B.
Wang
,
K.
Cheng
, and
Y. S.
Ang
, “
Switching Rashba spin-splitting by reversing electric-field direction
,”
Phys. Rev. Mater.
7
,
044604
(
2023
).
136.
T.
Das
and
A. V.
Balatsky
, “
Engineering three-dimensional topological insulators in Rashba-type spin-orbit coupled heterostructures
,”
Nat. Commun.
4
,
1972
(
2013
).
137.
I. A.
Nechaev
,
S. V.
Eremeev
,
E. E.
Krasovskii
,
P. M.
Echenique
, and
E. V.
Chulkov
, “
Quantum spin Hall insulators in centrosymmetric thin films composed from topologically trivial BiTeI trilayers
,”
Sci. Rep.
7
,
43666
(
2017
).
138.
N.
Shitrit
,
I.
Yulevich
,
V.
Kleiner
, and
E.
Hasman
, “
Spin-controlled plasmonics via optical Rashba effect
,”
Appl. Phys. Lett.
103
,
211114
(
2013
).
139.
M. P.
Walser
,
U.
Siegenthaler
,
V.
Lechner
,
D.
Schuh
,
S. D.
Ganichev
,
W.
Wegscheider
, and
G.
Salis
, “
Dependence of the Dresselhaus spin-orbit interaction on the quantum well width
,”
Phys. Rev. B
86
,
195309
(
2012
).
140.
A. V.
Koudinov
,
Y. G.
Kusrayev
,
D.
Wolverson
,
L. C.
Smith
,
J. J.
Davies
,
G.
Karczewski
, and
T.
Wojtowicz
, “
Giant modulation of resonance Raman scattering from (Cd, Mn)Te quantum wells by secondary illumination
,”
Phys. Rev. B
79
,
241310
(
2009
).
141.
R.
Völkl
,
M.
Schwemmer
,
M.
Griesbeck
,
S. A.
Tarasenko
,
D.
Schuh
,
W.
Wegscheider
,
C.
Schüller
, and
T.
Korn
, “
Spin polarization, dephasing, and photoinduced spin diffusion in (110)-grown two-dimensional electron systems
,”
Phys. Rev. B
89
,
075424
(
2014
).
142.
H.
Ma
,
Y.
Zhu
,
Y.
Chen
, and
C.
Jiang
, “
Tuning spin-orbit coupling and realizing inverse persistent spin helix by an extra above-barrier radiation in a GaAs/Al 0.3Ga 0.7As heterostructure
,”
Opt. Express
31
,
14473
(
2023
).
143.
M.
Michiardi
,
F.
Boschini
,
H.-H.
Kung
,
M. X.
Na
,
S. K. Y.
Dufresne
,
A.
Currie
,
G.
Levy
,
S.
Zhdanovich
,
A. K.
Mills
,
D. J.
Jones
,
J. L.
Mi
,
B. B.
Iversen
,
P.
Hofmann
, and
A.
Damascelli
, “
Optical manipulation of Rashba-split 2-dimensional electron gas
,”
Nat. Commun.
13
,
3096
(
2022
).
144.
S.
Rossi
,
E.
Talamas Simola
,
M.
Raimondo
,
M.
Acciarri
,
J.
Pedrini
,
A.
Balocchi
,
X.
Marie
,
G.
Isella
, and
F.
Pezzoli
, “
Optical manipulation of the Rashba effect in germanium quantum wells
,”
Adv. Opt. Mater.
10
,
2201082
(
2022
).
145.
J.
Chen
,
K.
Wu
,
W.
Hu
, and
J.
Yang
, “
High-throughput inverse design for 2D ferroelectric Rashba semiconductors
,”
J. Am. Chem. Soc.
144
,
20035
(
2022
).
146.
D.
Zhang
,
P.
Schoenherr
,
P.
Sharma
, and
J.
Seidel
, “
Ferroelectric order in van der Waals layered materials
,”
Nat. Rev. Mater.
8
,
25
(
2023
).
147.
J.
Gou
,
H.
Bai
,
X.
Zhang
,
Y. L.
Huang
,
S.
Duan
,
A.
Ariando
,
S. A.
Yang
,
L.
Chen
,
Y.
Lu
, and
A. T. S.
Wee
, “
Two-dimensional ferroelectricity in a single-element bismuth monolayer
,”
Nature
617
,
67
(
2023
).
148.
X.
Zhang
,
Y.
Lu
, and
L.
Chen
, “
Ferroelectricity in 2D elemental materials
,”
Chin. Phys. Lett.
40
,
067701
(
2023
).
149.
C.
Xiao
,
F.
Wang
,
S. A.
Yang
,
Y.
Lu
,
Y.
Feng
, and
S.
Zhang
, “
Elemental ferroelectricity and antiferroelectricity in group-V monolayer
,”
Adv. Funct. Mater.
28
,
1707383
(
2018
).
150.
Y.
Wang
,
C.
Xiao
,
M.
Chen
,
C.
Hua
,
J.
Zou
,
C.
Wu
,
J.
Jiang
,
S. A.
Yang
,
Y.
Lu
, and
W.
Ji
, “
Two-dimensional ferroelectricity and switchable spin-textures in ultra-thin elemental Te multilayers
,”
Mater. Horiz.
5
,
521
(
2018
).
151.
P.
Man
,
L.
Huang
,
J.
Zhao
, and
T. H.
Ly
, “
Ferroic phases in two-dimensional materials
,”
Chem. Rev.
123
,
10990
(
2023
).
152.
X.
Jin
,
Y.-Y.
Zhang
, and
S.
Du
, “
Recent progress in the theoretical design of two-dimensional ferroelectric materials
,”
Fund. Res.
3
,
322
(
2023
).
153.
Z.
Fei
,
W.
Zhao
,
T. A.
Palomaki
,
B.
Sun
,
M. K.
Miller
,
Z.
Zhao
,
J.
Yan
,
X.
Xu
, and
D. H.
Cobden
, “
Ferroelectric switching of a two-dimensional metal
,”
Nature
560
,
336
(
2018
).
154.
P.
Sharma
,
F.-X.
Xiang
,
D.-F.
Shao
,
D.
Zhang
,
E. Y.
Tsymbal
,
A. R.
Hamilton
, and
J.
Seidel
, “
A room-temperature ferroelectric semimetal
,”
Sci. Adv.
5
,
eaax5080
(
2019
).
155.
Y.
Ma
,
Y.
Dai
,
W.
Wei
,
X.
Li
, and
B.
Huang
, “
Emergence of electric polarity in BiTeX (X = Br and I) monolayers and the giant Rashba spin splitting
,”
Phys. Chem. Chem. Phys.
16
,
17603
(
2014
).
156.
L.
Zhang
,
Y.
Gu
, and
A.
Du
, “
Two-dimensional Janus antimony selenium telluride with large Rashba spin splitting and high electron mobility
,”
ACS Omega
6
,
31919
(
2021
).
157.
W.
Shi
and
Z.
Wang
, “
Mechanical and electronic properties of Janus monolayer transition metal dichalcogenides
,”
J. Phys.: Condens. Matter
30
,
215301
(
2018
).
158.
M.
He
,
X.
Li
,
X.
Liu
,
L.
Li
,
S.
Wei
, and
C.
Xia
, “
Ferroelectric control of band structures in the two-dimensional Janus WSSe/In 2Se 3 van der Waals heterostructures
,”
Phys. E: Low-Dimens. Syst. Nanostruct.
142
,
115256
(
2022
).
159.
Y. M.
Koroteev
,
G.
Bihlmayer
,
J. E.
Gayone
,
E. V.
Chulkov
,
S.
Blügel
,
P. M.
Echenique
, and
P.
Hofmann
, “
Strong spin-orbit splitting on Bi surfaces
,”
Phys. Rev. Lett.
93
,
046403
(
2004
).
160.
S.
LaShell
,
B. A.
McDougall
, and
E.
Jensen
, “
Spin splitting of an Au(111) surface state band observed with angle resolved photoelectron spectroscopy
,”
Phys. Rev. Lett.
77
,
3419
(
1996
).
161.
T.
Koga
,
J.
Nitta
,
T.
Akazaki
, and
H.
Takayanagi
, “
Rashba spin-orbit coupling probed by the weak antilocalization analysis in InAlAs/InGaAs/InAlAs quantum wells as a function of quantum well asymmetry
,”
Phys. Rev. Lett.
89
,
046801
(
2002
).
162.
H.
Choi
,
T.
Kakegawa
,
M.
Akabori
,
T.
kazu Suzuki
, and
S.
Yamada
, “
Spin–orbit interactions in high in-content InGaAs/InAlAs inverted heterojunctions for Rashba spintronic devices
,”
Phys. E: Low-Dimens. Syst. Nanostruct.
40
,
2823
(
2008
).
163.
A. D.
Caviglia
,
M.
Gabay
,
S.
Gariglio
,
N.
Reyren
,
C.
Cancellieri
, and
J.-M.
Triscone
, “
Tunable Rashba spin-orbit interaction at oxide interfaces
,”
Phys. Rev. Lett.
104
,
126803
(
2010
).
164.
L.
Cheng
,
L.
Wei
,
H.
Liang
,
Y.
Yan
,
G.
Cheng
,
M.
Lv
,
T.
Lin
,
T.
Kang
,
G.
Yu
,
J.
Chu
,
Z.
Zhang
, and
C.
Zeng
, “
Optical manipulation of Rashba spin–orbit coupling at SrTiO 3-based oxide interfaces
,”
Nano Lett.
17
,
6534
(
2017
).
165.
A. C.
Garcia-Castro
,
P.
Ghosez
,
E.
Bousquet
, and
A. H.
Romero
, “
Oxyfluoride superlattices KTaO 3/KmF 3 ( M = Zn , Ni): Structural and electronic phenomena
,”
Phys. Rev. B
102
,
235140
(
2020
).
166.
S.
Takahashi
and
S.
Maekawa
, “
Spin current, spin accumulation and spin Hall effect*
,”
Sci. Technol. Adv. Mater.
9
,
014105
(
2008
).
167.
S.
Sugahara
,
Y.
Takamura
,
Y.
Shuto
, and
S.
Yamamoto
, “Field-effect spin-transistors,” in Handbook of Spintronics, edited by Y. Xu, D. D. Awschalom, and J. Nitta (Springer Netherlands, Dordrecht, 2016), pp. 1243–1279.
168.
Y.
Ma
,
S.
Miura
,
H.
Honjo
,
S.
Ikeda
,
T.
Hanyu
,
H.
Ohno
, and
T.
Endoh
, “
A 600- μW ultra-low-power associative processor for image pattern recognition employing magnetic tunnel junction-based nonvolatile memories with autonomic intelligent power-gating scheme
,”
Jpn. J. Appl. Phys.
55
,
04EF15
(
2016
).
169.
A.
Mizrahi
,
T.
Hirtzlin
,
A.
Fukushima
,
H.
Kubota
,
S.
Yuasa
,
J.
Grollier
, and
D.
Querlioz
, “
Neural-like computing with populations of superparamagnetic basis functions
,”
Nat. Commun.
9
,
1533
(
2018
).
170.
W. A.
Borders
,
A. Z.
Pervaiz
,
S.
Fukami
,
K. Y.
Camsari
,
H.
Ohno
, and
S.
Datta
, “
Integer factorization using stochastic magnetic tunnel junctions
,”
Nature
573
,
390
(
2019
).
171.
J.
Slonczewski
, “
Currents and torques in metallic magnetic multilayers
,”
J. Magn. Magn. Mater.
247
,
324
(
2002
).
172.
L.
Berger
, “
Emission of spin waves by a magnetic multilayer traversed by a current
,”
Phys. Rev. B
54
,
9353
(
1996
).
173.
A. D.
Kent
and
D. C.
Worledge
, “
A new spin on magnetic memories
,”
Nat. Nanotechnol.
10
,
187
(
2015
).
174.
T.
Kawahara
,
K.
Ito
,
R.
Takemura
, and
H.
Ohno
, “
Spin-transfer torque RAM technology: Review and prospect
,”
Microelectron. Reliab.
52
,
613
(
2012
).
175.
D.
Apalkov
,
B.
Dieny
, and
J. M.
Slaughter
, “
Magnetoresistive random access memory
,”
Proc. IEEE
104
,
1796
(
2016
).
176.
T.
Jungwirth
,
J.
Wunderlich
, and
K.
Olejník
, “
Spin Hall effect devices
,”
Nat. Mater.
11
,
382
(
2012
).
177.
J.
Sinova
,
S. O.
Valenzuela
,
J.
Wunderlich
,
C. H.
Back
, and
T.
Jungwirth
, “
Spin Hall effects
,”
Rev. Mod. Phys.
87
,
1213
(
2015
).
178.
G.
Vignale
, “
Ten years of spin Hall effect
,”
J. Supercond. Novel Magn.
23
,
3
(
2010
).
179.
J.
Maciejko
,
T. L.
Hughes
, and
S.-C.
Zhang
, “
The quantum spin Hall effect
,”
Annu. Rev. Condens. Matter Phys.
2
,
31
(
2011
).
180.
A.
Manchon
,
H. C.
Koo
,
J.
Nitta
,
S. M.
Frolov
, and
R. A.
Duine
, “
New perspectives for Rashba spin–orbit coupling
,”
Nat. Mater.
14
,
871
(
2015
).
181.
P.
Němec
,
E.
Rozkotová
,
N.
Tesařová
,
F.
Trojánek
,
E.
De Ranieri
,
K.
Olejník
,
J.
Zemen
,
V.
Novák
,
M.
Cukr
,
P.
Malý
, and
T.
Jungwirth
, “
Experimental observation of the optical spin transfer torque
,”
Nat. Phys.
8
,
411
(
2012
).
182.
G. E. W.
Bauer
,
E.
Saitoh
, and
B. J.
van Wees
, “
Spin caloritronics
,”
Nat. Mater.
11
,
391
(
2012
).
183.
A.
Manchon
,
J.
Železný
,
I. M.
Miron
,
T.
Jungwirth
,
J.
Sinova
,
A.
Thiaville
,
K.
Garello
, and
P.
Gambardella
, “
Current-induced spin-orbit torques in ferromagnetic and antiferromagnetic systems
,”
Rev. Mod. Phys.
91
,
035004
(
2019
).
184.
S.
Sugahara
and
J.
Nitta
, “
Spin-transistor electronics: An overview and outlook
,”
Proc. IEEE
98
,
2124
(
2010
).
185.
M.
Johnson
, “
Bipolar spin switch
,”
Science
260
,
320
(
1993
).
186.
D.
Di Sante
,
P.
Barone
,
R.
Bertacco
, and
S.
Picozzi
, “
Electric control of the giant Rashba effect in bulk GeTe
,”
Adv. Mater.
25
,
509
(
2013
).
187.
H.
Djani
,
A. C.
Garcia-Castro
,
W.-Y.
Tong
,
P.
Barone
,
E.
Bousquet
,
S.
Picozzi
, and
P.
Ghosez
, “
Rationalizing and engineering Rashba spin-splitting in ferroelectric oxides
,”
npj Quantum Mater.
4
,
51
(
2019
).
188.
S.
Picozzi
, “
Ferroelectric Rashba semiconductors as a novel class of multifunctional materials
,”
Front. Phys.
2
,
10
(
2014
).
189.
S.
Singh
,
A. C.
Garcia-Castro
,
I.
Valencia-Jaime
,
F.
Muñoz
, and
A. H.
Romero
, “
Prediction and control of spin polarization in a Weyl semimetallic phase of BiSb
,”
Phys. Rev. B
94
,
161116
(
2016
).
190.
S.
Singh
,
W.
Ibarra-Hernández
,
I.
Valencia-Jaime
,
G.
Avendaño-Franco
, and
A. H.
Romero
, “
Investigation of novel crystal structures of Bi–Sb binaries predicted using the minima hopping method
,”
Phys. Chem. Chem. Phys.
18
,
29771
(
2016
).
191.
E.
Bruyer
,
D.
Di Sante
,
P.
Barone
,
A.
Stroppa
,
M.-H.
Whangbo
, and
S.
Picozzi
, “
Possibility of combining ferroelectricity and Rashba-like spin splitting in monolayers of the 1 T-type transition-metal dichalcogenides M X 2 ( M = Mo , W ; X = S , Se , Te )
,”
Phys. Rev. B
94
,
195402
(
2016
).
192.
F.-T.
Huang
,
S.
Joon Lim
,
S.
Singh
,
J.
Kim
,
L.
Zhang
,
J.-W.
Kim
,
M.-W.
Chu
,
K. M.
Rabe
,
D.
Vanderbilt
, and
S.-W.
Cheong
, “
Polar and phase domain walls with conducting interfacial states in a Weyl semimetal MoTe 2
,”
Nat. Commun.
10
,
4211
(
2019
).
193.
S.
Singh
,
J.
Kim
,
K. M.
Rabe
, and
D.
Vanderbilt
, “
Engineering Weyl phases and nonlinear Hall effects in T d-MoTe 2
,”
Phys. Rev. Lett.
125
,
046402
(
2020
).
194.
W.
Hou
,
A.
Azizimanesh
,
A.
Dey
,
Y.
Yang
,
W.
Wang
,
C.
Shao
,
H.
Wu
,
H.
Askari
,
S.
Singh
, and
S. M.
Wu
, “
Strain engineering of vertical molybdenum ditelluride phase-change memristors
,”
Nat. Electron.
7
,
8
(
2024
).
195.
B.
Zhou
, “
Ferroelectric Rashba semiconductors, AgBiP 2X 6 (X = S, Se, and Te), with valley polarization: An avenue towards electric and nonvolatile control of spintronic devices
,”
Nanoscale
12
,
5533
(
2020
).
196.
X.
Fu
,
C.
Jia
,
L.
Sheng
,
Q.
Li
,
J.
Yang
, and
X.
Li
, “
Bipolar Rashba semiconductors: A class of nonmagnetic materials for electrical spin manipulation
,”
J. Phys. Chem. Lett.
14
,
11292
(
2023
).
197.
Y.
Aharonov
and
A.
Casher
, “
Topological quantum effects for neutral particles
,”
Phys. Rev. Lett.
53
,
319
(
1984
).
198.
X.
Wang
and
A.
Manchon
, “
Rashba diamond in an Aharonov–Casher ring
,”
Appl. Phys. Lett.
99
,
142507
(
2011
).
199.
F.
Trier
,
P.
Noël
,
J.-V.
Kim
,
J.-P.
Attané
,
L.
Vila
, and
M.
Bibes
, “
Oxide spin-orbitronics: Spin–charge interconversion and topological spin textures
,”
Nat. Rev. Mater.
7
,
258
(
2022
).
200.
W.
Zhang
and
K. M.
Krishnan
, “
Epitaxial exchange-bias systems: From fundamentals to future spin-orbitronics
,”
Mater. Sci. Eng.: R: Rep.
105
,
1
(
2016
).
201.
Y.
Dong
,
T.
Xu
,
H.-A.
Zhou
,
L.
Cai
,
H.
Wu
,
J.
Tang
, and
W.
Jiang
, “
Electrically reconfigurable 3D spin-orbitronics
,”
Adv. Funct. Mater.
31
,
2007485
(
2021
).
202.
G.
Finocchio
,
M.
Di Ventra
,
K. Y.
Camsari
,
K.
Everschor-Sitte
,
P. K.
Amiri
, and
Z.
Zeng
, “
The promise of spintronics for unconventional computing
,”
J. Magn. Magn. Mater.
521
,
167506
(
2021
).
203.
B.
Wang
,
S.
Shan
,
X.
Wu
,
C.
Wang
,
C.
Pandey
,
T.
Nie
,
W.
Zhao
,
Y.
Li
,
J.
Miao
, and
L.
Wang
, “
Picosecond nonlinear spintronic dynamics investigated by terahertz emission spectroscopy
,”
Appl. Phys. Lett.
115
,
121104
(
2019
).
204.
L.
Guo
,
S.
Hu
,
X.
Gu
,
R.
Zhang
,
K.
Wang
,
W.
Yan
, and
X.
Sun
, “
Emerging spintronic materials and functionalities
,”
Adv. Mater.
36
,
2301854
(
2023
).
205.
J. F.
Sierra
,
J.
Fabian
,
R. K.
Kawakami
,
S.
Roche
, and
S. O.
Valenzuela
, “
Van der Waals heterostructures for spintronics and opto-spintronics
,”
Nat. Nanotechnol.
16
,
856
(
2021
).
206.
H.
Nakayama
,
B.
Xu
,
S.
Iwamoto
,
K.
Yamamoto
,
R.
Iguchi
,
A.
Miura
,
T.
Hirai
,
Y.
Miura
,
Y.
Sakuraba
,
J.
Shiomi
, and
K. I.
Uchida
, “
Above-room-temperature giant thermal conductivity switching in spintronic multilayers
,”
Appl. Phys. Lett.
118
,
042409
(
2021
).
207.
K.-I.
Uchida
and
R.
Iguchi
, “
Spintronic thermal management
,”
J. Phys. Soc. Jpn.
90
,
122001
(
2021
).
208.
E.
Beaurepaire
,
J.-C.
Merle
,
A.
Daunois
, and
J.-Y.
Bigot
, “
Ultrafast spin dynamics in ferromagnetic nickel
,”
Phys. Rev. Lett.
76
,
4250
(
1996
).
209.
J. H.
Mentink
,
J.
Hellsvik
,
D.
Afanasiev
,
B.
Ivanov
,
A.
Kirilyuk
,
A.
Kimel
,
O.
Eriksson
,
M.
Katsnelson
, and
T.
Rasing
, “
Ultrafast spin dynamics in multisublattice magnets
,”
Phys. Rev. Lett.
108
,
057202
(
2012
).
210.
Y.
He
,
X.
Li
,
J.
Yang
,
W.
Li
,
G.
Li
,
T.
Wu
,
W.
Yu
, and
L.
Zhu
, “
High-throughput screening giant bulk spin-split materials
,”
Results Phys.
49
,
106490
(
2023
).