Indirect exchange interaction between magnetic impurities in a gapped graphene structure

We study exchange interaction between two magnetic impurities in doped gapped graphene (the Ruderman-Kittel-Kasuya-Yosida [RKKY]) interaction by directly computing Green’s function beyond Dirac approximation. Tight binding model Hamiltonian in the presence of magnetic long range ordering has been applied to describe electron dynamics. RKKY interaction as a function of distance between localized moments has been analyzed. It has been shown that a magnetic ordering along the z-axis mediates two different interactions for spin directions which corresponds to a XXZ model interaction between two magnetic moments. The exchange interaction along arbitrary direction between two magnetic moments, has been obtained using the static spin susceptibilities of gapped graphene structure. The effects of spin polarization on the the dependence of exchange interaction on distance between moments are investigated via calculating correlation function of spin density operators. Our results show the chemical potential impacts ...


I. INTRODUCTION
2][3][4] Graphene consists of a single atomic layer of graphite, which can also be viewed as a sheet of unrolled carbon nanotube.Several anomalous phenomena ranging from half integer quantum Hall effect, nonzero Berry's phase, 5 to minimum conductivity 2 have been observed in experiments.The carriers in graphene behave as massless relativistic Dirac fermions with an effective speed of light c = 10 6 m/s within the low energy range ( < 0.5eV ). 5 The charge and spin oscillatory interactions in metals has attracted considerable attention both on the theoretical and experimental sides. 6,7Ruderman and Kittel 8 suggested that the spin oscillatory interaction in metals could provide a long-range interaction between nuclear spins in metals.Kasuya and Yosida extended the theory to include the long-range interaction between magnetic impurities and the combined refers to RKKY interaction.The Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction is one of the mechanisms of interaction between two magnetic impurities mediated by the conduction electrons of the host and is a fundamental a quantity of interest. 8,9Interactions between magnetic moments introduced to graphene lattice focus considerable interest.The form of RKKY interaction, originally derived for bulk, metallic, three-dimensional systems, is sensitive to the geometry and dimensionality of the underlying system and closely connected with the electronic structure of the host material.The RKKY interaction in pristine graphene has been studied by several groups. 10,11Due to the particle symmetry of graphene, the RKKY interaction induces ferromagnetic correlation between magnetic impurities on the same sublattice, while anti-ferromagnetic correlation between the ones on different sublattices.The peculiar dispersion relation for charge carriers in monolayer graphene, 12 being a zero-gap semiconductor, together with a bipartite nature of the underlying graphene crystalline lattice with dominant nearest neighbors electron hopping, causes the indirect RKKY coupling in graphene to differ from that found in 2 dimensional metals.4][15] Moreover, in undoped graphene, the coupling magnetic impurities in the same sublattice is always ferromagnetic, while for the impurities in different sublattices, it is antiferromagnetic.The dependence of the interaction on the distance R between two local magnetic moments, at the Dirac point, is found to be R 3 , whereas it behaves as R 2 in conventional two-dimensional systems. 19Such a fast decay rate denotes that the interaction is rather short ranged.In doped graphene, on the other hand, the spacial dependence of the interaction is predicted to be similar to conventional 2 dimensional systems, but this still remains to be experimentally verified.Due to the fact that the RKKY interaction is originated by the exchange coupling between the impurity moments and the spin of itinerant electrons in the bulk of the system, spin polarization of electrons is expected to influence directly this interaction 20 Specially, combination of the spin-dependence with a Driac-like spectrum can mediate a much richer collective behavior of magnetic adatoms. 21Graphene with imbalanced chemical potentials of spinup and spin-down electrons, presents a unique spin chiral material in which the interplay between the spin polarization, gapless spectrum, and the chiral nature of electrons have been shown to result intriguing phenomena. 22However RKKY iteration in gapless graphene has been studied in linear approximation for band structure, 14 in the present work the spacial dependence of this interaction has been investigated in full band method.Providing capability to control a type and density of charge carriers by gate voltage or by the chemical doping 23 made graphene instructive for novel nano-electronic devices.However, a gapped semiconducting behavior would be more suitable for electronic application.There have been some proposed in literature for a gap generation in graphene due to breaking of the sublattice symmetry by some substrates, 24 to adsorb some molecules, spinorbit interaction and finite size effect.Using Green's function approach under Dirac approximation, the RKKY interaction between two localized impurity atoms has been calculated. 14n this work, we calculate the RKKY interaction mediated by spin-polarized electron gas in an gapped graphene structure using Green's function approach beyond Dirac approximation.Our theory for the spin polarization dependence of RKKY interaction is motivated not only by fundamental transport considerations, but also by application and potential future experiments in graphene spintronic field.With a spin-polarization along the z-axis, we show that the RKKY interaction is different for different spin directions corresponding to a XXZ model interaction between the two magnetic moments when their spin orientations are fixed.
We study the effects of magnetization and electronic concentration on spatial dependence of RKKY interaction in the context of spin polarized tight binding model Hamiltonian.The dependence of RKKY interaction on spatial distance between two magnetic impurities has obtained via calculating the static spin susceptibilities using Green's function method.Both longitudinal and transverse components of static spin susceptibilities are necessary to be calculated in order to study exchange interactions between local moments.By means Green's function approach under full band method and Wick theorem the spin susceptibilities have been found based on the time ordered spin components density correlations.The study of RKKY interaction on honeycomb lattice in the presence of gap parameter, spin polarization and electronic concentration is the novelty of our work.However the RKKY interaction in graphene lattice has been presented in the previous theoretical works such as Refs.14-18, the effects of temperature, magnetization of graphene structure and gap parameter have not been discussed in those.Such cases have been focused in the present work.In the last section we discuss and analyze our results to show how longitudinal magnetization and chemical potential affect the spatial dependence of RKKY interaction.

II. THEORETICAL FORMALISM
Spin-polarized gapped graphene system is identified by a spin dependent chemical potential µ σ (σ = ±).Such magnetic ordering or spin polarization can be injected, for instance, by ferromagnetic electrodes on top graphene sheet.Intrinsic ferromagnetic correlations are also predicted to exist in graphene sheets 25 under certain conditions.The dynamics of noninteracting π electrons with spin σ on the honeycomb lattice as gapped graphene structure can be described by the following tight binding model (1) ) annihilates an electron with spin σ on sublattice A(B) of unit cell i and t ≈ 2.9eV implies the nearest neighbor hopping integral.The sum i, j in Eq.( 1) runs over distinct nearest neighbors.
The band gap, 2∆ has a nonzero value as a result of breaks the symmetry between sublattices, A and B. µ σ is spin dependent chemical potential implying a nonzero spin polarization parameter for gapped graphene structure.It is worthwhile to mention the point regarding the impurity impacts on the electronic structure.Impurities substantially modify the electronic structure of graphene, that leads to affect the exchange interaction. 26The main reason is the corrugation of graphene lattice, which involves sigma-states into play.However in this work we have approximately applied tight binding model Hamiltonian.According to Fig.
(1) the lattice structure of graphene is shown and the primitive unit cell vectors are given by where a is the length of lattice translational vector.We consider unit vector j along armchair direction.
In terms of Fourier transformation of electronic operators, the hamiltonian in Eq.(1) gets the following form within nearest neighbor approximation forms The Fourier transformation of each above Green's function elements is obtained by where ω m = (2m + 1)πk B T is Fermionic Matsubara frequency.T introduces the equilibrium temperature of the system.After some algebraic calculations, the following expression is obtained for Green's functions in Fourier presentation where α, β refer to the each atomic basis of honeycomb lattice and E j (k) is the band structure of gapped graphene like structure.Moreover coefficients C αβ j=± (k) are given by , Wick's theorem has been applied to write the charge response in terms of matrix elements of noninteracting electronic Green's function.The spin dependence of each component Green's function in Eq.( 6) arises from chemical potential µ σ .This spin dependent chemical potential, µ σ , is determined by the concentration of electrons with spin σ (n σ ) To determine µ σ , we use the definition of spin polarization and total occupation of electrons.Spin polarization is given by m = |n ↑ − n ↓ |/n and electronic concentration is expressed as Based on the values of magnetization m and electronic concentration n, the chemical potential for each spin degree of freedom, µ σ , can be obtained by numerical solving Eq. (8).In order to obtain the spatial dependence of RKKY interaction for transverse and longitudinal spin components, static spin susceptibilities needs to be calculated.Linear response theory gives us the noninteracting spin response functions based on the correlation function of both transverse and longitudinal components of spin operators.We introduce χ S z S z and χ S + S − as longitudinal and transverse spin susceptibilities, respectively 035320-5 Rezania, Naseri, and Shahrestani AIP Advances 7, 035320 (2017) ω n = 2nπk B T denotes the Bosonic Matsubara frequency.The Fourier transformations of z and ladder components of spin density operators are given as After substitution of operator form of z-component of spin density into the correlation function we arrive the following expression for χ (0) Since long range magnetic ordering has been considered for graphene nanoribbon, the contribution of electrons with spin down to the response function is different from that of electrons with spin up.In order to calculate the correlation function in Eq.( 11), one particle spin dependent Green's function presented in Eq.( 6) should be exploited.Wick's theorem has been applied to express the charge response in terms of matrix elements of non-interacting electronic Green's function.After using the Fourier transformation in Matsubara's representation, 27 the expression for χ (0) S z S z (q, iω m ) is given by where n F (x) = 1 e x/k B T +1 is the Fermi-Dirac distribution function.In a similar way, the transverse spin susceptibility can be written as By implementing Wick's theorem and performing Matsubara frequency summation rules, one can find the following relation for transverse spin susceptibility in terms of noninteracting Green's function Our system incorporates two localized magnetic moments whose interaction is mediated through a spin polarized electron liquid.We assume that the graphene nanoribbon is spin polarized first, and 035320-6 Rezania, Naseri, and Shahrestani AIP Advances 7, 035320 (2017) then we add the magnetic moments.The contact interaction between the spin of itinerant electrons and two magnetic impurities with magnetic moments S 1 and S 2 , located respectively at R 1 and S 2 , is given by where λ is the coupling constant between conduction electrons and impurity, s(r) is the spin density operator of electrons.The RKKY interaction which arises from the quantum effects is obtained by using a second order perturbation 8,28 which is the honored XXZ model.R denotes the spatial distance between two local moments S 1 and S 2 .In other hand the coupling exchange constants are related to longitudinal and transverse static spin susceptibilities of the electron gas 8,28 as where χ zz (R) and χ xx (R) are named longitudinal and in plane exchange coupling constants, respectively.The effects of spin polarization on the RKKY interaction originates from factor spin dependent chemical potential which has been defined in Eq.( 8).It should be mentioned that we were not able to find simple analytic expression for the in plane and longitudinal exchange coupling constants and in the next section, we will present the numerical results of them for gapped graphene structure.

III. NUMERICAL RESULTS AND DISCUSSION
In this section, we turn to our main numerical results for the RKKY exchange coupling in the presence of a spin polarization along z axis for electrons on gapped graphene sheet lattice along z-axis by analyzing the calculated χ S x S x and χ S z S z .In order to obtain the numerical results for RKKY interactions, both longitudinal and transverse spin susceptibilities in zero frequency limit should be calculated according to Eqs. ( 12)- (14).By inserting these static susceptibilities into Eq.(17) we find the dependence of in plane and longitudinal RKKY interactions on distance between two impurities.The amounts of chemical potential for each spin component are obtained by the values of spin polarization and concentration of electron gas.Moreover the equilibrium temperature for all following numerical results is to be 0.05.The distance dependence of in plane exchange coupling χ S x S x for doped gapless graphene for 0.0 < µ/t < 1.0 are illustrated in Fig. (2).The magnetization is assumed to be zero.For any value of µ/t, the exchange coupling χ S x S x (R) exhibits an oscillatory behavior as a function of R so that its amplitude grows with chemical potential.However the period of this oscillation is independent of chemical potential.Also the amplitude decreases with distance for all values of chemical potential µ/t.Based on Fig. (2), we see the amplitude of the oscillation has no considerable dependence on chemical potential at large distance.For R/a < 5, it is clearly observed that the width of antiferromagnetic exchange values ( χ S x S x (R) > 0) is more than that of ferromagnetic (J S x S x (R) < 0) ones.Although this is not the case of R/a > 5 where the widths for both ferromagnetic and antiferromagnetic exchange values are the same.
In Fig.
(3) we plot χ S x S x (R) versus normalized distance between impurities for undoped gapless graphene at different values of longitudinal magnetization, m.The oscillation amplitude of χ S x S x (R) decreases with magnetization as shown in Fig. (3) although there is no remarkable dependence on magnetization for J S x S x (R) at distance values above 2.5.In fact all plots fall on each other on the whole range of normalized distance above 2.5.The effect of spin polarization on the spatial behavior of longitudinal coupling exchange constant between localized moments has been also studied.magnetization up to 0.5 as we see in Fig. (4).Upon more increasing magnetization above 0.5 up to 1.0, it is clearly observed that oscillation amplitude of χ S z S z (R) decreases.In addition, a drastic reduction for χ zz (R → 0) is observed when electronic system becomes full polarized, i.e. m = 1.The frequency of oscillation of χ S z S z (R) versus R is approximately independent of spin polarization parameter.Also the spatial behavior of longitudinal exchange constant has no remarkable dependence on m for distances above 5.0.For normalized distances R/a < 2.5, we deal with antiferromagnetic  4) shows a considerable difference between χ S z S z (R) and J S x S x (R) for each value of µ/t at each distance R/a.In Fig. (5) we have plotted in plane exchange coupling constant χ S x S x between two impurities localized as a function of R/a in undoped gapped graphene structure for different gap parameters ∆/t.Spin polarization parameter is set to As shown in this figure, the amplitude of oscillation decreases with gap parameter and distance between two external moments.However the plots for all values of ∆/t fall on each other on the whole range of distance above 5.0.For R/a < 5, it is clearly observed that the width of antiferromagnetic exchange values ( χ S x S x (R) > 0) is more than that of ferromagnetic ( χ S x S x (R) < 0) ones.We have also studied the spatial behaviors of both in plane and longitudinal RKKY interactions in gapped graphene structure for ∆/t = 0.8.The spatial dependence of in plane and longitudinal exchange constant between impurities in undoped gapped graphene structure for different spin polarization m have been presented in Figs.(6,7), respectively.Fig. (6)  indicates the oscillation amplitude of χ S x S x (R) increases with magnetization up to 0.25.Afterwards we observe the reduction of oscillation amplitude for magnetization above 0.25 up to full polarization.According to Fig. (7) the curves of χ S z S z (R) for m = 0.098, 0.25, 0.5, 0.8 fall on each other on the whole range of distance between localized moments.Moreover the plots of longitudinal RKKY interaction for nonmagnetic and full polarized gapped graphene structure fall on each other as shown in Fig. (7).However the oscillation frequency of RKKY interaction in Fig. (7) is independent of magnetization parameter, the amplitude of oscillation of longitudinal RKKY interaction for m = 0.0, 1.0 is lower than that for the other magnetization parameter values.In Fig. (7) we present the spatial dependence FIG. 6.In plane RKKY interaction (J xx (R)) between two localized moments in undoped gapped graphene structure as a function of the distance R with fixed gap parameter, ∆/t = 0.8, for different spin polarizations for fixed temperature k B T /t = 0.05. of in plane RKKY interaction χ S x S z (R) in gapped graphene structure with ∆/t = 0.8 for different chemical potentials for zero magnetization.This figure shows the oscillation amplitude for lower values of R has the same value for chemical potentials µ/t = 0.2, 0.6, 0.8.However we clearly observe that a sudden increase takes place when µ/t gets the values 0.9 and 1.0.The numerical results for in plane RKKY interaction between two localized moments in nonmagnetic gapped graphene with ∆/t = 0.8 for different chemical potentials have been presented in Fig. (8).This figure shows the spatial dependence of longitudinal RKKY interaction for µ/t = 0.2, 0.6, 0.8 fall on each other.According to Fig. (8), a drastic increase for oscillation amplitude of χ S x S x (R) is clearly observed when normalized chemical potential gets the values 0.9 and 1.0.The frequency of the spatial oscillation of χ S x S x (R) is independent of the chemical value.Moreover there is no remarkable dependence of χ S x S x (R) on chemical potential for at high values of R/a as shown in Fig. (8).
FIG. 1. Structure of an armchair graphene nanoribbon, consisting of sublattices, A and B. n is the width of the ribbon.Every unit cell contains n number of A and B sublattices.Two additional hard walls (j = 0, n + 1)are imposed on both edges.
FIG.2.In plane RKKY interaction (χ Sx Sx (R)) between two localized moments in doped gapless graphene as a function of the distance R for values of chemical potential into interval 0.0 < µ/t < 1.0 for fixed temperature k B T /t = 0.05.The magnetization is set to zero.

FIG. 4 .FIG. 5 .
FIG.4.Longitudinal RKKY interaction (χ SzSz (R)) between two localized moments in undoped spin polarized gapless graphene as a function of the distance R for values of magnetization, m, for fixed temperature k B T /t = 0.05.

FIG. 7 .FIG. 8 .
FIG.7.Longitudinal RKKY interaction (J zz between two localized moments in magnetic gapped graphene structure as a function of the distance with fixed gap parameter, ∆/t = 0.8, for different spin polarizations for fixed temperature k B T /t = 0.05.