RKKY interaction in the zigzag-edge silicene-like nanoflake

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RKKY interaction in the zigzag-edge silicene-like nanoflake Fatemeh Mazhari Mousavi, Rouhollah Farghadan ∗ Department of Physics, University of Kashan, Kashan, 87317-51167, Iran

a r t i c l e

i n f o

Article history: Received 8 July 2019 Received in revised form 9 September 2019 Accepted 12 September 2019 Available online xxxx Communicated by R. Wu Keywords: RKKY interaction Zigzag-edge Silicene-like nanoflake Kane-mele-Hubbard Spin-orbit interactions

a b s t r a c t By considering the Kane-Mele-Hubbard approximation on the honeycomb lattice, we investigate the spin-spin correlation for two magnetic impurities in zigzag edge silicene-like nanoflake (ZSiLF). The dependence of the spatial behaviors of RKKY interaction on the electron-electron (e-e), intrinsic spinorbit interactions (ISOI) and, electric field are systematically investigated. Generally, the spatial behaviors of the RKKY interaction sensitively change by changing e-e interaction and electric field strengths in the presence of ISOI. The ISOI in a ZSiLF result in long-range in-plane and Ising interactions. Moreover, e-e interaction induces non-zero Dzyaloshinsky-Moriya (DM) term and nearly distance-independent Ising interaction (similar to graphene nanoflake) in the presence of ISOI. Furthermore, with considering e-e interaction, the in-plane DM interaction increases by increasing the strengths of the electric field and ISOI. © 2019 Elsevier B.V. All rights reserved.

1. Introduction In the spintronics, the information carriers are charge and spins; therefore, it is crucial to managing the spin degree of freedom [1]. In this regard, two-dimensional structures such as graphene, silicene, germanene, and stanene [2] have shown enormous potential for spintronic applications [3]. Basically, proximity with a ferromagnetic material, edge magnetism and using external magnetic and electric field are some ways which split two spin subbands in these structures [4–6]. Moreover, one of the efficient ways is using magnetic impurities, which is to induce local magnetic moments in nonmagnetic materials. In graphene, the magnetic properties for impurities which are created by removing a carbon atom are different than those of traditional Anderson [7]. The magnetic properties which are controlled by spin correlations between magnetic impurities and itinerant electrons, known as the indirect RKKY interaction (Ruderman-Kittel-KasuyaYosida) [8–10]. Recently, RKKY interaction has been studied in a variety of 2D materials and nanostructures. In this regard, various types of graphene nanostructures, such as bilayer and monolayer sheet [11–15], nanoribbon [16–18], nanotube [19–22] and nanoflake [23–26], and even in the presence of electron-electron (e-e) interaction [25–28] are investigated. Moreover, the effect of spin exchange energy and charge doping on the RKKY interaction in zigzag edge triangular graphene nanoflake is investigated [26].

*

Corresponding author. E-mail address: [email protected] (R. Farghadan).

https://doi.org/10.1016/j.physleta.2019.125991 0375-9601/© 2019 Elsevier B.V. All rights reserved.

Moreover, considering e-e interaction in graphene, show that the RKKY coupling (Ising term) becomes nearly distance-independent [26–28]. On the other hand, the investigations indicate that the magnetic adatoms such as (Ni, Au, Ti, and Os) and Rshaba spin-orbit interaction (SOI) could induce large SOI in graphene [29]. In the presence of intrinsic spin-orbit interaction (ISOI) the degeneracy of in-plane and out-of-plane spin correlation is broken. Moreover, ISOI causes a twisted RKKY coupling between magnetic impurities with three different terms: Ising, Heisenberg, and Dzyaloshinsky-Moriya (DM) interactions [30]. Other two dimensional structures such as silicene, germanene and stanene could have a relatively large ISOI rather to graphene about λ S O = 3.9, 43, 100 meV respectively [1,7,31–33]. In this regard, silicene exhibited as a magnetic semiconductor in the presence of magnetic impurity and has more compatibility with contemporary silicon-based technology [34]. ISOI [29,35] and Rashba SOI [23,36] in silicene layer [37], silicene nanoribbon [38] and in transition metal dichalcogenides (TMDs) [39–41] have been investigated. Experimentally also showed that SOI twists the orientation between the magnetic impurities in a bulk silicene [37]. The investigation and the ability to control the RKKY interaction is important to quantum computing and spintronic applications. The zigzag edge nanoflakes are especially crucial due to their net magnetization [42]. Moreover, the nanoflakes behave in a quantum-dot manner [43]. In zigzag-edge silicene nanostructures due to the existence of nearly-zero-energy states, by placing impurities at the zigzag edge, the RKKY interaction is more significant than the impurities are in the bulk region [16,38]. Therefore, we investigate

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the spatial behaviors of indirect RKKY interaction in zigzag-edge of silicene-like nanoflake (ZSiLF) with consideration the effect of ISOI (λ S O ), the Coulombic on-site interactions (U ) and in the presence of external electric field (perpendicular field E z ). Our results show that these interactions have a significant influence on the spatial properties of the RKKY interaction in silicene-like structures. ISOI solely could emphasize the Ising and in-plane interactions. Moreover, the e-e interaction increases the amplitude of the Ising term, and this interaction becomes nearly distance-independent, even with considering the effect of ISOI similar to graphene nanoflake without ISOI [26]. Moreover, increasing the strength of ISOI or electric field in the presence of e-e interaction could effectively enhance the in-plane DM term of the RKKY interactions. These results may be useful for silicene-like spintronic applications. 2. Model and method We study the embedded magnetic impurity at ZSiLF with using of Kane-Mele-Hubbard Hamiltonian and the single-band tightbinding method. By ignoring Rashba SOI, the tight binding Hamiltonian involves the ISOI, the perpendicular electric field, the Hubbard term, and the magnetic impurity term. The Hamiltonian that describes the system is as follows:

H tot = H 0 + H imp ,

(1) Where H imp is Anderson-Kondo Hamiltonian and shows the spin interaction between impurity S i and conduction electrons. Moreover, J 0 = 0.1t is the effective coupling constant between the conduction electrons and impurities. Using a second-order perturbation theory, can replace the H imp by the RKKY interaction [39–41]:

where [44–46]

H 0 = −t



†

λS O



†

ciα c jα + i √ νi j c iα σαz β c jβ 3 3 < i j >α <>α β

 +U [ ni ↑ ni ↓  + ni ↓ ni ↑  − ni ↑ ni ↓ ] i

+elE z



Fig. 1. Outline zigzag-edge triangular nanoflake. (a) Two impurities are placed at the on-site position on the nanoflake edge, the first impurity with the red circle and the second impurity with the green circle are marked. (b) and (c) show the same and different sublattices, respectively. M is number hexagonal rings forming each side of the triangular structure and N is total number of electrons (N = M 2 + 4M + 1) [26]. a0 = 3.86 A ◦ is the lattice constant [33]. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)

y

μi c i†α c iα .

(2)

+ J D M (S1 × S2 )z .

iα

In the above expression, the first term is tight binding Hamiltonian that t is the hopping parameter for silicene equal to 1.6 eV † [32,33], c i α (c j β ) is the creates (annihilates) operator an electron at site i ( j ) with the spin α (β). The second term relates to the ISOI, and λ S O represents the strength of the ISOI and σ z is the Pauli matrix in the z-direction. νi j according to formula νi j =

y

H R K K Y = J xx ( S 1x S 2x + S 1 S 2 ) + J zz S 1z S 2z (5)

Where J xx = J y y (in-plane), J zz (Ising), and J D M (in-plane Dzyaloshinskii-Moriya) terms are indirect exchange component. Moreover, the difference between triplet and singlet impurity configurations calculate the effective J ’s in silicene nanoflakes [16,25, 26,39,40]:

E (↑ γ , ↓ η) − E (↑ γ , ↑ η)

− → − → − → − → ( d j × d i )/| d j × d i | takes values ±1 [32,33], where di and

Jγ η =

d j are two bonds connecting next-nearest neighbor atoms. In the first and second term, < i j > and << i j >> are the summation over all the nearest or next-nearest neighbor hopping sites, respectively. The third term refers to the perpendicular electric field. Due to the buckled structure silicene the two sublattice planes are separated by a perpendicular distance 2l, that l = 0.23 A ◦ , and μi = ±1 for the A (B) site [32,33]. The fourth term is Hubbard

where γ , η ∈ x, y , z display the direction of the spin for the two magnetic impurity.

†

Hamiltonian where ni ,↑ = c i ↑ c i ↑ and U is the Coulomb interaction and equal to U = t [47]. The expectation value n and s is calculated by the eigenfunction u  of the Hamiltonian in H 0 [44]:

ni ,↑  =

 E
u ∗i , E u i , E ; ni ,↓  =



u ∗i + N , E u i + N , E .

(3)

E
The two magnetic impurities in the system with conduction electrons have an indirect exchange interaction, which is described by H imp as follows [16,25,26]:

H imp = (

J0 2

) S az (na,↑  − na,↓ ) + (

J0 2

) S az (na,↑  − na,↓ ).

(4)

2

,

(6)

3. Results and discussion Here, we investigate the effect of e-e interaction, SOI, and electric field on the interaction between two magnetic impurities. Our results show that magnetic interaction can be tuned using these interactions. For this purpose, we consider a triangular silicene nanoflake (see Fig. 1) and locate impurities on a zigzag-edge, and calculate the RKKY interaction as a function of the distance between magnetic impurities. To show the effect of SOI on the RKKY interaction, we consider three different values of SOI (λ S O = 0, λ, 4λ; λ = 3.9 meV) and shown the RKKY interaction as a function of the distance between two impurities (R /a0 ) in Fig. 2. Our results show that there is a direct dependence between the RKKY interaction and the SOI strength, and with increasing SOI strength, the RKKY interaction increases, also, there is an inverse dependence between the distance and the RKKY interaction size when two impurities are separated from each other. Generally, the increasing of SOI strength causes the increasing of

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Fig. 2. The logarithmic representation of the RKKY interaction in the z-direction ( J zz ) and x-direction (=y-direction) ( J xx = J y y ) as a function of the distances between two impurities for the ZSiLF with M = 12 and λ = 3.9 meV. (a) and (c) the same sublattice and (b) and (d) the different sublattice in the position on-site on the edge of nanoflake. The first impurity is fixed in its place and the second impurity moved in the edge.

the J xx = J y y and J zz terms which this increasing becomes more notable with increasing the distance between two impurities especially at (R /a0 > 3). Note that J zz and J xx have the same behavior for same (A-A) and different sublattices (A-B) but the magnitude of RKKY interaction for different sublattice (A-B) is sensitively lesser than another one (A-A) and J zz shows a little fluctuations at some distance (for example R /a0 = 5 in the different sublattice case). Also, the results show with increasing the SOI strength, J D M are almost zero. Moreover, the behavior of J zz in the absence of ISOI, have a good agreement with the results of the graphene nanoflake [26]. Generally, the ISOI result in long-range RKKY interaction between two magnetic impurities. Furthermore, in similarity to the results obtained for honeycomb structures at half filling [48], for silicene nanoflake in the presence of the ISOI, the RKKY interaction is ferromagnetic ( J R K K Y > 0) between two impurities in the same sublattice and is antiferromagnetic ( J R K K Y < 0) between two impurities in different sublattice [48] (not shown here). Now, we mainly focus on the exchange interaction between impurities and conduction electrons in the presence of e-e interaction. In this regard, we consider three different e-e interaction strengths (U = 0, 0.2t , t), and then investigate the dependence of RKKY interactions on the distance between impurities. Interestingly, even considering the small value for the Hubbard term (U = 0.2t) increases the J zz size, which with increasing distance does not decrease and becomes nearly distance-independent as presented in Figs. 3(a) and 3(b) for the same and different sublattices, respectively. With more increasing Hubbard term to (U = t) the Ising term more increasing. Moreover, in the absence of ISOI, our results are consistent with the results of the work done on graphene [25–27]. The RKKY terms with and without e-e interaction for the same sublattice more than different sublattice. Moreover, the J xx decreases with increasing e-e interaction in the same sublattice that shown in Fig. 3(c), but in different sublattice, J xx gener-

ally increases and shows some fluctuations with complex behavior (Fig. 3(d)). Therefore, e-e interaction increases the Ising term for two types of sublattice cases, but the in-plane interaction between magnetic impurities for the same sublattice in spite of different sublattice decreases. Our results in Fig. 3(e) and 3(f) show that with considering the e-e interaction, the J D M becomes non zero but shows different behavior for two different types of sublattices. In fact, by increasing the strength of e-e interaction from U = 0.2t to U = t for the impurities in the same sublattice, the J D M decreases but for the different sublattice, increases. Finally, our results show that the increasing of Hubbard parameter increases the Ising term ( J zz ) and induced in-plane DM magnetization. Therefore the e-e interaction has a significant influence on the spatial properties of the RKKY interaction ( J xx , J zz , and J D M ) in silicene based nanostructures. The ISOI strength in the silicene-like structure such as germanene and stanene could change. For more clarification, we investigate the dependence of RKKY interaction on the variation of ISOI strength in Fig. 4. The spatial behavior of RKKY interaction as a function of the distance between two magnetic impurities at a fixed Hubbard interaction U = t for three different values of ISOIs λ S O = 0, λ, 4λ is shown in Fig. 4. By increasing the ISOI at U = t, there is a minimal change in the J zz and J xx (Figs. 4(a), 4(b), 4(c) and 4(d)), but the J D M for impurities in the same and different sublattices increases with increasing λ S O (Figs. 4(e) and 4(f)). Moreover, the J zz shows minimal changes versus distance between impurities and preserves nearly distance-independent even in the presence of ISOI and generally the J D M sensitively increases by increasing ISOI strength even with considering e-e interaction in the ZSiLF. The electrical control of magnetic properties in the crucial aspect of spin-dependent silicene-like applications. Moreover, the

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Fig. 3. The effect of variation of e-e interaction in the presence of ISOI on the various terms of the RKKY interaction for the ZSiLF with M = 12. λ S O = λ, U = 0, 0.2t , t, (a), (c) and (e) for the same sublattice and (b), (d) and (f) for the different sublattice in the position on-site on the edge of nanoflake.

Fig. 4. The effect of variation of SOI in the presence of e-e interaction on the various terms of the RKKY interaction for the ZSiLF with M = 12. λ S O = 0, λ, 4λ, U = t, (a), (c) and (e) for the same sublattice and (b), (d) and (f) for the different sublattice in the position on-site on the edge of nanoflake.

electric field could largely tune the band dispersion of silicene and convert silicene from topological insulator to band insulator [44,49]. Furthermore, in silicene sheet, the various magnetic phase

such as spiral, ferromagnetic, and antiferromagnetic can be tuned by electric field [38]. For investigation the ability of electric control of magnetic properties of triangular silicene like structure, we

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Fig. 5. The effect of variation of the electric field in the presence of e-e and ISOIs on the various terms of the RKKY interaction for the ZSiLF with M = 12. λ S O = λ, U = t, elE z = 0, 0.2t , 0.5t, (a), (c) and (e) for the same sublattice and (b), (d) and (f) for the different sublattice in the position on-site on the edge of nanoflake.

investigate the effect of the perpendicular electric field (E z ) in the presence of e-e and ISOIs in Fig. 5. In this case, we set λ S O = λ and U = t and investigate spatial behavior of RKKY interaction for three different values of elE z = 0, 0.2t , 0.5t which experimentally possible to produce a large external electric field [50]. Our results show that applying the perpendicular electric field also increases the J D M (Figs. 5(e) and 5(f)), but the J zz (Figs. 5(a) and 5(b)) and J xx (Figs. 5(c) and 5(d)) terms decrease for both same and different sublattice. Interestingly the Ising term even in the presence of e-e interaction by increasing the electric field strength shows slightly decreases but still preserve nearly distance-independent. 4. Conclusion In this work, we mainly focus on the role of e-e interaction and electric field on the RKKY interaction intermediated by itinerant electrons in a ZSiLF. This simulation has been done using the Kane-Mele-Hubbard Hamiltonian and the single-band tightbinding method by considering the effect of the electric field strength. Our results show that intrinsic spin-orbit interaction causes long range RKKY interaction for Ising and in-plane components. By taking e-e interaction, the amplitude of the interaction in the z-direction increases and become nearly distance-independent even in the presence of ISOI. Moreover, the applied electric field and e-e interactions induce the non-zero in-plane DM interaction with ISOI. In detail, increasing SOI could enhance RKKY interaction. However, the in-plane DM term still preserves zero in the absence of e-e interaction. Moreover, increasing the SOI strength in the presence of e-e interaction could not sensitively change the inplane and Ising term but increasing the DM term in the same and different sublattices. Moreover, increasing the perpendicular electric field could sensitively reduce the Ising term and increasing the in-plane DM term for both sublattices in ZSiLF. These results could pave the way to control the magnetic interaction on the silicene-

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