Ruderman-Kittel-Kasuya-Yosida Interaction Research Articles

Equilibrium and real-time properties of the spin correlation function in the two-impurity Kondo model

We investigate the equilibrium and real-time properties of the spin correlation function $\langle \vec{S}_1\vec{S}_2 \rangle$ in the two-impurity Kondo model for different distances $R$ between the two-impurity spins. It is shown that the competition between the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction and the Kondo effect governs the amplitude of $\langle \vec{S}_1\vec{S}_2 \rangle$. For distances $R$ exceeding the Kondo length scale, the Kondo effect also has a profound effect on the sign of the correlation function. For ferromagnetic Heisenberg couplings $J$ between the impurities and the conduction band, the Kondo effect is absent and the correlation function only decays for distances beyond a certain length scale introduced by finite temperature. The real-time dynamics after a sudden quench of the system reveals that correlations propagate through the conduction band with Fermi velocity. We identify two distinct timescales for the long time behavior which reflects that for small $J$ the system is driven by the RKKY interaction while for large $J$ the Kondo effect dominates. Interestingly, we find that at certain distances a one-dimensional dispersion obeying $\epsilon(k)=\epsilon(-k)$ may lead to a local parity conservation of the impurities such that $\langle \vec{S}_1\vec{S}_2 \rangle$ becomes a conserved quantity for long times and does not decay to its equilibrium value.

Probing decoupled edge states in a zigzag phosphorene nanoribbon via RKKY exchange interaction

Phosphorene is an anisotropic puckered two-dimensional ($2$D) hexagonal lattice of phosphorus atoms. The edge modes in a zigzag phosphorene nanoribbon (ZPNR) are quasi-flat in nature and fully isolated from the bulk states, which are unique in comparison to the other hexagonal lattices like graphene, silicene etc. We theoretically investigate the Ruderman-Kittel-Kasuya-Yosida (RKKY) exchange interaction between two magnetic impurities placed on the nanoribbon, and extract the signatures of the flat edge states via the behavior of it. Due to the complete separation of the edge states from the bulk, we can isolate the edge mode contribution via the RKKY interaction from that of the bulk by tuning the external gate potential when both the impurities are placed at the same edge. The bulk induced RKKY interaction exhibits very smooth oscillation with the distance between the two impurities, whereas for edge modes it fluctuates very rapidly. We also explore the effect of tensile strain both in absence and presence of gate voltage and reveal that the RKKY interaction strength can be boosted under suitable doping, when both the impurities are within the bulk.

Bulk RKKY signatures of topological phase transition in silicene

Silicene offers an ideal platform for exploring the phase transition due to strong spin-orbit interaction and its unique structure with strong tunability. With applied electric field and circularly polarized light, silicone is predicted to exhibit rich phases. We propose that these intricate phase transitions can be detected by measuring the bulk Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction. We have in detail analyzed the dependence of RKKY interaction on phase parameters for different impurity configurations along zigzag direction. Importantly, we present an interesting comparison between different terms of RKKY interaction with phase diagram. It is found that the in-plane and out-of-plane terms can exhibit the local extreme value or change of sign at the phase critical point and remarkable difference in magnitude for different phase regions. Consequently, the magnetic measurement provides unambiguous signatures to identify various types of phase transition simultaneously, which can be carried out with present technique.

Ionic Gel Modulation of RKKY Interactions in Synthetic Anti-Ferromagnetic Nanostructures for Low Power Wearable Spintronic Devices.

To meet the demand of developing compatible and energy-efficient flexible spintronics, voltage manipulation of magnetism on soft substrates is in demand. Here, a voltage tunable flexible field-effect transistor structure by ionic gel (IG) gating in perpendicular synthetic anti-ferromagnetic nanostructure is demonstrated. As a result, the interlayer Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction can be tuned electrically at room temperature. With a circuit gating voltage, anti-ferromagnetic (AFM) ordering is enhanced or converted into an AFM-ferromagnetic (FM) intermediate state, accompanying with the dynamic domain switching. This IG gating process can be repeated stably at different curvatures, confirming an excellent mechanical property. The IG-induced modification of interlayer exchange coupling is related to the change of Fermi level aroused by the disturbance of itinerant electrons. The voltage modulation of RKKY interaction with excellent flexibility proposes an application potential for wearable spintronic devices with energy efficiency and ultralow operation voltage.

Monte Carlo simulations of magnetic properties of Kekulene structure bilayers separate by a nonmagnetic with RKKY interactions

The magnetic properties of magnetic bilayers of Kekulene structure separate by a nonmagnetic layer with Ruderman-Kittel-Kasuya-Yosida (RKKY) exchange interactions with Ising spin model have been studied using Monte Carlo simulations. The RKKY interaction between the bilayers of Kekulene is considered for different distances. The transition temperature has been deduced from the magnetizations and magnetic susceptibilities partial for a fixed value of nonmagnetic layer. The reduced transition temperatures are also deduced from the total magnetization and total magnetic susceptibilities with different values of L. The magnetic hysteresis cycles of systems have been determined.

Kondo Impurities Coupled to a Helical Luttinger Liquid: RKKY-Kondo Physics Revisited.

We show that the paradigmatic Ruderman-Kittel-Kasuya-Yosida (RKKY) description of two local magnetic moments coupled to propagating electrons breaks down in helical Luttinger liquids when the electron interaction is stronger than some critical value. In this novel regime, the Kondo effect overwhelms the RKKY interaction over all macroscopic interimpurity distances. This phenomenon is a direct consequence of the helicity (realized, for instance, at edges of a time-reversal invariant topological insulator) and does not take place in usual (nonhelical) Luttinger liquids.

Ionic liquid gating control of RKKY interaction in FeCoB/Ru/FeCoB and (Pt/Co)2/Ru/(Co/Pt)2 multilayers

To overcome the fundamental challenge of the weak natural response of antiferromagnetic materials under a magnetic field, voltage manipulation of antiferromagnetic interaction is developed to realize ultrafast, high-density, and power efficient antiferromagnetic spintronics. Here, we report a low voltage modulation of Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction via ionic liquid gating in synthetic antiferromagnetic multilayers of FeCoB/Ru/FeCoB and (Pt/Co)2/Ru/(Co/Pt)2. At room temperature, the distinct voltage control of transition between antiferromagnetic and ferromagnetic ordering is realized and up to 80% of perpendicular magnetic moments manage to switch with a small-applied voltage bias of 2.5 V. We related this ionic liquid gating-induced RKKY interaction modification to the disturbance of itinerant electrons inside synthetic antiferromagnetic heterostructure and the corresponding change of its Fermi level. Voltage tuning of RKKY interaction may enable the next generation of switchable spintronics between antiferromagnetic and ferromagnetic modes with both fundamental and practical perspectives.

Strongly anisotropic RKKY interaction in monolayer black phosphorus

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 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 tuned by changing the rotational configuration of impurities. Owing to the anisotropic band dispersion, the oscillatory behavior with respect to the angle of the rotation and the distance of two magnetic impurities, R is well-described by sin(2kFR), where the Fermi wavelength kF changes in different directions. We also find that the tail of the RKKY oscillations falls off as 1/R2 at large distances.

Stabilization of magnetic skyrmions by RKKY interactions

We study the stabilization of an isolated magnetic skyrmion in a magnetic monolayer on a nonmagnetic conducting substrate via the Ruderman-Kittel-Kasuya-Yosida (RKKY) exchange interaction. Two different types of the substrate are considered, usual normal metal and single-layer graphene. While the full stability analysis for skyrmions in the presence of the RKKY coupling requires a separate effort that is outside the scope of this work, we are able to study the radial stability (stability of a skyrmion against collapse) using variational energy estimates obtained within first-order perturbation theory, with the unperturbed Hamiltonian describing the isotropic Heisenberg magnet, and the two perturbations being the RKKY exchange and the easy-axis anisotropy. We show that a proper treatment of the long-range nature of the RKKY interaction leads to a qualitatively different stabilization scenario compared to previous studies, where solitons were stabilized by the frustrated exchange coupling (leading to terms with the fourth power of the magnetization gradients) or by the Dzyaloshinskii-Moriya interaction (described by terms linear in the magnetization gradients). In the case of a metallic substrate, the skyrmion stabilization is possible under restrictive conditions on the Fermi surface parameters, while in the case of a graphene substrate the stabilization is naturally achieved in several geometries with a lattice-matching of graphene and magnetic layer.

The role of electronic dopant on full band in-plane RKKY coupling in armchair graphene nanoribbons-magnetic impurity system

Motivated by the growing interest in solving the obstacles of spintronics applications, we study the Ruderman-Kittel-Kasuya-Yosida (RKKY) effective pairwise interaction between magnetic impurities interacting through the π-electrons embedded in both electronically doped-semiconducting and metallic armchair graphene nanoribbons. In terms of the Green’s function formalism, treated in a tight-binding approximation with hopping beyond Dirac cone approximation, the RKKY coupling is an attraction or a repulsion depending on the magnetic impurities distances. Our results show that the RKKY coupling in semiconducting nanoribbons is much more affected by doping than metallic ones. Furthermore, we found that the RKKY coupling increases with ribbon width, while there exist some critical electronic concentrations in RKKY interaction oscillations. On the other hand, we find an unusual incoming wave-vector direction for electrons which describes more clearly the ferro- and antiferromagnetic spin configurations in such system. Also, the RKKY coupling at low and high-temperature regions has been addressed for both ferro- and antiferromagnetic spin arrangements.

Long range of indirect exchange interaction on the edges of MoS2 flakes

We study the Ruderman–Kittel–Kasuya–Yosida interaction between two magnetic impurities connected to the edges of zigzag-terminated MoS2 flakes. When the impurities lie on the edges of the flake, the effective exchange interaction exhibits sizable noncollinear Dzyaloshinskii–Moriya character that competes with a strong Ising coupling. We analyze the characteristic decay exponent for doping levels inside the band gap of the infinite layer, corresponding to edge states of the flake at the Fermi level. The characteristic exponents show sub-two-dimensional behavior for these band fillings, with decays much slower than quadratic. The Ising interaction has effectively one-dimensional long range, while the noncollinear component that grows for short impurity separation becomes comparable in magnitude. The resulting tunable exchange interaction on these systems opens the way for the study of interesting phases of impurity arrays with long-range stable helical order.

Thermodynamics of two-impurity Anderson model with Dzyaloshinskii–Moriya interaction

In this work, we use the numerical renormalization group (NRG) theory to study the thermodynamics of the two-impurity Anderson model. Two different methods are used to estimate the effect of Dzyaloshiskii–Moriya (DM) interaction on the variation of the Kondo temperature. When the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction is vanishing, the two different estimations give different tendencies. If we use the peak of the specific heat to identify the variation of the Kondo temperature versus the DM interaction, we get an almost linear function. However, if we use the low temperature universal curve of the impurity entropy, we get a quadratic function. These results indicate that previous debates about the influence of spin–orbit coupling on the Kondo temperature may come from the different definitions of the Kondo temperature. When the RKKY interaction is ferromagnetic, there are two stages of Kondo screening. Both estimations demonstrate that the second stage of the Kondo temperature is exponentially dependent on the DM interaction. There results are dramatically different from those calculated via perturbation theory.

Interelectron interactions and the RKKY potential between H adatoms in graphene

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 analyze the stability of several regular adatom superlattices with respect to small displacements of a single adatom, distinguishing the cases of adatoms which populate either both or only one sublattice of the graphene lattice.

RKKY interaction of magnetic impurities in multi-Weyl semimetals

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 , 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.

Anisotropic RKKY interaction and modulation with mechanical strain in phosphorene

We investigate the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction between two magnetic impurities placed on a phosphorene surface based on the tight-binding model. It is found that the RKKY interaction exhibits strong anisotropy along different lattice directions and is sensitive to the deformed direction. The RKKY interaction is largest for impurities distributed in the armchair direction while the strain effect is strongest for the deformation exerted along the zigzag direction. Applied linear strain can increase the RKKY magnitude nonlinearly, and observably prolong the decay rate from the exponent to the law with the impurity distance. Most interestingly, near the strain-induced closing point of the bandgap, we find that the RKKY interaction is no longer simply ferromagnetic or antiferromagnetic, but presents oscillatory behavior, exhibiting the transition from ferromagnetism to antiferromagnetism. This originates from the combination effect of negative energy in the conduction band due to modification of the Fermi surface and the narrowing bandgap, and importantly both of them are reached simultaneously just by tuning the strain. Therefore, the strain effect proves to be an alternative approach to engineering impurity interactions in phosphorene materials.

Suppressed Kondo effect and Kosterlitz–Thouless-type phase transition induced by level difference in a triple dot device

Quantum dot system provides an ideal platform for quantum information processing, within which to demonstrate the quantum states is one of the most important issue for quantum simulation and quantum computation. In this paper, we report a peculiar electron state in a parallel triple dot device where the Ruderman–Kittel–Kasuya–Yosida interaction is invalid when the level differences of the dots sweep into appropriate regime. This extraordinary tendency then results in an antiferromagnetic spin coupling between two of the dots and may lead to zero or full conductance, relying deeply on the relation of the two level spacings. e.g. when the level differences are kept equal, the Kondo effect is totally suppressed although the dots are triply occupied, since in this case a local inter-dot transport loop is found to play an important role in the transmission coefficient. By contrast, when the differences are retained symmetric, the Kondo peak reaches nearly to its unitary limit, owing to that the inter-dot transport process is significantly suppressed. To approach these problems, voltage controllable quantum phase transitions of Kosterlitz-Thouless type and first order are shown, and possible pictures related to the many-body effect and the effective Kondo model are given.

Effect of the Rashba splitting on the RKKY interaction in topological-insulator thin films

We investigate the effect of Rashba splitting on the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction in a topological-insulator (TI) thin film both at finite and zero chemical potential. We show that the spin susceptibility of the TI thin film depends strongly on the direction of the distance vector between impurities. In addition to the well-known Heisenberg-, Ising-, and Dzyaloshinskii-Moria (DM)-like terms reported before in TIs, we find another term in the off-diagonal part of the spin-susceptibility tensor which is symmetric in contrast to the DM term. Furthermore, we show how one can tune the RKKY interaction by using electric field applied perpendicularly to the surface plane of the TI, where in the presence of such a field the RKKY interaction can be enhanced drastically for small chemical doping. We present our results for two different situations, namely intersurface pairing of magnetic impurities as well as intrasurface pairing. The behavior of these two situations is completely different, which we describe by mapping the density of states of each surface on the band dispersion.

Study of RKKY Interactions in a Bilayer Graphene Structure with Non-equivalent Planes

In this paper, we study the magnetic properties and the effect of the Ruderman-Kittel-Kasuya-Yosida (RKKY) interactions in a bilayer graphene structure with non-equivalent planes of a ferromagnetic consisting of the spins σ = 1 / 2 and S = 1. The studied system is formed by two magnetic planes (A and B), separated by a number of non-magnetic planes. In fact, the influence of the coupling exchange interactions, the external magnetic field, and the crystal field is investigated and presented as well as the ground-state phase diagrams. Using Monte Carlo simulations, we examine the behavior of the magnetizations and the total magnetization as a function of the values of system parameters. The effect of the RKKY interactions increases for increasing values of the number of non-magnetic planes; for this reason, the studied system becomes disorder when the number of non-magnetic planes increased. When the susceptibility peaks correspond to the reduced critical temperature (T / JA), this critical temperature is displaced towards the lower temperatures.

Strongly Correlated Effects in a Triangular Dot System Induced by Inter-Dot Hopping and Coulomb Repulsion

We study the transport property and the phase transition for a triple quantum dot system in a triangular geometry with one centered dot and two side dots by adopting the Wilson’s numerical renormalization group method. When three dots are decoupled with no coupling, the Kondo peak turns to reach the unitary limit, and the spin-1/2 Kondo effect is found. When the hopping between the centered dot and two side dots increases, two side dots form a local spin triplet due to the Ruderman–Kittel–Kasuya–Yosida interaction mediated by the centered dot, and the Kondo peak is suppressed. As the hopping between two side dots increases, a first order quantum phase transition occurs. In this case, two side dots form a spin singlet and the Kondo peak is recovered. When the inter-dot Coulomb repulsion between two side dots is taken into account and the hopping between two side dots is small, the charge on each side dot drops to 0.5 through a first-order quantum phase transition. As a result, one of the side dots is singly occupied, while the other one is empty. The linear conductance is suppressed due to the strong coupling between the centered dot and the singly occupied side dot. When the hopping between two side dots is large enough, the charge on each side dot decreases to 0.5 continuously and a crossover is found, for in this case the symmetry of spin configurations within two side dots is broken.

Oscillation and suppression of Kondo temperature by RKKY coupling in two-site Kondo systems

We apply our recently developed, selfconsistent renormalization group (RG) method [] to STM spectra of a two-impurity Kondo system consisting of two cobalt atoms connected by a one-dimensional Cu chain on a Cu surface []. This RG method was developed to describe local spin screening in multi-impurity Kondo systems in presence of the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction. Using the RKKY interaction of a one-dimensional chain, we explain the experimentally observed suppression and oscillation of the Kondo temperature, TK(y), as a function of the length of the chain and the corresponding RKKY interaction parameter y, regardless of the RKKY coupling being ferromagnetic or antiferromagnetic.