We consider RKKY interaction between two magnetic impurities in graphene. The consideration is based on the perturbation theory for the thermodynamic potential in the imaginary time representation. We analyze the symmetry of the RKKY interaction on the bipartite lattice at half filling. Our analytical calculation of the interaction is based on direct evaluation of real space spin susceptibility. We show in the Appendix, added to the published version, that the approach can be easily generalized to the case of finite temperature.
In our previous work (E. Kogan, Phys. Rev. B {\bf 84}, 115119 (2011)) we calculated RKKY interaction between two magnetic impurities in pristine graphene using the Green's functions (GF) in the coordinate -- imaginary time representation. Now we show that the calculations of the GF in this representation can be simplified by using the Feynman's trick, which allows to easily calculate RKKY interaction in gapped graphene. We also present calculations of the RKKY interaction in gapped or doped graphene using the coordinate -- imaginary frequency representation. Both representations, corresponding…
We show explicitly that two approaches to calculations of RKKY interaction between two magnetic impurities in graphene based on Green's functions: one using coordinate--imaginary time representation, and the other using momentum--frequency representation, give exactly the same results and none of them has to operate with diverging integrals and implement the cut-off procedure. Together with this, we claim that the first approach is more convenient for the calculations than the second one.
In our previous work (E. Kogan, Phys. Rev. B 84, 115119 (2011)) we presented calculation of RKKY interaction between two magnetic impurities in graphene based on Matsubara Green's functions (MGF) in the coordinate -- imaginary time representation. Now we present the calculation based on MGF in the coordinate -- frequency representation. We claim that both approaches have an important advantage over those based on zero temperature Green's functions (ZTGF), which are very briefly reviewed in the beginning of the present work. The MGF approaches, in distinction to the ZTGF approaches, operate onl…
The carrier-mediated RKKY interaction between local spins plays an important role for the application of magnetically doped graphene in spintronics and quantum computation. Previous studies largely concentrate on the influence of electronic states of uniform systems on the RKKY interaction. Here we reveal a very different way to manipulate the RKKY interaction by showing that the anomalous focusing - a well-known electron optics phenomenon in graphene P-N junctions - can be utilized to refocus the massless Dirac electrons emanating from one local spin to the other local spin. This gives rise t…
We demonstrate that the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction in graphene can be strongly modified by a time-periodic driving field even in the weak drive regime. This effect is due to the opening of a dynamical band gap at the Dirac points when graphene is exposed to circularly polarized light. Using Keldysh-Floquet Green's functions, we develop a theoretical framework to calculate the time-averaged RKKY coupling under weak periodic drives and show that its magnitude in undoped graphene can be decreased controllably by increasing the driving strength, while mostly maintaining its f…
We present a theory of both the itinerant carrier-mediated RKKY interaction and the virtual excitations-mediated Bloembergen-Rowland (BR) interaction between magnetic moments in graphene induced by proximity effect with a ferromagnetic film. We show that the RKKY/BR interaction consists of the Heisenberg, Ising, and Dzyaloshinskii-Moriya (DM) terms. In the case of the nearest distance, we estimate the DM term from the RKKY/BR interaction is about 0.13 meV for the graphene/Co interface, which is consistent with the experimental result of DM interaction $0.16 \pm 0.05$ meV. Our calculations indi…
In our publication from 8 years ago (Phys. Rev. B {\bf 84}, 115119 (2011)) we calculated RKKY interaction between two magnetic impurities adsorbed on graphene at zero temperature. We show in this short paper that the approach based on Matsubara formalism and perturbation theory for the thermodynamic potential in the imaginary time and coordinate representation which was used then, can be easily generalized, and calculate RKKY interaction between the magnetic impurities at finite temperature.
The growing interest in carbon-based spintronics has stimulated a number of recent theoretical studies on the RKKY interaction in graphene, based on which the energetically favourable alignment between magnetic moments embedded in this material can be calculated. The general consensus is that the strength of the RKKY interaction in graphene decays as 1/D3 or faster, where D is the separation between magnetic moments. Such an unusually fast decay for a 2-dimensional system suggests that the RKKY interaction may be too short ranged to be experimentally observed in graphene. Here we show in a mat…
The growing interest in carbon-based spintronics has stimulated a number of recent theoretical studies on the RKKY interaction in graphene, with the aim of determining the most energetically favourable alignments between embedded magnetic moments. The RKKY interaction in undoped graphene decays faster than expected for conventional two-dimensional materials and recent studies suggest that the adsorption configurations favoured by many transition-metal impurities may lead to even shorter ranged decays and possible sign-changing oscillations. Here we show that these features emerge in a mathemat…
We study Rudermann-Kittel-Kasuya-Yosida (RKKY) interaction in carbon nanotubes (CNTs) and graphene nanoribbons in the presence of spin orbit interactions and magnetic fields. For this we evaluate the static spin susceptibility tensor in real space in various regimes at zero temperature. In metallic CNTs the RKKY interaction depends strongly on the sublattice and, at the Dirac point, is purely ferromagnetic (antiferromagnetic) for the localized spins on the same (different) sublattice, whereas in semiconducting CNTs the spin susceptibility depends only weakly on the sublattice and is dominantly…
We calculate the screening function in bilayer graphene (BLG) both in the intrinsic (undoped) and the extrinsic (doped) regime within random phase approximation, comparing our results with the corresponding single layer graphene (SLG) and the regular two dimensional electron gas (2DEG). We find that the Kohn anomaly is strongly enhanced in BLG. We also discuss the Friedel oscillation and the RKKY interaction, which are associated with the non-analytic behavior of the screening function at $q=2k_F$. We find that the Kohn anomaly, the Friedel oscillation, and the RKKY interaction are all qualita…
Of fundamental interest in the field of spintronics is the mechanism of indirect exchange coupling between magnetic impurities embedded in metallic hosts. A range of physical features, such as magnetotransport and overall magnetic moment formation, are predicated upon this magnetic coupling, often referred to as the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction. Recent theoretical studies on the RKKY in graphene have been motivated by possible spintronic applications of magnetically doped graphene systems. In this work a combination of analytic and numerical techniques are used to examine t…
We show that the carrier-mediated exchange interaction, the so-called RKKY coupling, between two magnetic impurity moments in graphene is significantly modified in the presence of electron-electron interactions. Using the mean-field approximation of the Hubbard-$U$ model we show that the $(1+\cos(2{\bf k}_D\cdot {\bf R})$-oscillations present in the bulk for non-interacting electrons disappear and the power-law decay becomes more long ranged with increasing electron interactions. In zigzag graphene nanoribbons the effects are even larger with any finite $U$ rendering the long-distance RKKY cou…
We study the carrier-mediated exchange interaction, the so-called RKKY coupling, between two magnetic moments in graphene using exact diagonalization on the honeycomb lattice. By using the tight-binding nearest neighbor band structure of graphene we avoid the use of a momentum cut-off which plagues results in the Dirac continuum model formulation. We extract both the short and long impurity-impurity distance behavior and show several corrections to earlier long distance results. In the bulk the RKKY coupling is proportional to $1/|{\bf R}|^3$ and displays $(1+\cos(2{\bf k}_D\cdot {\bf R})$-typ…
Magnetically-doped graphene systems are potential candidates for application in future spintronic devices. A key step is to understand the pairwise interactions between magnetic impurities embedded in graphene that are mediated by the graphene conduction electrons. A large number of studies have been undertaken to investigate the indirect exchange, or RKKY, interactions in graphene. Many of these studies report a decay rate faster than expected for a 2-dimensional material and the absence of the usual distance dependent oscillations. In this review we summarize the techniques used to calculate…
We study the RKKY interaction between two magnetic impurities located on same layer (intralayer case) or on different layers (interlayer case) in undoped bilayer graphene in the four-bands model, by directly calculating the Green functions in the eigenvalues and eigenvectors representation. Our results show that both intra- and interlayer RKKY interactions between two magnetic impurities located on same (opposite) sublattice are always ferromagnetic (antiferromagnetic). Furthermore we find unusual long-distance decay of the RKKY interaction in BLG. The intralyer RKKY interactions between two m…
We report the dependence of Ruderman-Kittel-Kasuya-Yoshida\,(RKKY) interaction on nonmagmetic disorder and gate voltage in grapheme. First the semiclassical method is employed to reserve the expression for RKKY interaction in clean graphene. Due to the pseudogap at Dirac point, the RKKY coupling in undoped grapheme is found to be proportional to $1/R^3$. Next, we investigate how the RKKY interaction depends on nonmagnetic disorder strength and gate voltage by studying numerically the Anderson tight-binding model on a honeycomb lattice. We observe that the RKKY interaction along the armchair di…
We use first-principles Quantum Monte-Carlo simulations to study the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between hydrogen adatoms attached to a graphene sheet. We find that the pairwise RKKY interactions at distances of a few lattice spacings are strongly affected by inter-electron interactions, in particular, the potential barrier between widely separated adatoms and the dimer configuration becomes wider and thus harder to penetrate. We also point out that anti-ferrromagnetic and charge density wave orderings have very different effects on the RKKY interaction. Finally, we analyz…
We study the Ruderman-Kittle-Kasuya-Yosida (RKKY) interaction in the presence of spin polarized two dimensional Dirac fermions. We show that a spin polarization along the z-axis mediates an anisotropic interaction which corresponds to a XXZ model interaction between two magnetic moments. For undoped graphene, while the $x$ part of interaction keeps its constant ferromagnetic sign, its $z$ part oscillates with the distance of magnetic impurities, $R$. A finite doping causes that both parts of the interaction oscillate with $R$. We explore a beating pattern of oscillations of the RKKY interactio…
We investigate the effect of vacancy defects on the electronic and magnetic properties of zigzag graphene nanoribbons (zGNRs) by making use of the Green's function formalism in combination with the tight-binding Hamiltonian. The evolution of the indirect exchange coupling, known as Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction, including single, double, and multiple 5-8-5 divacancy defects is explained. Our numerical calculations show that the changes in the electronic structure and the exchange coupling of zGNRs depend significantly on the location of the divacancy defects with respect to …
Graphene has been identified as a promising material with numerous applications, particularly in spintronics. In this paper we investigate the peculiar features of spin excitations of magnetic units deposited on graphene nanoribbons and how they can couple through a dynamical interaction mediated by spin currents. We examine in detail the spin lifetimes and identify a pattern caused by vanishing density of states sites in pristine ribbons with armchair borders. Impurities located on these sites become practically invisible to the interaction, but can be made accessible by a gate voltage or dop…
The indirect exchange interaction is one of the key factors in determining the overall alignment of magnetic impurities embedded in metallic host materials. In this work we examine the range of this interaction in magnetically-doped graphene systems in the presence of armchair edges using a combination of analytical and numerical Green function (GF) approaches. We consider both a semi-infinite sheet of graphene with a single armchair edge, and also quasi-one-dimensional armchair edged graphene nanoribbons (GNRs). While we find signals of the bulk decay rate in semi-infinite graphene and signal…
We study the exchange interaction $J$ between two magnetic impurities in graphene (the RKKY interaction) by directly computing the lattice Green's function for the tight-binding band structure for the honeycomb lattice. The method allows us to compute $J$ numerically for much larger distances than can be handled by finite-lattice calculations as well as for small distances. % avoids the use of a cutoff function often invoked in the literature to curtail the diverging contributions from the linear bands and yields results that are valid for all distances. In addition, we rederive the analytical…
We investigate the effects of nonmagnetic disorder on the Ruderman-Kittel-Kasuya-Yoshida (RKKY) interaction in graphene by studying numerically the Anderson model with on-site and hopping disorder on a honeycomb lattice at half filling. We evaluate the strength of the interaction as a function of the distance R between two magnetic ions, as well as their lattice positions and orientations. In the clean limit, we find that the strength of the interaction decays as 1/R^3, with its sign and oscillation amplitude showing strong anisotropy. With increasing on-site disorder, the mean amplitude decre…
We calculate the indirect charge carrier mediated Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between magnetic impurities for two selected graphene nanoflakes containing four hexagonal rings in their structure, differing by their geometry. We describe the electronic structure of either charge-neutral or doped nanoflakes using the tight-binding approximation with the Hubbard term, which is treated within the molecular-field approximation. We find pronounced differences in the RKKY coupling energies, dependent on the placement of the pair of magnetic moments in the nanostructure and on the …
We obtain an analytical expression for the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction $J$ in electron or hole doped graphene for linear Dirac bands. The results agree very well with the numerical calculations for the full tight-binding band structure in the regime where the linear band structure is valid. The analytical result, expressed in terms of the Meijer G-function, consists of a product of two oscillatory terms, one coming from the interference between the two Dirac cones and the second coming from the finite size of the Fermi surface. For large distances, the Meijer G-function be…
A form of an indirect Ruderman-Kittel-Kasuya-Yosida (RKKY)-like coupling between magnetic on-site impurities in armchair graphene nanoribbons is studied theoretically. The calculations are based on a tight-binding model for a finite nanoribbon system with periodic boundary conditions. A pronounced Friedel-oscillation-like dependence of the coupling magnitude on the impurity position within the nanoribbon resulting from quantum size effects is found and investigated. In particular, the distance dependence of coupling is analysed. For semiconducting nanoribbons, this dependence is exponential-li…
We study RKKY interactions for magnetic impurities on graphene in situations where the electronic spectrum is in the form of Landau levels. Two such situations are considered: non-uniformly strained graphene, and graphene in a real magnetic field. RKKY interactions are enhanced by the lowest Landau level, which is shown to form electron states binding with the spin impurities and add a strong non-perturbative contribution to pairwise impurity spin interactions when their separation $R$ no more than the magnetic length. Beyond this interactions are found to fall off as $1/R^3$ due to perturbati…
Dirac electrons in clean graphene can mediate the interactions between two localized magnetic moments. The functional form of the RKKY interaction in pristine graphene is specified by two main features: (i) an atomic scale oscillatory part determined by a wave vector $\vec Q$ connecting the two valleys. Furthermore with doping another longer range oscillation appears which arise from the existence of an extended Fermi surface characterized by a single momentum scale $k_F$. (ii) $R^α$ decay in large distances where the exponent $α=-3$ is a distinct feature of undoped Dirac sea (with a linear di…
For low-dimensional metallic structures, such as nanotubes, the exchange coupling between localized magnetic dopants is predicted to decay slowly with separation. The long-range character of this interaction plays a significant role in determining the magnetic order of the system. It has previously been shown that the interaction range depends on the conformation of the magnetic dopants in both graphene and nanotubes. Here we examine the RKKY interaction in carbon nanotubes in the presence of uniaxial strain for a range of different impurity configurations. We show that strain is capable of am…
We study the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between two contact magnetic impurities placed on bilayer graphene (BLG). We compute the interaction mediated by the carriers of the pristine and biased BLG as well as the conduction electrons of the doped system. The results are obtained from the linear-response expression for the susceptibility written in terms of the integral over lattice Green's functions. For the unbiased system, we obtain some analytical expressions in terms of the Meijer G-functions, which consist of the product of two oscillatory terms, one coming from the i…
We present a theoretical study of indirect exchange interaction between magnetic adatoms in graphene. The coupling between the adatoms to a graphene sheet is described in the framework of tunneling Hamiltonian. We account for the possibility of this coupling being of resonant character if a bound state of the adatom effectively interacts with the continuum of 2D delocalized states in graphene. In this case the indirect exchange between the adatoms mediated by the 2D carriers appears to be substantially enhanced compared to the results known from Ruderman-Kittel-Kasuya-Yosida (RKKY) theory. Mor…
The RKKY interaction between two impurities on a curved surface is investigated. Practical curved sample parameters are considered for investigating curvatures, similar to those observed experimentally in available samples such as graphene. For this type of two-dimensional curved systems and for two types of Schrödinger and Dirac carriers, the RKKY interaction and local density of states (LDOS) are calculated. Results show that the strength of the RKKY interaction on a curved surface depends significantly on the nature of particles embedded on this surface, indicating that the geometry has a d…
We theoretically investigate the indirect signatures of the tilted anisotropic Dirac cones on Ruderman-Kittel-Kasuya-Yosida (RKKY) exchange interaction in a two dimensional polymorph of boron atoms. Unlike the case of isotropic non-tilted Dirac material-graphene, here we observe that the tilting of the Dirac cones exhibits a significant impact on the RKKY exchange interaction in terms of the suppression of oscillation frequency. The reason can be attributed to the behavior of the Fermi level and the corresponding density of states with respect to the tilting parameter. The direct measurement o…
We have systematically investigated the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between two magnetic impurities in Weyl semimetals with arbitrary monopole charge $Q$. We find that the RKKY interaction becomes intrinsically anisotropic for $Q\geq2$, and its dependence on Fermi energy and impurity separation is directly controlled by the monopole charge. With the increase of $Q$, the RKKY interaction becomes more long-ranged and more anisotropic, which makes interesting magnetic orders easier to form and thus may have important applications in spintronics.
We theoretically investigate the features of Ruderman-Kittel-Kasuya-Yosida (RKKY) exchange interaction between two magnetic impurities, mediated by the interfacial bound states inside a domain wall (DW). The latter separates the two regions with oppositely signed inversion symmetry broken terms in graphene and Weyl semimetal. The DW is modelled by a smooth quantum well which hosts a number of discrete bound states including a pair of gapless, metallic zero-energy modes with opposite chiralities. We find clear signatures of these interfacial chiral bound states in spin response (RKKY exchange i…
We theoretically study decoherence of two localized spins interacting via the RKKY interaction in one-, two-, and three-dimensional electron gas. We derive the kinetic equation for the reduced density matrix of the localized spins and show that energy relaxation caused by singlet-triplet transition is suppressed when the RKKY interaction is ferromagnetic. We also estimate the decoherence time of the system consisting of two quantum dots embedded in a two dimensional electron gas.
We study the changes in the electronic structure induced by lattice defects in graphene planes. In many cases, lattice distortions give rise to localized states at the Fermi level. Electron-electron interactions lead to the existence of local moments. The RKKY interaction between these moments is always ferromagnetic, due to the semimetallic properties of graphene
We develop the theory of conductance of a quantum dot which carries a spin and is coupled via RKKY interaction to another spin-carrying quantum dot. The found dependence of the differential conductance on bias and magnetic field at fixed RKKY interaction strength may allow one to distinguish between the possible ground states of the system. Transitions between the ground states are achieved by tuning the RKKY interaction, and the nature of these transitions can be extracted from the temperature dependence of the linear conductance. The feasibility of the corresponding measurements is evidenced…
A discussion on the interaction between skyrmions in a bi-layer system connected by a non-magnetic metal is presented. From considering a free charge carrier model, we have shown that the Ruderman-Kittel-Kasuya-Yosida (RKKY ) inter- action can induce attractive or repulsive interaction between the skyrmions depending on the spacer thickness. We have also shown that due to an increasing in RKKY energy when the skyrmions are far from each other, their widths are diminished. Finally, we have obtained analytical solutions to the skyrmion position when the in-plane distance between the skyrmions is…
We consider Mn-doped bulk zinc-blende semiconductors described by the 4 $ \times$ 4 Luttinger Hamiltonian. In these semiconductors, Mn atom acts as an acceptor providing the system a mobile hole, and also acts like a magnetic impurity of spin $S=5/2$. We obtain exact analytical expressions of the hole mediated Ruderman-Kittel-Kasuya-Yoshida (RKKY) exchange interaction between two Mn${}^{2+}$ ions. The RKKY interaction of the Luttinger system consists of collinear Heisenberg-like and Ising-like interactions. The characteristic beating patterns appear in the range functions of the RKKY interacti…
We study the interaction between two magnetic adatom impurities in graphene using the Anderson model. The two-impurity Anderson Hamiltonian is solved numerically by using the quantum Monte Carlo technique. We find that the inter-impurity spin susceptibility is strongly enhanced at low temperatures, significantly diverging from the well-known Ruderman-Kittel-Kasuya-Yoshida (RKKY) result which decays as $R^{-3}$.
We investigate theoretically the transport properties of two independent artificial Kondo impurities. They are coupled together via a tunable Ruderman-Kittel-Kasuya-Yoshida (RKKY) interaction. For strong enough antiferromagnetic RKKY interaction, the impurity density of states increases with the applied in-plane magnetic field. This effect can be used to distinguish between antiferromagnetic and ferromagnetic RKKY interactions. These results may be relevant to explain some features of recent experiments by Craig et al. (cond-mat/0404213).
We theoretically study the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction in two-dimensional black phosphorus, phosphorene. The RKKY interaction enhances significantly for the low levels of hole doping owing to the nearly valence flat band. Remarkably, for the hole-doped phosphorene, the highest RKKY interaction occurs when two impurities are located along the zigzag direction and it tends to a minimum value with changing the direction from the zigzag to the armchair direction. We show that the interaction is highly anisotropic and the magnetic ground-state of two magnetic adatoms can be tun…
We theoretically study the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between magnetic impurities in both Dirac and Weyl semimetals (SMs). We find that the internode process, as well as the unique three-dimensional spin-momentum locking, has significant influences on the RKKY interaction, resulting in both a Heisenberg and an Ising term, and an additional Dzyaloshinsky-Moriya term if the inversion symmetry is absent. These interactions can lead to rich spin textures and possible ferromagnetism in Dirac and time-reversal symmetry-invariant Weyl SMs. The effect of anisotropic Dirac and Wey…
A dilute distribution of magnetic impurities is assumed to be present in doped graphene. We calculate the interaction energy between two magnetic impurities which are coupled via the indirect-exchange or Ruderman-Kittel-Kasuva-Yosida (RKKY) interaction by the doped conduction electrons. The current model is a half-filled $AB$-lattice structure. Our calculations are based on the retarded lattice Green's function formalism in momentum-energy space which is employed in linear response theory to determine the magnetic susceptibility in coordinate space. Analytic results are obtained for gapped gra…
Ruderman-Kittel-Kasuya-Yosida (RKKY) is an essential long range magnetic interaction between magnetic impurities or magnetic layered structures, the magnitude of which oscillates with the distance ($\mathrm{R}$) between them. We have investigated the RKKY interaction between two magnetic impurities in both time-reversal and inversion symmetry broken Weyl semimetals (WSMs) where the energy dispersion is tilted in momentum space and the momentum of the conduction electron is locked with the pseudo-spin. Two important features are revealed, firstly, at the small tilt limit, we show that the RKKY …
The two dimensionality plus the linear band structure of graphene leads to new behavior of the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction, which is the interaction between two magnetic moments mediated by the electrons of the host crystal. We study this interaction from linear response theory. There are two equivalent methods both of which may be used for the calculation of the susceptibility, one involving the integral over a product of two Green's functions and the second that involves the excitations between occupied and unoccupied states, which was followed in the original work of Ru…
Carrier-mediated Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction plays an important role in itinerant magnetism. There have been intense interest on its general trend on bipartite lattice with particle-hole symmetry. In particular, recently fabricated graphene is well described by the honeycomb lattice within tight-binding approximation. We use SUSY quantum mechanics to study the RKKY interaction on bipartite lattices. The SUSY structure naturally differentiate the zero modes and those paired states at finite energies. The significant role of zero modes is largely ignored in previous literatu…
Clusters of magnetic atoms lying on a metallic substrate can exhibit Kondo effect due to the exchange interaction with mobile electrons in the metal, which can be observed in STM measurements. The same interaction results in the induced RKKY interaction between the magnetic moments in the cluster. For different geometries of the cluster, it can enhance or suppress Kondo-correlations. We calculate the tunneling density of states for trimer clusters of different geometries and reveal the signatures of competition between the Kondo effect and RKKY interaction that can be seen in STM experiments.
A new formalism to investigate RKKY interaction in aperiodic materials is presented. Particularly useful in the tight-binding scheme, it is used in this paper to probe the role of local environment and long range quasiperiodic potential on this mesoscopic coupling. Interesting features are revealed and discussed within the context of anomalous localization and transport
We calculated the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between the magnetic impurities mediated by electrons in nanoribbons. It was shown that the RKKY interaction is strongly dependent on the width of the nanoribbon and the transverse positions of the impurities. The transverse confinement of electrons is responsible for the above size effect of the RKKY interaction. It provides a potential way to control the RKKY interaction by changing nanostructure geometry.
As a new two-dimensional Dirac material, 8-\textit{Pmmn} borophene hosts novel anisotropic and tilted massless Dirac fermions (MDFs) and has attracted increasing interest. However, the potential application of 8-\textit{Pmmn} borophene in spin fields has not been explored. Here, we study the long-range RKKY interaction mediated by anisotropic and tilted MDFs in magnetically-doped 8-\textit{Pmmn} borophene. To this aim, we carefully analyze the unique real-space propagation of anisotropic and tilted MDFs with noncolinear momenta and group velocities. As a result, we analytically demonstrate the…
