Two-dimensional van der Waals materials offer exceptional tunability in their electronic properties. In this paper, we explore how twisting and hydrostatic pressure can be leveraged to engineer the electronic and optical characteristics of bilayer PtSe2. Using state-of-the-art first-principles density functional methods, we calculate the electronic band structure and the imaginary part of the dielectric function across multiple twist angles and pressure values. We find, that at the twist angle theta=13.17 degrees, bilayer PtSe2, which is intrinsically an indirect semiconductor, transforms into a direct-gap semiconductor. Moreover, we demonstrate that hydrostatic out-of-plane pressure boosts near-infrared optical activity, further expanding the functional potential of PtSe2 bilayers. The demonstrated high tunability of electronic and optical properties by twisting and pressure opens new application directions of PtSe2 in optoelectronics.

By utilizing the proximity effect, we introduce a platform that exploits ferroelectric switching to modulate spin currents in graphene proximitized by a ferroelectric In2Se3 monolayer. Through first-principles calculations and tight-binding modeling, we studied the electronic structure of graphene/In2Se3 heterostructure for twist angles of 0 degrees and 17.5 degrees, considering both ferroelectric polarizations. We discover that switching the ferroelectric polarization reverses the sign of the charge-to-spin conversion coefficients, acting as a chirality switch of the in-plane spin texture in graphene. For the twisted heterostructure, we observed the emergence of an unconventional radial Rashba field for one ferroelectric polarization direction. Additionally, we demonstrated that the Rashba phase can be directly extracted from the ratio of conversion efficiency coefficients, providing a straightforward approach to characterize the in-plane spin texture in graphene. All the unique features of the studied graphene/In2Se3 heterostructure can be experimentally detected, offering a promising approach for developing advanced spintronic devices with enhanced performance and efficiency.

By utilizing the proximity effect, we introduce a platform that exploits ferroelectric switching to modulate spin currents in graphene proximitized by a ferroelectric In 2 Se 3 monolayer. Through first-principles calculations and tight-binding modeling, we studied the electronic structure of graphene/In 2 Se 3 heterostructure for twist angles of 0 ∘ and 17.5 ∘ , considering both ferroelectric polarizations. We discover that switching the ferroelectric polarization reverses the sign of the charge-to-spin conversion coefficients, acting as a chirality switch of the in-plane spin texture in graphene. For the twisted heterostructure, we observed the emergence of an unconventional radial Rashba field for one ferroelectric polarization direction. Additionally, we demonstrated that the Rashba phase can be directly extracted from the ratio of conversion efficiency coefficients, providing a straightforward approach to characterize the in-plane spin texture in graphene. All the unique features of the studied graphene/In 2 Se 3 heterostructure can be experimentally detected, offering a promising approach for developing advanced spintronic devices with enhanced performance and efficiency.

According to the Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) theory, the transition from a solid to liquid in two dimensions proceeds through an orientationally ordered liquid-like hexatic phase. However, alternative mixed melting scenarios, in which melting proceeds through the hexatic phase with both continuous and discontinuous transitions, have also been observed in some two-dimensional systems. In this study, we imaged silver iodide embedded in multilayer graphene using time- and temperature-resolved in situ atomic-resolution scanning transmission electron microscopy and nanobeam electron diffraction. We observed the hexatic phase and provide evidence supporting a mixed melting scenario.

In 1965 Kohn and Luttinger proposed a genuine electronic mechanism for superconductivity. Despite the bare electrostatic interaction between two electrons being repulsive, in a metal electron-hole fluctuations can give rise to Friedel oscillations of the screened Coulomb potential. Cooper pairing among the electrons then emerges when taking advantage of the attractive regions. The nature of the leading pairing mechanism in some two-dimensional transition metal dichalcogenides is still debated. Focusing on NbSe_2, we show that superconductivity can be induced by the Coulomb interaction when accounting for screening effects on the trigonal lattice with multiple orbitals. Using ab initio-based tight-binding parametrizations for the relevant low-energy d-bands, we evaluate the screened interaction microscopically, in a scheme including Bloch overlaps. In the direct space, we find long-range Friedel oscillations alternating in sign, a key to the Kohn-Luttinger mechanism. The momentum-resolved gap equations predict two degenerate solutions at the critical temperature Tc, signaling the unconventional nature of the pairing. Their complex linear combination, i.e., a chiral gap with p-like symmetry, provides the ground state of the system. Our prediction of a fully gapped chiral phase well below Tc is in excellent agreement with the spectral function extracted from tunneling spectroscopy measurements of single-layer NbSe_2.

Materials and systems that exhibit persistent spin texture provide a platform for creating robust spin states that can be used in quantum computing, information storage, and other advanced technological applications. Using first-principles calculations, we demonstrate that a persistent spin texture can be obtained in monolayer phosphorene with broken space inversion symmetry and subjected to external strain. We calculate the effects of strain on the intrinsic and extrinsic spin–orbit coupling in phosphorene and show that for the tensile strain of about 1.2% in the zigzag direction, conduction electrons near the Γ point maintain a consistent and uniform spin polarization direction. The anisotropy of spin polarization is reflected in the parameters of an effective symmetry-adapted spin–orbit Hamiltonian. Furthermore, the calculated spin relaxation rates due to the Dyakonov–Perel mechanism display a giant anisotropy, up to 10 ^5 , which is directly related to the discovered persistent spin texture. We also show that strain can reverse the anisotropy of spin mixing parameter b ^2 connected to the Elliott–Yafet spin relaxation mechanism. Our results suggest that spin texture in phosphorene can be modulated by strain, enabling its potential usage in spintronics.

Spin–orbit coupling (SOC) is fundamental to many phenomena in solid-state physics. Two-dimensional materials and van der Waals heterostructures provide researchers with exquisite control over this interaction; the ability to fine-tune SOC has impacts on spin transport and relaxation, topological states, optoelectronics, magnetization dynamics and even superconductivity and other correlated states. This Technical Review covers both the theoretical methodology and experimentally relevant phenomenology of SOC in 2D materials, by providing essential insights into the process of extracting the spin interactions from the underlying electronic structure obtained from first-principles density functional theory calculations. This Technical Review begins with graphene. Its SOC has a surprisingly complicated origin yet graphene remains the benchmark for other elemental centrosymmetric 2D materials in which SOC leads to a mixing of spin-up and spin-down components of the Bloch states. We then discuss spin–orbit materials, such as transition-metal dichalcogenides, in which strong SOC and the lack of space-inversion symmetry yield large spin splittings of the valence and conduction bands. This enables highly efficient optical spin orientation or robust valley Hall effect in transition-metal dichalcogenides. Next, we give guidelines for extracting the spin–orbit characteristics of van der Waals heterostructures, such as graphene/WSe2, which serve as a platform for SOC engineering. For these representative systems, we highlight the essentials of first-principles-based methodology, including supercell formation, strain artefacts, twisting, gating and lattice relaxation. Finally, we briefly discuss the effects of proximity exchange coupling, which is another relevant spin interaction for spintronics. This Technical Review demonstrates how first-principles calculations and effective modelling provide realistic insights into spin–orbit interactions and their engineering in 2D materials and van der Waals heterostructures.

We study spin-orbit proximity effects in an armchair (4,4) carbon nanotube on the Pt(111) surface. By employing first-principles calculations, we show that the Dirac cone of the metallic nanotube is altered due to strong hybridization with the Pt substrate. Inserting a monolayer hexagonal boron nitride (hBN) between the nanotube and the substrate limits the hybridization effects leading to recovering the Dirac cone. The Dirac bands display asymmetric spin splitting, 0.7 meV for the right movers and 1.7 meV for the left movers at the K valley, due to the proximity to the Pt substrate. We find that the Dirac states exhibit almost perfect spin polarization, transverse to the nanotube axis and to the stacking direction, forming a proper condition for a charge-to-spin conversion with coherent spin transport in the nanotube. We propose an effective Hamiltonian describing the proximity-induced effects on the Dirac electrons and their spin texture.

Effective control of interlayer interactions is a key element in modifying the properties of van der Waals heterostructures and the next step toward their practical applications. Focusing on the phosphorene-WSe2 heterostructure, we demonstrate, using first-principles calculations, proximity-induced amplification of the spin-orbit coupling in phosphorene by applying vertical pressure. We simulate external pressure by changing the interlayer distance between bilayer constituents and show that it is possible to tune the spin-orbit field of phosphorene holes in a controllable way. By fitting effective electronic states of the proposed Hamiltonian to the first-principles data, we reveal that the spin-orbit coupling in phosphorene hole bands is enhanced more than two times for experimentally accessible pressures up to 17 kbar. Correspondingly, we find that the pressure-enhanced spin-orbit coupling boosts the Dyakonov-Perel spin relaxation mechanism, reducing the spin lifetime of phosphorene holes by factor 4. We further explore the role of the lateral shift on the spin-orbit field and reveal that the spin-orbit strength of phosphorene holes can be sizably modulated when strong pressure is applied. We also found that the thermopower is governed mainly by the phosphorene and pressure reduces the overall thermoelectric efficiency of the heterostructure.

We analyze, using first-principles calculations and the method of invariants, the spin-orbit proximity effects in trilayer heterostructures comprising phosphorene and encapsulating WSe$_2$ monolayers. We focus on four different configurations, in which the top/bottom WSe$_2$ monolayer is twisted by 0 or 60 degrees with respect to phosphorene, and analyze the spin splitting of phosphorene hole bands around the $\Gamma$ point. Our results show that the spin texture of phosphorene hole bands can be dramatically modified by different encapsulations of phosphorene monolayer. For a symmetrically encapsulated phosphorene, the momentum-dependent spin-orbit field has the out-of-plane component only, simulating the spin texture of phosphorene-like group-IV monochalcogenide ferroelectrics. Furthermore, we reveal that the direction of the out-of-plane spin-orbit field can be controlled by switching the twist angle from 0 to 60 degrees. Finally, we show that the spin texture in asymmetrically encapsulated phosphorene has the dominant in-plane component of the spin-orbit field, comparable to the Rashba effect in phosphorene with an applied sizable external electric field. Our results confirm that the significant modification and control of the spin texture is possible in low common-symmetry heterostructures, paving the way for using different substrates to modify spin properties in materials important for spintronics.

We analyze the spin-orbit coupling effects in a three-degree twisted bilayer heterostructure made of graphene and an in-plane ferroelectric SnTe, with the goal of transferring the spin-orbit coupling from SnTe to graphene, via the proximity effect. Our results indicate that the point-symmetry breaking due to the incompatible mutual symmetry of the twisted monolayers and a strong hybridization has a massive impact on the spin splitting in graphene close to the Dirac point, with the spin splitting values greater than 20 meV. The band structure and spin expectation values of graphene close to the Dirac point can be described using a symmetry-free model, triggering different types of interaction with respect to the threefold symmetric graphene/transition-metal dichalcogenide heterostructure. We show that the strong hybridization of the Dirac cone's right movers with the SnTe band gives rise to a large asymmetric spin splitting in the momentum space. Furthermore, we discover that the ferroelectricity-induced Rashba spin-orbit coupling in graphene is the dominant contribution to the overall Rashba field, with the effective in-plane electric field that is almost aligned with the (in-plane) ferroelectricity direction of the SnTe monolayer. We also predict an anisotropy of the in-plane spin relaxation rates. Our results demonstrate that the group-IV monochalcogenides MX (M=Sn, Ge; X=S, Se, Te) are a viable alternative to transition-metal dichalcogenides for inducing strong spin-orbit coupling in graphene.

The immunity of Ising superconductors to external magnetic fields originates from a spin locking of the paired electrons to an intrinsic Zeeman-like field. The spin-momentum locking in non-centrosymmetric crystalline materials leads to type-I Ising pairing in which the direction of the intrinsic field can be deduced from the spin expectation values. Conversely, in centrosymmetric crystals the electron spins locked to the orbitals can form Ising type-II pairs consisting of spin-orbit split doublets. Due to time-reversal symmetry, the doublets are spin degenerate, making it difficult to read the spin polarization of bands and the direction of spin-orbit fields. Here we present an efficient approach to determine the direction of the intrinsic field using the spin-mixing parameter $b^2$. Using first principles calculations based on the density functional theory, we study monolayer transition metal dichalcogenide superconductors PdTe$_2$, NbTe$_2$, and TiSe$_2$ with the 1T structure. We calculate $b^2$ for individual Fermi pockets and provide a general picture of possible Ising type-II pairing within the full Brillouin zone. In order to complement our first principles results, we use group theory to provide a detailed picture of spin-orbit coupling and spin mixing in the relevant bands forming Fermi pockets. We demonstrate that contrary to the anticipated effects of spin-orbit locking, not every spin-orbit split spin doublet actively participates in Ising pairing. Finally, by connecting the spin-mixing parameter $b^2$ with the intrinsic out-of-plane Zeeman field we estimate the upper in-plane critical magnetic field.

Recently, the learning by confusion (LbC) approach has been proposed as a machine learning tool to determine the critical temperature Tc of phase transitions without any prior knowledge of its even approximate value. The method has been proven effective, but it has been used only for continuous phase transitions, where the confusion results only from deliberate incorrect labeling of the data. However, in the case of a discontinuous phase transition, additional confusion can result from the coexistence of different phases. To verify whether the confusion scheme can also be used for discontinuous phase transitions, we apply the LbC method to three microscopic models, the Blume-Capel, the q-state Potts, and the Falicov-Kimball models, which undergo continuous or discontinuous phase transitions depending on model parameters. With the help of a simple model, we predict that the phase coexistence present in discontinuous phase transitions can indeed make the neural network more confused and thus decrease its performance. However, numerical calculations performed for the models mentioned above indicate that other aspects of this kind of phase transition are more important and can render the LbC method even less effective. Nevertheless, we demonstrate that in some cases the same aspects allow us to use the LbC method to identify the order of a phase transition.

We study spin-orbit proximity effects in a hybrid heterostructure build of a one-dimensional (1D) armchair carbon nanotube and two-dimensional (2D) buckled monolayer bismuthene. We show, by performing first-principles calculations, that Dirac electrons in the nanotube exhibit large spin-orbit coupling due to a close vicinity of bismuthene. The calculated low-energy band structures of the proximized nanotube display a strong dependence on the position of the nanotube on the substrate, similar to twist-angle dependence found in 2D heterostructures. Based on the first-principles results, we formulate an effective low-energy Hamiltonian of the nanotube and identify key interactions governing the proximity spin-orbit coupling. The proximity-induced spin splitting of Dirac cone bands is in meV range, confirming an efficient transfer of spin-orbit coupling from bismuthene to the nanotube.

We study the spin-orbit proximity effects in a hybrid heterostructure built from a one-dimensional (1D) armchair carbon nanotube and two-dimensional (2D) buckled monolayer bismuthene. We show, by performing first-principles calculations, that Dirac electrons in the nanotube exhibit large spin-orbit coupling due to the close vicinity of bismuthene. The calculated low-energy band structure and the spin texture of the proximitized nanotube display a strong dependence on the position of the nanotube on the substrate, similar to the twist-angle dependence found in 2D heterostructures. Based on the first-principles results, we formulate an effective low-energy Hamiltonian of the nanotube, and we identify key interactions governing the proximity spin-orbit coupling. The proximity-induced spin splitting of Dirac cone bands is in the meV range, confirming an efficient transfer of spin-orbit coupling from bismuthene to the nanotube.

We investigate, using first-principles methods and effective-model simulations, the spin-orbit coupling proximity effects in a bilayer heterostructure comprising phosphorene and WSe$_2$ monolayers. We specifically analyze holes in phosphorene around the $\Gamma$ point, at which we find a significant increase of the spin-orbit coupling that can be attributed to the strong hybridization of phosphorene with the WSe$_2$ bands. We also propose an effective spin-orbit model based on the ${\bf C}_{1{\rm v}}$ symmetry of the studied heterostructure. The corresponding spin-orbit field can be divided into two parts: the in-plane field, present due to the broken nonsymmorphic horizontal glide mirror plane symmetry, and the dominant out-of-plane field triggered by breaking the out-of-plane rotational symmetry of the phosphorene monolayer. Furthermore, we also demonstrate that a heterostructure with 60$^\circ$ twist angle exhibits an opposite out-of-plane spin-orbit field, indicating that the coupling can effectively be tuned by twisting. The studied phosphorene/WSe$_2$ bilayer is a prototypical low common-symmetry heterostructure in which the proximity effect can be used to engineer the spin texture of the desired material.

Abstract Buckled monolayer nitrogene has been recently predicted to be stable above the room temperature. The low atomic number of nitrogen atom suggests, that spin–orbit coupling in nitrogene is weak, similar to graphene or silicene. We employ first principles calculations and perform a systematic study of the intrinsic and extrinsic spin–orbit coupling in this material. We calculate the spin mixing parameter $$b^2$$ b 2 , reflecting the strength of the intrinsic spin–orbit coupling and find, that $$b^2$$ b 2 is relatively small, on the order of $$10^{-6}$$ 10 - 6 . It also displays a weak anisotropy, opposite for electrons and holes. To study extrinsic effects of spin–orbit coupling we apply a transverse electric field enabling spin–orbit fields $$\Omega$$ Ω . We find, that $$\Omega$$ Ω are on the order of a single $$\mu$$ μ eV in the valence band, and tens to a hundred of $$\mu$$ μ eV in the conduction band, depending on the applied electric field. Similar to $$b^2$$ b 2 , $$\Omega$$ Ω is also anisotropic, in particular for the conduction electrons.

Transition metal dichalcogenides (TMDCs) are ideal candidates to explore the manifestation of spin-valley physics under external stimuli. In this study, we investigate the influence of strain on the spin and orbital angular momenta, effective g -factors, and Berry curvatures of several monolayer TMDCs (Mo and W based) using a full ab initio approach. At the K -valleys, we find a surprising decrease of the conduction band spin expectation value for compressive strain, consequently increasing the dipole strength of the dark exciton by more than one order of magnitude (for ∼ 1 % – 2 % strain variation). We also predict the behavior of direct excitons g -factors under strain: tensile (compressive) strain increases (decreases) the absolute value of g -factors. Strain variations of ∼1% modify the bright (A and B) excitons g -factors by ∼0.3 (0.2) for W (Mo) based compounds and the dark exciton g -factors by ∼0.5 (0.3) for W (Mo) compounds. Our predictions could be directly visualized in magneto-optical experiments in strained samples at low temperature. Additionally, our calculations strongly suggest that strain effects are one of the possible causes of g -factor fluctuations observed experimentally. By comparing the different TMDC compounds, we reveal the role of spin–orbit coupling (SOC): the stronger the SOC, the more sensitive are the spin-valley features under applied strain. Consequently, monolayer WSe 2 is a formidable candidate to explore the role of strain on the spin-valley physics. We complete our analysis by considering the side valleys, Γ and Q points, and by investigating the influence of strain in the Berry curvature. In the broader context of valley- and strain-tronics, our study provides fundamental microscopic insights into the role of strain in the spin-valley physics of TMDCs, which are relevant to interpret experimental data in monolayer TMDCs as well as TMDC-based van der Waals heterostructures.

Recently, the learning by confusion (LBC) approach has been proposed as a machine learning tool to determine the critical temperature Tc of phase transitions without any prior knowledge of its even approximate value. However, the effectiveness of the method has been demonstrated only for continuous phase transitions, where confusion can result only from a deliberate incorrect labeling of the data and not from the coexistence of different phases. To verify whether the confusion scheme can also be used for discontinuous phase transitions, in this work, we apply the LBC method to three microscopic models, the Blume-Capel, the q-state Potts, and the Falicov-Kimball models, which undergo continuous or discontinuous phase transitions depending on model parameters. With the help of a simple model, we predict that the phase coexistence present in discontinuous phase transitions can make the neural network more confused and thus decrease its performance. However, numerical calculations performed for the models mentioned above indicate that other aspects of this kind of phase transition are more important and can render the LBC method less effective. Nevertheless, we demonstrate that in some cases the same aspects allow us to use the LBC method to identify the order of a phase transition

Monolayer PtSe$_2$ is a semiconducting transition metal dichalcogenide characterized by an indirect band gap, space inversion symmetry, and high carrier mobility. Strong intrinsic spin-orbit coupling and the possibility to induce extrinsic spin-orbit fields by gating make PtSe$_2$ attractive for fundamental spin transport studies as well as for potential spintronics applications. We perform a systematic theoretical study of the spin-orbit coupling and spin relaxation in this material. Specifically, we employ first principles methods to obtain the basic orbital and spin-orbital properties of PtSe$_2$, also in the presence of an external transverse electric field. We calculate the spin mixing parameters $b^2$ and the spin-orbit fields $\Omega$ for the Bloch states of electrons and holes. This information allows us to predict the spin lifetimes due to the Elliott-Yafet and D'yakonov-Perel mechanisms. We find that $b^2$ is rather large, on the order of $10^{-2}$ and $10^{-1}$, while $\Omega$ varies strongly with doping, being about $10^{3} - 10^{4}$\,ns$^{-1}$ for %typical Fermi levels in the interval $(10-100)$ meV, carrier density in the interval $10^{13}-10^{14}$\,cm$^{-2}$ at the electric field of 1 V/nm. We estimate the spin lifetimes to be on the picosecond level.