1. Introduction
For the past few years, two-dimensional (2D) van der Waals crystals have usually been regarded as an important arena for studying 2D magnetism [1–14]. Due to the development of the exfoliation method and various other methods [15–17], the thickness of 2D magnetic materials can be precisely controlled experimentally, and these layered 2D materials exhibit novel properties different from those of the three-dimensional system [18–27]. For example, CrI3 bilayers with small twist angle exist with both ferromagnetic and antiferromagnetic ground states [28–30]. The spin prefers an in-plane direction in 2D room-temperature ferromagnetic 1T-CrTe2 [31–34]. Monolayer Fe3GeTe2 and Fe3GaTe2 exhibit robust intrinsic 2D ferromagnetism with strong perpendicular anisotropy [35, 36]. Using scanning magneto-optic Kerr microscopy, an unexpected intrinsic long-range ferromagnetic order is found in pristine layered Cr2Ge2Te6 [3]. At room temperature, 2H-phase vanadium diselenide (VSe2) shows magnetic order in the 2D form only, in which the 3d orbitals of V4+ ions split into different groups due to structural anisotropy [37]. An exceptionally sizable orbital moment of the V3+ ion is displayed in the VI3 monolayer [38]. The physical mechanism of intrinsic 2D magnetism remains elusive, although many novel and unique magnetic phenomena have been discovered experimentally.
The Mermin–Wagner–Hohenberg theorem proves that there is no long-range magnetic order at any nonzero temperature for a one- or two-dimensional isotropic Heisenberg model (meaning the long-range magnetic order is destroyed by thermal fluctuations), however, magnetic anisotropy can suppress the effect of thermal fluctuations [39, 40]. It is obvious that magnetic anisotropy plays a critical role in 2D magnetic materials, and hence a generalized Heisenberg spin Hamiltonian model is established [3, 7, 41]. Bilayer Cr2Ge2Te6 maintains long-range magnetic order at cryogenic temperatures and displays complex magnetic interactions with considerable magnetic anisotropy [42]. Intrinsic ferromagnetism and perpendicular magnetic anisotropy at room temperature in few-monolayer CrTe2 films have been unambiguously evidenced by superconducting quantum interference and x-ray magnetic circular dichroism [43]. Sizable magnetic anisotropy energy and intrinsic antiferromagnetism have been discovered in 2D transition-metal borides [44]. By employing density functional theory calculations, 2D ferromagnetic rare-earth material GdB2N2 shows large perpendicular magnetic anisotropy at room temperature [45].
On the one hand, the edge effect on the magnetic and electronic properties of the layered 2D magnetic material is very significant [46–49]. For zigzag- or armchair-edged structures, the single-layer 2D MnO crystal has a degenerate antiferromagnetic ground state and a relatively less favorable ferromagnetic state [50]. Ta2S3 possesses a large out-of-plane magnetic anisotropy energy and high Curie temperature, and especially as the spin and orbital degrees of freedom couple, the Ta2S3 monolayer becomes a Chern insulator with a fully spin-polarized half-metallic edge state [51]. A new class of nonreciprocal edge spin waves exists in 2D antiferromagnetic nanoribbons [52]. On the other hand, controlling the magnetism of 2D magnetic materials by purely electrical means is also important, contributing to the development of information technology. The magnetic ground state of 2D antiferromagnet CrSBr bilayers [53] and CrI3 bilayers [54–56] can be switched from antiferromagnetic to ferromagnetic with electric field. Electric-field-controlled magnetic phase transitions are achieved in the metamagnetic intermetallic alloy FeRh [57].
The surface anisotropy layer, to a certain degree, can influence the magnetization behavior of semi-infinite ferromagnets or thick films. However, the surface effect is very weak in the system, which is masked by the intrinsic bulk magnetic order [58, 59]. The properties of nanostructure Ising or Ising–Heisenberg spin systems are obviously changed when the edge of the system is modified [47, 60–62]. Half-metallicity in nanometer-scale graphene ribbons and scaling rules for the band gaps of graphene nanoribbons as a function of their widths are predicted by using first-principles calculations [63, 64]. Zigzag graphene nanoribbons exhibit the presence of nonbonding edge states that were predicted to realize a peculiar type of magnetic ordering [65, 66]. Magnetic zigzag edges of graphene exist despite the fact that no true long-range magnetic order is possible in one dimension [67, 68]. When one or both sides of the isotropic Heisenberg spin nanoribbon are modified with anisotropic spins, the system, whether or not long-range magnetic order is established, is worthy of further study. Therefore, we theoretically investigate Heisenberg spin nanoribbons decorated by magnetic anisotropy of side edge spins. The magnetic order and transition temperature of the system show some phenomena. The arrangement of the manuscript is as follows. In section 2 , we will introduce the model and spin Green's function method in detail [69–72]. In section 3 , the long-range magnetic order and transition temperature of the system are analyzed and discussed. Finally, a brief conclusion is given in section 4 .
2. Model and method
In this paper, the magnetic order of nanoribbons with decoration on one side or two sides are investigated by use of the spin Green's function method. The schematic geometry of the system is depicted in figure 1, in which the blue and black balls denote two different Heisenberg spins with the spin quantum number μ = 1 and S = 1/2 respectively. The structure is effectively infinite in the y direction and has a finite width in the x direction, where the lines of spins are labeled by an integer n (= 1, 2, 3, ⋯, N).
Figure 1. Schematic representations of (a) the nanoribbon with one side edge decoration and (b) the nanoribbon with double side edge decoration. |
The system is described by a Heisenberg model, and its Hamiltonian is written as,
$\begin{eqnarray}\begin{array}{c}H=-2{J}_{{\rm{s}}}\displaystyle \sum _{\left\langle {ij}\right\rangle }{{\boldsymbol{\mu }}}_{i}{\rm{\cdot }}{{\boldsymbol{\mu }}}_{j}-2{J}_{{\rm{\perp }}}\displaystyle \sum _{\left\langle {ij}\right\rangle }{{\boldsymbol{\mu }}}_{i}{\rm{\cdot }}{{\boldsymbol{S}}}_{j}\\ \,-2J\displaystyle \sum _{\left\langle {ij}\right\rangle }{{\boldsymbol{S}}}_{i}{\rm{\cdot }}{{\boldsymbol{S}}}_{j}-D\displaystyle \sum _{i}{\left({\mu }_{i}^{z}\right)}^{2},\end{array}\end{eqnarray}$
where the index ⟨ij⟩ represents a pair of nearest-neighbor spins. The first three terms represent exchange interaction energies of the system, and Js, J⊥ and J denote the exchange interaction between nearest-neighbor spins μ and μ, μ and S, and S and S respectively. The last term presents the single-ion anisotropy energy of spin μ in the system, and D is the single-ion anisotropy parameter of spin μ. The retarded Green's function is introduced [71], $\begin{eqnarray}{G}_{{ij}}^{{\epsilon }_{j}}\left(t-{t}^{{\prime} }\right)=\left\langle \left\langle {\epsilon }_{i}^{+}\left(t\right){\rm{;}}{B}_{j}^{{\epsilon }_{j}}\left({t}^{{\prime} }\right)\right\rangle \right\rangle \end{eqnarray}$
where ${B}_{j}^{{\epsilon }_{j}}=e\left({\alpha }^{{\epsilon }_{j}}{\epsilon }_{j}^{z}\right){\epsilon }_{j}^{-},$ the spin raising and lowering operators of spin ε are defined as ${\epsilon }_{i}^{\pm }={\epsilon }_{i}^{x}\pm i{\epsilon }_{i}^{y},$ and ${\alpha }^{{\epsilon }_{i}}$ is Callen's parameter [73], εi = Si or μi dependent on the site position of i. Using time Fourier transforms and the equation of motion for Green's functions, $\begin{eqnarray}\omega {\left\langle \left\langle A{\rm{;}}B\right\rangle \right\rangle }_{\omega }=\left\langle \left[A,B\right]\right\rangle {\delta }_{{ij}}+{\left\langle \left\langle \left[A,H\right]{\rm{;}}B\right\rangle \right\rangle }_{\omega },\end{eqnarray}$
where A and B are two arbitrary operators respectively. For the transverse spin operator at site i little correlated with the longitudinal spin operator at site j (i≠j), the higher order Green's function can be decoupled by the random-phase approximation [74], $\begin{eqnarray}\left\langle \left\langle {{\epsilon }_{{i}^{{\rm{{\prime} }}}}^{z}\epsilon }_{i}^{+};{B}_{j}^{\epsilon }\right\rangle \right\rangle \cong \left\langle {\epsilon }_{{i}^{{\rm{{\prime} }}}}^{z}\right\rangle \left\langle \left\langle {\epsilon }_{i}^{+};{B}_{j}^{\epsilon }\right\rangle \right\rangle .\end{eqnarray}$
Using Anderson–Callen decomposition [75], the higher order Green's function generated by single-ion anisotropy at the same lattice site can be expressed as, $\begin{eqnarray}\left\langle \left\langle \left({\mu }_{i}^{z}{\mu }_{i}^{+}+{\mu }_{i}^{+}{\mu }_{i}^{z}\right);{B}_{j}^{\mu }\right\rangle \right\rangle \approx 2\left\langle {\mu }_{i}^{z}\right\rangle {{\rm{\Theta }}}_{i}\left\langle \left\langle {\mu }_{i}^{+};{B}_{j}^{\mu }\right\rangle \right\rangle \end{eqnarray}$
where ${{\rm{\Theta }}}_{i}=1-\frac{1}{2{\mu }^{2}}\left[\mu \left(\mu +1\right)-\left\langle {\mu }_{i}^{z}{\mu }_{i}^{z}\right\rangle \right]$.By using spatial Fourier transforms [76, 77], we can obtain a group of equations about the Fourier component of the Green's function16 ) can be converted into the integral, namely $\frac{1}{{N}_{y}}{\sum }_{{k}_{y}}\to \frac{1}{2\pi }{\int }_{-\pi /a}^{\pi /a}\text{d}{k}_{y}.$ ${R}_{n,n}$ is the algebraic cofactor of element in matrix $\omega {\boldsymbol{I}}-{\boldsymbol{W}}.$ The values of ${\omega }_{i}({k}_{y})$ define the spin wave spectrum which can be obtained from the coefficient matrix of equation (6 ). For convenience, JS, J⊥, J, D and T are reduced by J (JS/J, J⊥/J, D/J and T/J), J = 1.0 and kB = 1.
$\begin{eqnarray}\left(\omega {\boldsymbol{I}}-{\boldsymbol{W}}\right){\boldsymbol{G}}={\boldsymbol{F}}\end{eqnarray}$
where I is the unit matrix. F is a one-dimensional column matrix, $\begin{eqnarray}\begin{array}{c}{\boldsymbol{F}}=\begin{array}{c}\left[\begin{array}{c}2\left\langle {\mu }_{1}^{z}\right\rangle {\delta }_{1,n}\\ 2\left\langle {S}_{2}^{z}\right\rangle {\delta }_{2,n}\\ \vdots \\ 2\left\langle {S}_{N-1}^{z}\right\rangle {\delta }_{N-1,n}\\ 2\left\langle {\epsilon }_{N}^{z}\right\rangle {\delta }_{N,n}\end{array}\right]\,\end{array}\end{array}\end{eqnarray}$
where the index 1, ⋯, N denotes the order of the lines of spins, and the spatial Fourier component G in the y direction of the Green's function is $\begin{eqnarray}\begin{array}{c}{\boldsymbol{G}}=\left[\begin{array}{c}{G}_{1,n}^{\mu }\\ {G}_{2,n}^{s}\\ \vdots \\ {G}_{N-1,n}^{s}\\ {G}_{N,n}^{\epsilon }\end{array}\right].\end{array}\end{eqnarray}$
The coefficient matrix W is expressed as $\begin{eqnarray}\begin{array}{c}{\boldsymbol{W}}=\left[\begin{array}{ccccccc}{X}_{1}^{\mu } & {-\left({p}_{1}^{\mu }\right)}^{{\rm{{\prime} }}} & 0 & 0 & \cdots & 0 & 0\\ -{p}_{2}^{s} & {X}_{2}^{s} & {-\left({p}_{2}^{s}\right)}^{{\rm{{\prime} }}} & 0 & \cdots & 0 & 0\\ 0 & {-p}_{3}^{s} & {X}_{3}^{s} & -{\left({p}_{3}^{s}\right)}^{{\rm{{\prime} }}} & \cdots & 0 & 0\\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & 0 & \cdots & {X}_{N-1}^{s} & -{\left({p}_{N-1}^{s}\right)}^{{\rm{{\prime} }}}\\ 0 & 0 & 0 & 0 & \cdots & -{p}_{N}^{\epsilon } & {X}_{N}^{\epsilon }\end{array}\right]\end{array}.\end{eqnarray}$
When one side edge of the nanoribbon is decorated, the matrix elements of W are, $\begin{eqnarray}\left\{\begin{array}{c}\begin{array}{c}{X}_{1}^{\mu }=2{J}_{s}\left\langle {\mu }_{1}^{z}\right\rangle \left(z-z{\gamma }_{{k}_{y}}\right)+2\left\langle {\mu }_{1}^{z}\right\rangle {{\rm{\Theta }}}_{1}D+2{J}_{{\rm{\perp }}}\left\langle {S}_{2}^{z}\right\rangle ,n=1\\ {X}_{2}^{s}=2{J}_{{\rm{\perp }}}\left\langle {\mu }_{1}^{z}\right\rangle +2J\left\langle {S}_{2}^{z}\right\rangle \left(z-z{\gamma }_{{k}_{y}}\right)+2J\left\langle {S}_{3}^{z}\right\rangle ,n=2\\ {X}_{n}^{s}=2J\left\langle {S}_{n-2}^{z}\right\rangle +2J\left\langle {S}_{n-1}^{z}\right\rangle \left(z-z{\gamma }_{{k}_{y}}\right)+2J\left\langle {S}_{n}^{z}\right\rangle ,3\leqslant n\leqslant N-1\end{array}\\ {X}_{N}^{s}=2J\left\langle {S}_{N-1}^{z}\right\rangle +2J\left\langle {S}_{N}^{z}\right\rangle \left(z-z{\gamma }_{{k}_{y}}\right),n=N\end{array}\right.\end{eqnarray}$
$\begin{eqnarray}\left\{\begin{array}{c}\begin{array}{c}{p}_{2}^{s}=2{J}_{{\rm{\perp }}}\left\langle {S}_{2}^{z}\right\rangle ,n=2\\ {p}_{n}^{s}=2J\left\langle {S}_{n}^{z}\right\rangle ,3\leqslant n\leqslant N\\ {\left({p}_{1}^{\mu }\right)}^{{\prime} }=2{J}_{{\rm{\perp }}}\left\langle {\mu }_{1}^{z}\right\rangle ,n=1\end{array}\\ {\left({p}_{n}^{s}\right)}^{{\prime} }=2J\left\langle {S}_{n}^{z}\right\rangle ,2\leqslant n\leqslant N-1\end{array}\right..\end{eqnarray}$
When both side edges of the nanoribbon are decorated, the four nonzero elements in the last two rows of the matrix W are replaced by $\begin{eqnarray}\left\{\begin{array}{c}{X}_{N-1}^{s}=2J{S}_{N-2}^{z}+2J{S}_{N-1}^{z}\left(z-z{\gamma }_{{k}_{y}}\right)+2{J}_{\perp }{\mu }_{N}^{z},\,n=N-1\\ {X}_{N}^{\mu }=2{J}_{\perp }{S}_{N-1}^{z}+2{J}_{s}{\mu }_{N}^{z}\left(z-z{\gamma }_{{k}_{y}}\right)+2{\mu }_{N}^{z}{{\rm{\Theta }}}_{N}D,\,n=N\end{array}\right.\end{eqnarray}$
$\begin{eqnarray}\left\{\begin{array}{c}{\left({p}_{N-1}^{s}\right)}^{{\prime} }=2{J}_{{\rm{\perp }}}\left\langle {S}_{n}^{z}\right\rangle ,n=N-1\\ \,{p}_{N}^{\mu }=2{J}_{{\rm{\perp }}}\left\langle {\mu }_{N}^{z}\right\rangle ,n=N\end{array}\right.\end{eqnarray}$
where z = 2 and ${\gamma }_{{k}_{y}}=\cos {\rm{}}\left({k}_{y}\right),$ ky representing the wave vector in the y axial direction within the first Brillouin zone. By means of the spectral theorem and Callen's technique [73], the magnetization of the nth line can be written as, $\begin{eqnarray}\left\langle {\epsilon }_{n}^{z}\right\rangle =\frac{\left({{\rm{\Phi }}}_{n}+1+{\epsilon }_{n}\right){{\rm{\Phi }}}_{n}^{2{\epsilon }_{n}+1}-\left({{\rm{\Phi }}}_{n}-{\epsilon }_{n}\right){\left({{\rm{\Phi }}}_{n}+1\right)}^{2{\epsilon }_{n}+1}}{{\left({{\rm{\Phi }}}_{n}+1\right)}^{2{\epsilon }_{n}+1}-{{\rm{\Phi }}}_{n}^{2{\epsilon }_{n}+1}}\end{eqnarray}$
and the spin correlation function $\left\langle {\epsilon }_{n}^{z}{\epsilon }_{n}^{z}\right\rangle ,$ is given by, $\begin{eqnarray}\left\langle {\epsilon }_{n}^{z}{\epsilon }_{n}^{z}\right\rangle ={\epsilon }_{n}\left({\epsilon }_{n}+1\right)-\left(1+2{{\rm{\Phi }}}_{n}\right)\left\langle {\epsilon }_{n}^{z}\right\rangle ,\end{eqnarray}$
where εn denotes the spin quantum number of the spin in the nth line. The auxiliary function is written as, $\begin{eqnarray}{{\rm{\Phi }}}_{n}=\frac{1}{{N}_{y}}\displaystyle \sum _{{k}_{y}}\displaystyle \sum _{i=1}^{N}\frac{{R}_{n,n}\left({\omega }_{i}\left({k}_{y}\right)\right)}{\left({\text{e}}^{\left.\beta {\omega }_{i}({k}_{y}\right)}-1\right){\prod }_{j=1\left(j\ne i\right)}^{N}\left({\omega }_{i}\left({k}_{y})-{\omega }_{j}({k}_{y}\right)\right)},\end{eqnarray}$
where β = 1/(kBT), ${k}_{y}=\frac{2\pi m}{{N}_{y}a},$ a is the lattice constant and the integer m takes $-\frac{{N}_{y}}{2}\lt m\leqslant \frac{{N}_{y}}{2}.$ Ny denotes the number of the spin in a period in an infinite spin line. As Ny is infinity, the summation of ${k}_{y}$ in equation (3. Results and discussions
Firstly, we discuss the influence of the exchange interaction J⊥/J on the magnetization of the one side edge decorated nanoribbon, as shown in figures 2(a)–(c). For JS/J = 0.8, D/J = 0.4 and N = 7, the transition temperature TC/J of the system is enhanced with increasing exchange interaction J⊥/J. For the quasi-2D isotropic Heisenberg spin system, the system has a finite phase transition temperature when a very weak interaction is exhibited between two layers [78]. A similar phenomenon is also observed for the isotropic Heisenberg nanoribbon decorated by side spins with single-ion anisotropy. In figure 2(a), when the nanoribbon is not decorated (J⊥/J = 0), the red line represents the behavior of the magnetization ⟨μz⟩ with increasing temperature, and the phase transition temperature TC/J is about 0.81, but $\left\langle {S}^{z}\right\rangle $ = 0. When J⊥/J = 0.0001, the black line indicates the change of ⟨μz⟩ with increasing temperature, and the magnetization $\left\langle {S}^{z}\right\rangle $ decreases faster and then forms a trail with increasing temperature. The phase transition temperature TC/J of the nanoribbon is about 0.89. Compared with the red line, the black line decreases faster with the increase of temperature but has a larger transition temperature. As J⊥/J ≠ 0, the coordination number of the edge decorated spin increases, where the spin of the edge decorated chain affects the magnetic order of the nanoribbon, resulting in a faster decline of the magnetization of the edge decorated spin. And the magnetic order of the nanoribbon in turn influences the magnetic order of the edge decorated spin, which leads to a larger phase transition temperature of the edge decorated spin. The result means that spontaneous magnetization of the nanoribbon will disappear, once no exchange interaction exists between the host spins and the decorated edge spins. The magnetic order of the system is extraordinarily sensitive to the decorated edge spins. The magnetic order of the nanoribbon shows immediately as J⊥/J increases from 0 to 0.0001 in figure 2(a). Below the phase transition temperature, the magnetization of the lines of spins away from the decorated edge decreases in figures 2(b)–(c), which shows that the influence of the decorated edge on the magnetic order of the system decreases gradually with increasing distance from the decorated edge. The transition temperature TC/J of the system increases rapidly from a certain value, and then slowly with strengthening J⊥/J in figure 2(d) (the parameter J⊥/J starts from 0.0001). The certain value is the transition temperature of the decorated edge spins. For comparison, the transition temperatures of a 2D square lattice and simple cubic lattice are given respectively [73, 79].
Figure 2. Temperature dependence of the magnetization of the nanoribbon with one side edge decoration for different coupling strengths between decorated spin and the nanoribbon with N = 7, JS/J = 0.8, D/J = 0.4, for (a) J⊥/J = 0 and J⊥/J = 0.0001, (b) J⊥/J = 0.1 and (c) J⊥/J = 0.6, and (d) transition temperature TC/J versus J⊥/J. |
The effect of the exchange interaction JS/J on the magnetization of the one side edge decorated nanoribbon is shown in figures 3(a)–(c). With J⊥/J = 0.6, D/J = 0.4 and N = 7, the transition temperature TC/J of the system strengthens as the exchange interaction JS/J increases from zero in figure 3(d). A remarkable case is that, as the side edge decorated spins separate from each other (JS/J = 0.0), the long-range magnetic order of the system still exists at nonzero temperature, as shown in figure 3(a). The system possesses spontaneous magnetization, whether the side edge decorated spins link together or separate from each other, even when they are arranged randomly.
Figure 3. Temperature dependence of the magnetization of the nanoribbon with one side edge decoration for different coupling strengths between decorated spins with N = 7, J⊥/J = 0.6, D/J = 0.4, for (a) JS/J = 0.0, (b) JS/J = 0.4 and (c) JS/J = 0.8, and (d) transition temperature TC/J versus Js/J. |
The effect of the single-ion anisotropy D/J on the magnetization of the one side edge decorated nanoribbon is shown in figures 4(a)–(c). With J⊥/J = 0.6, JS/J = 0.8 and N = 7, the transition temperature TC/J of the system strengthens rapidly from zero as D/J increases from zero in figure 4(d). In particular, when D/J = 0.0, no spontaneous magnetization in the system appears at nonzero temperature, as shown in figure 4(a), which abides by the Mermin–Wagner theorem [39]. The nonzero-temperature magnetic order of the system emerges immediately as existing magnetic anisotropy in the system, which is also consistent with Hohenberg's study [40]. These theoretical results can provide a comprehensive understanding of electric-field-induced magnetic phases in 2D van der Waals magnetic materials [53–57] and explain well the magnetism in graphene nanoribbons with zigzag edge [61, 80].
Figure 4. Temperature dependence of the magnetization of the nanoribbon with one side edge decoration for different coupling strengths between decorated spins with N = 7, J⊥/J = 0.6, JS/J = 0.8, for (a) D/J = 0.01, (b) D/J = 0.2 and (c) D/J = 0.4, and (d) transition temperature TC/J versus D/J. |
In figure 5, with D/J = 0.4, J⊥/J = 0.6 and JS/J = 0.8, the influence of the number of lines of spins N (width of the nanoribbon) on the magnetization of the one side edge decorated nanoribbon is shown. The magnetization of the lines of spins, away from the decorated edge, obviously decreases faster and then forms a trail with increasing temperature. For N < 10, the transition temperature TC/J of the system enhances slowly with increasing width of the system, as shown in figure 5(d). However, when the number of lines of spins N ≥ 10, the transition temperature TC/J of the system hardly changes as the width of the system increases. It can be inferred that, as the width of the nanoribbon approaches infinite limit N→∞, the nanoribbon becomes a semi-infinite 2D spin system and still maintains the same stable magnetic order. The results are quite different from those of the multilayer ferromagnetic films, where the influence of surface anisotropy on the long-range magnetic order in multilayer ferromagnetic films decreases with increasing film thickness, until the magnetic order is hidden by the intrinsic bulk magnetic order in multilayer ferromagnetic films [58, 59].
Figure 5. Temperature dependence of the magnetization of the nanoribbon with one side edge decoration for different widths of the nanoribbon with D/J=0.4, J⊥/J = 0.6 and JS/J = 0.8, for (a) N = 7, (b) N = 15 and (c) N = 30, and (d) transition temperature TC/J versus N. |
For a comprehensive understanding of the influence of various decorated edges on the nanoribbon, the double side edge decorated nanoribbon is also investigated. In figure 6, with J⊥/J = 0.6, D/J = 0.4 and N = 7, the transition temperature TC/J of the system is enhanced as the exchange interaction JS/J increases from zero, which is similar to the result for the one side edge decorated nanoribbon. According to two-fold symmetry of the system (shown in figure 1(b)), the magnetization of the system shows four temperature behaviors, $\left\langle {\mu }_{1}^{z}\right\rangle $=$\left\langle {\mu }_{7}^{z}\right\rangle ,$ $\left\langle {S}_{2}^{z}\right\rangle $=$\left\langle {S}_{6}^{z}\right\rangle ,$ $\left\langle {S}_{3}^{z}\right\rangle $=$\left\langle {S}_{5}^{z}\right\rangle $ and $\left\langle {S}_{4}^{z}\right\rangle $ respectively. The middle line of spins, away from the decorated edge, shows the smallest magnetization.
Figure 6. Temperature dependence of the magnetization of the nanoribbon with double side edge decoration for different coupling strengths between decorated spins with N = 7, J⊥/J = 0.6 and D/J = 0.4, for (a) Js/J = 0.0, (b) Js/J = 0.4, (c) Js/J = 0.8 and (d) Js/J = 1.6. |
The transition temperatures influenced by J⊥/J, Js/J, D/J and N are investigated both in the one side edge decorated nanoribbon and the double side edges decorated nanoribbon. In figure 7(a), with N = 7, D/J = 0.4 and Js/J = 0.8, the transition temperature of the double side edges decorated nanoribbon increases faster and eventually achieves a larger value with increasing J⊥/J than that of the one side edge decorated nanoribbon. In figure 7(b), with N = 7, D/J = 0.4 and J⊥/J = 0.6, and increasing Js/J, the transition temperature of the one side edge decorated nanoribbon approaches gradually that of the double side edges decorated nanoribbon. In figure 7(c), with N = 7, Js/J = 0.8 and J⊥/J = 0.6, an unexpected phenomenon is found in that the behaviors of transition temperature are almost the same with increasing D/J in both the one side edge decorated nanoribbon and the double side edges decorated nanoribbon. In figure 7(d), with D/J = 0.4, Js/J = 0.8 and J⊥/J = 0.6, when the number of lines of spins N increases, the transition temperature of the one side edge decorated nanoribbon increases very slowly and then tends to a constant value, but the transition temperature of the double side edge decorated nanoribbon decreases and then tends to the same constant value. The results reveal that, as N→∞, the long-range magnetic orders are similar both in the one side edge decorated nanoribbon and the double side edge decorated nanoribbon.
Figure 7. Transition temperature TC/J versus J⊥/J (a), Js/J (b), D/J (c) and N (d) respectively, (a) for N = 7, D/J = 0.4 and Js/J = 0.8, (b) for N = 7, D/J = 0.4 and J⊥/J = 0.6, (c) for N = 7, Js/J = 0.4 and J⊥/J = 0.6 and (d) for Js/J=0.8, D/J = 0.4 and J⊥/J = 0.6. |
4. Conclusion
For the one side edge decorated and double side edge decorated Heisenberg spin nanoribbons, five main cases are summarized: (I) the transition temperature of the system is enhanced by increasing the exchange interaction J⊥/J between the host spins and decorated edge spins. In particular, no long-range magnetic order appears in the system (excluding decorated edge spins) at nonzero temperature when J⊥/J = 0. (II) A remarkable case is that, as the decorated edge spins separate from each other, the long-range magnetic order of the system still exists at nonzero temperature. The system possesses spontaneous magnetization, whether the decorated side edge spins link together or separate from each other, even when they are arranged randomly. (III) The transition temperature of the system strengthens rapidly from zero with single-ion anisotropy D/J increasing from zero. In particular, when D/J = 0, no magnetic order of the system appears at nonzero temperature. (IV) Below phase transition temperature, the magnetization of the lines of spins away from the decorated side edge decreases, which illustrates that the influence of the decorated edge on the magnetic order of the system decreases gradually with increasing distance from the decorated edge. (V) When the number of lines of spins N increases, the transition temperature of the one side edge decorated nanoribbon increases very slowly and then tends to a constant value, but the transition temperature of the double side edge decorated nanoribbon decreases and then tends to the same constant value. In particular, as N→∞, the long-range magnetic orders are similar both in the one side edge decorated nanoribbon and the double side edge decorated nanoribbon.