Single-walled Ising nanotube with opposite sign of interactions using Wang-Landau algorithm

The effect of opposite sign of interactions in a single-walled Ising nanotube is investigated using the Wang-Landau algorithm. The thermodynamic observables are calculated from the estimated density of states (DOS) with and without the presence of an external magnetic field. Irrespective of the applied magnetic field, a symmetric trend of DOS is observed for opposite sign of interactions which is in contrast to the asymmetric trend for same sign of interactions. Further, two types of anti-ferromagnetic (AFM) orderings, namely A-type and C-type anti-ferromagnetic order, are observed for opposite sign of interactions. These AFM spin orientations are switched to ferromagnetic (FM) phase by increasing the applied magnetic field ($B$). However, the spin ordering changes from the ordered AFM/FM phase to a disordered paramagnetic phase by increasing the temperature. Phase diagram shows that these three phases coexist around $B=2.0$. This study indicates that, by properly tuning the magnetic properties, the single-walled nanotube can be used for fabrication of new types of magnetic storage nano materials.
Key words: Single-walled Ising nanotube, opposite sign of interactions, Wang-Landau algorithm, and phase transition.

     In recent decades, the low dimensional magnetic systems has been intensively studied due to their exotic magnetic properties and vast technological applications including biomedicines [1], biomolecular motors [2], spintronics devices [3], energy storage [4], magneto-optic recording [5], and nanocomposite magnets [6]. The magnetic nanosystems has various structures viz, nanoparticles, nanotubes, nanowires, nanoribben, nanographene, etc,. Many experimental techniques are used to synthesis nanomaterials like, CoNi nanotubes [7], FeNi nanotubes [8], and carbon nanotubes [9]. Theoretical studies also play a vital role in understanding the magnetic nanosystems via different techniques including Green function technique [10], differential operator technique [11], mean field approximation [12], effective field theory [13] and Monte Carlo technique [14]. Among the interacting models available in the literature, the Ising model is one of the simplest model to study the phase transition in magnetic systems. The main advantage of this model is that any interacting system can be easily mapped to this model [15, 16], and it is more convenient to study the physical properties of low dimensional systems.

Several studies are reported in the context of Ising modelled nanosystems. In particular, Mn doped ZnO nanotubes were investigated using the Monte Carlo techniques, and analyzed for the effect of Mn concentration on magnetic properties [17]. Some of the hexagonal Ising nanotubes with core-shell structures were reported to have ferrimagnetic mixed spins [18]. Core-shell nanotube was also studied using metropolis algorithm to analyze the compensation temperature, magnetic properties, and phase diagram [19]. The pseudocritical magnetic properties for a single-walled ferromagnetic nanotube with varying system size and varying topology were reported [20]. Some studies explored the magnetic properties of hexagonal Ising nanotube by applying the effective field theory coupled with differential operator technique. These techniques show that some systems with opposite sign of exchange interaction suffer magnetic frustration [21, 22, 23]. Our interest is to study the magnetic properties and the phase behavior of single walled Ising nanotube (SWINT) influenced by the opposite sign of interaction between the inter-layer and the intra-layer using the Monte Carlo technique.

Apart from the studies of ferromagnetic interaction, several studies of Anti Ferro Magnetic (AFM) interactions in the layered lattice systems were also investigated and found to have different types of ground state [24, 25]. The AFM interactions in the alternative layers of the Ising system exhibit phase changes [26, 27]. The experimental and DFT studies of the magnetic materials exhibit different types of magnetic spin orientations. The A-type AFM order was observed for the layered $EuMg_{2}Sb_{2}$ single crystals [28, 29], its crystallographic, magnetic, and electronic properties are indicating a narrow gap semiconducting property.

In the case of nanotube system, Salazar et al. have studied the magnetic properties of ferromagnetic single walled nanotube using metropolis algorithm [30, 31]. However, there are few simulation studies of AFM ordering for SWINT [32]. In our previous study, the ferromagnetic (FM) and different types of AFM orderings were observed for SWINT. The metropolis algorithm is used to examine the system and to analyze its phase transition [32]. We have also observed the asymmetric trend of Density of States (DOS) of SWINT under the applied magnetic field with the same sign of interactions using Wang-Landau (WL) algorithm. The WL algorithm has been widely used among the non-Boltzmann sampling techniques [33]. The advantages of WL algorithm is that one can estimate DOS and then calculate the average thermodynamics properties at any finite temperature. The main focus of the present study is to investigate the effect of opposite sign of interactions in a SWINT with the absence/presence of a external magnetic field by employing WL algorithm.

The manuscript is presented as follows: Section 2 describes the SWINT model and WL algorithm. The discussion on the observed results with and without magnetic field is provided in the Section 3. Finally, the Conclusion section summarizes the important findings.

    The spin-half Ising model is tailored as a single-walled Ising nanotube (SWINT) with a layered graphite-like structure [32], as shown in Fig.1. Each layer comprise of six lattice sites with magnetic moments $S_{i}$ with values of $+1$ for spin up and $-1$ for spin down. The Hamiltonian of the system is,

$$H~{}=~{}-J_{1}\sum_{<i,j>}S_{i}S_{j}~{}-J_{2}\sum_{<i,k>}S_{i}S_{k}~{}-B\sum_{i}S_{i},$$

(1)

where, $J_{1}$ denotes strength of the intra-layer interaction, $J_{2}$ is strength of the inter-layer interaction, $<\cdots>$ represents the nearest neighbour spin sites, and $B$ is the external magnetic field. The system exhibits FM (parallel) spin ordering only if both the interactions $J_{1}$ and $J_{2}$ are greater than zero. The different types of AFM (anti-parallel) spin orderings are formed from the other combinations of signs of $J_{1}$ and $J_{2}$. Depending on such combination of interactions, the system is classified into two kinds: same sign ($J_{1}>0$; $J_{2}>0$, and $J_{1}<0$; $J_{2}<0$) and opposite sign of interactions ($J_{1}>0$; $J_{2}<0$, and $J_{1}<0$; $J_{2}>0$). Since some systems with opposite sign of exchange interaction suffers magnetic frustration [21, 22, 23], this work will pay special attention to the opposite sign of interactions of $J_{1}$ and $J_{2}$ in SWINT, which arranges spins in A-type and C-type AFM orderings. In A-type, the spin alignment is parallel within the layer and anti-parallel between the layer. In contrast, C-type AFM order is observed by anti-parallel alignment of spins within the layer and parallel alignment of spins between the layers [34]. The representation of these orderings are given in Fig. 1. The closed boundary condition is applied for the $x$ and $y$ axes, whereas the periodic boundary condition is applied for the $z$ axis. The Wang-Landau algorithm were applied to the SWINT system to compute the density of states ($g(E)$). It is a two step process: first step is the estimation of the DOS and second step is the calculation of average observables from production run.

The algorithm performs a random walk in an energy space with probability $P(E)\propto\frac{1}{g(E)}$ [35, 36, 37]. The algorithm begins with the initialization of DOS, $\ln g(E)=0$; histogram $H(E)=0$, and a modification factor $\ln f$. The initial configuration $C_{i}$ and the trail configuration $C_{t}$ are generated with energy $E_{i}$ and $E_{t}$, respectively. The transition (from $C_{i}$ to $C_{t}$) is accepted with transition probability,

$\displaystyle P(C_{i}\rightarrow C_{t})$

$\displaystyle=$

$\displaystyle min\Bigg{(}1,\Big{[}\ln g(E_{i})-\ln g(E_{t})\Big{]}\Bigg{)}.$

(2)

Further, the DOS and the histogram have been updated as $\ln~{}[g(E)]=\ln~{}[g(E)]+\ln f$ and $H(E)=H(E)+1$, respectively. If the histogram’s flatness reaches a certain level ($80\%$), the modification factor is updated to $\ln f=\ln f\times 0.5$, and the histogram is then reset to $H(E)=0$. The algorithm gets terminated when the $\ln f$ reaches a small enough value ($\approx 10^{-8}$). Finally, we obtain a converged DOS from this algorithm.

An entropic ensemble is generated during the production run, and the canonical ensemble averages of any quantities are calculated at a given finite temperature as [38, 39],

$\displaystyle\langle O\rangle_{\beta}$

$\displaystyle=$

$\displaystyle\frac{\sum\limits_{C}O(C)~{}g(E(C))~{}\exp[-\beta E(C)]}{\sum\limits_{C}g(E(C))~{}\exp[-\beta E(C)]},$

(3)

where $C$ represents the configuration, $\beta=\frac{1}{k_{B}T}$ ($k_{B}$ is a Boltzmann constant, taken as $1$ for entire simulation), $T$ is the temperature. It is to be noted that all the parameters used in this simulation has an appropriate units in terms of $k_{B}T$ and magnetic field. The system’s average magnetization, $\langle M\rangle$ and average energy, $\langle E\rangle$ are calculated using equation (3). The magnetization per spin is given by $M=\frac{1}{N}\sum_{i}S_{i}$. Considering there are $L$ number of layers, the total number of spins can be $N=6L$ as there are $6$ spins in each layer. The whole simulation is performed with $180$ spins in $30$ layers. The magnetic susceptibility is calculated as,

$\displaystyle\chi(T)$

$\displaystyle=$

$\displaystyle\frac{1}{k_{B}T}~{}(\langle M^{2}\rangle_{T}-\langle M\rangle^{2}_{T}).$

(4)

The free energy $F(T)=-k_{B}T\ln(Z)$, and entropy $S(T)=\frac{U(T)-F(T)}{T}$ are also calculated, where $Z=\sum\limits_{E}g(E)~{}\exp[-\beta E]$ is the partition function and $U(T)=\langle E\rangle_{T}$ is the internal energy [40]. The following section analyze the effect of opposite sign of interactions in a SWINT subjected to the external external magnetic field by employing WL algorithm.

    The ground state spin ordering is obtained in the absence of a magnetic field for opposite sign of interactions, depicted in Fig. 1. The A-type AFM ordering is obtained by applying the following interactions $J_{1}=+1$ and $J_{2}=-1$ (Fig. 1(a)). The spins follow parallel ordering within the layer, and anti-parallel ordering between the layers. The interactions $J_{1}=-1$ and $J_{2}=+1$ are applied to the system, yielding the C-type AFM (Fig. 1(b)). This ordering follow anti-parallel orientation within the layer, and a parallel orientation between the layers. So in this type of ordering, the spins are aligned similarly among all the layers.

In the entire simulation, the magnitude of the interaction $J_{2}$ kept fixed as ($J_{2}=|\pm 1|$) and its sign fixed depending on the ordering. Initially the system is investigated in the absence of magnetic field by varying the parameter $J_{1}$, and later by varying the magnetic field $B$. The simulation results are discussed in the proceeding section.

     The SWINT is investigated using the WL algorithm with opposite sign of interactions. The DOS for A-type and C-type AFM orderings are obtained in the absence and presence of an external magnetic field and its shape is observed to be always symmetrical for different magnetic fields. The external magnetic field $B$ is increased from $0.5$ to $3.0$ with fixed unit interaction strength for A-type and C-type AFM. The canonical ensemble averages ($\langle E\rangle$ and $\langle M\rangle$) for different magnetic fields are obtained from the estimated DOS. The susceptibility plots show distinct peaks that indicate the equilibrium phase transition.

The observed transition temperature increases for A-type AFM and decreases for C-type AFM with increase in $J_{1}$ in the absence of magnetic field and the system’s phase transits from A-type/C-type AFM (ordered phase) to paramagnetic order (disordered phase) with increasing temperature. When the applied magnetic field increases, the AFM orderings are rearranged to FM ordering at low temperatures. All FM, A-AFM, and C-AFM order curves transit to paramagnetic order at high temperatures. The transition temperature for both types of AFM orderings decreases with increasing magnetic field until $B=2.0$, and increases with increasing magnetic field for $B>2.0$. The system’s phase shifts from an A-type/C-type AFM to a paramagnetic phase when the external magnetic field is less than $2.0$. In the case of $B\geq 2.0$, the system switches from FM to a paramagnetic phase with increasing temperature.

The SWINT follows a second-order phase transition in the absence of a magnetic field. The second-order phase transition is also confirmed from the canonical entropy and the probability distribution in the presence of a magnetic field. However, the three phases (AFM, FM and Paramagnetic) coexisting around $B=2.0$ as shown Fig.7 indicates that there should be a tri-critical point in the vicinity of $B=2.0$. Since the spin system under study is computationally hard especially with the presence of opposite sign of interaction, the observation of the above phase coexistence with the presence of disorder and frustration [46] must needs further careful analysis in the future.

The above analysis indicates that, by properly tuning the magnetic properties, the single-walled nanotube can be used for fabrication of new types of magnetic storage nano materials. Further, this study will be useful for investigating the flexomagnetoelectric effect [47] of low dimensional anti-ferromagnetic magnetic systems for spintronics applications in the presence of external magnetic field.

Finally, in our WL simulation, the thermodynamic observables of SWINT are calculated and analyzed from the estimated DOS ($g(E)$) excluding the hysteresis behavior with the presence of magnetic field. In our future works, we will calculate the joint DOS ($g(E,M)$) which is the function of energy and magnetization for analyzing hysteresis behavior of the frustrated carbon nanotube structures [48, 49].

    In the absence of magnetic field, the net magnetization is a zero at low temperatures for all anti-ferromagnetic orderings as all the spins are aligned in anti-parallel direction. The total magnetization of varying interaction strengths is depicted in Figs. 10(a) and 11(a) for A-type and C-type AFM orderings, respectively. The increase in temperature can disrupt the anti-parallel alignment into random directions to produce non-zero magnetization values. Thus, the total magnetization gradually increases for all interactions and reaches its saturation value. Therefore, the anti-ferromagnetic order breaks with temperature and turns to the paramagnetic order. The magnetic susceptibility curves of various interaction strengths are shown in Figs. 10(b) and 11(b) for A-type and C-type AFM orderings. The transition temperature are calculated from the peak in susceptibility plot. The $T_{C}$ value increases with increase in interaction $J_{1}$ from $0.2$ to $1.0$ for A-type AFM ordering (Fig. 10(b)). For C-type ordering, the $T_{C}$ decreases while increasing the value of interaction $J_{1}$ from $-1.0$ to $-0.2$ (Fig. 11(b)).

Free energy and entropy are calculated directly from the obtained DOS for A-type and C-type AFM orderings (Fig. 12). As temperature increases, all free energy curves remain unchanged upto their transition temperatures. After that, the curves are gradually decreases and merged (Figs. 12(a) and 12(c)). The entropy curves for both AFM orderings are given in Figs. 12(b) and 12(d). The entropy value at low temperatures is close to zero as the number of accessible microstates is minimum. As the temperature increases, the number of accessible microstates increases, gradually raising the entropy curve. At high temperatures, the curves are merged and reach their saturation limit of the predicted magnetic entropy ($\ln 2$).

The author, A. Arul Anne Elden would like to thank T. Kiran for useful reading and valuable suggestions of the manuscript.