Magnon-lattice propagation in a Morse chain: the role played by the spin-lattice interaction and the initial condition

Abstract

Our research focuses on studying magnon dynamics in a Morse lattice. We used a Heisenberg Hamiltonian to represent the spins while a Morse formalism governed the lattice deformations. The strength of the spin-spin interaction depended on the distance between neighboring spins, which followed an exponential pattern. We explored various initial conditions for the lattice and spin wave function and observed how they affected the magnon-lattice propagation. Additionally, we analyzed the impact of the parameter that controlled the difference in time scales between spin and lattice deformation propagation.

Key words
localization; disorder; Anderson localization; nonlinear lattice

Introduction

The magnon dynamics under the effect of magnetoelastic coupling has attracted a high interest Masciocchi et al. (2022), Chen et al. (2023), Luo et al. (2023), Gries et al. (2022), Cong et al. (2022), Challali et al. (2023), Sun et al. (2022), Holanda et al. (2018), Xiong et al. (2017), Li et al. (2020), Hayashi & Ando (2018), Weiler et al. (2012), Zhang et al. (2019, 2020), Sasaki et al. (2021), Mingran et al. (2020), Morais et al. (2021), Sales et al. (2018, 2023). In Chen et al. (2023), the authors investigate the magnon propagation using a Boltzmann method framework which includes magnon-phonon interaction and diverse scattering terms. They solved the effective equations and detailed some abnormal phenomena observed in several experiments. Zhang et al. (2019) demonstrates that the magnon-phonon coupling controls the thermal Hall effect on a ferromagnetic square lattice featuring Dzyaloshinskii-Moriya interaction. Li et al. (2020) reported the existence of collective antiferromagnetic magnon-phonon pair formation in an insulator $C{r}_2{O}_3$. The results in ref. Hayashi & Ando (2018) indicate that the magnon-phonon coupling could amplify spin pumping in a Pt/YIG bi-layer film. Mingran et al. (2020) reported the first observation of the magneto-rotation coupling in a perpendicularly anisotropic magnetic film. They also introduce the theoretical background. Sales et al. (2018) investigated the magnon propagation in a Fermi-Pasta-Ulam. They studied the spin dynamics using a quantum Heisenberg Hamiltonian with the ferromagnetic ground state. The magnon-lattice interaction was introduced by considering the spin-spin interaction terms as a function of the distance between the spins. Solving the dynamics equations, they demonstrated the existence of a magnon-soliton mode. Morais et al. (2021) investigate the propagation of magnon states coupled to the harmonic modes of a linear lattice. It was considered an adiabatic approximation to deduce an effective quantum equation to describe the magnon dynamics. The authors demonstrate the existence of a self-trapping transition to the magnon state. Sales et al. (2018), the dynamics of magnon-lattice it was considered by writing the Heisenberg Hamiltonian in a nonlinear Morse chain. The authors demonstrate that the lattice deformation embodies a finite fraction of the spin wave function in the robust spin-lattice coupling regime, generating a mobile magnon-lattice excitation.

In this work, we revisit the problem of magnon dynamics in a Morse lattice. We designed the spin-spin coupling using a Heisenberg model. The intensity of the spin-spin interaction fits an exponential dependent on the distance between nearest-neighbor spins. This framework provides an effective spin-lattice interaction, and a single tunable parameter controls the intensity of this interaction. We will investigate the magnon-lattice dynamics considering a wide range of initial conditions. Our calculations indicate that the magnon-lattice pair formation strongly depends on the initial condition’s width. We will also investigate the dependence of the magnon-lattice propagations as a function of the time-scales difference between spin and lattice propagation. Our results suggest that magnon-lattice pair formation occurs for a small amount of magnon-lattice interaction as the time scale difference increases.

Model

Our model is a quantum one-dimensional Heisenberg model with $N$ spin $1/2$ on a nonlinear Morse chain. The spin-spin interaction is strongly dependent on the distance between nearest-neighbor spins. The complete quantum Hamiltonian is given by Evangelou & Katsanos (1992), de Moura et al. (2002):

$${\mathcal{H}}_{\mathcal{S}}=-\sum _{y=1}^N\{J_{y,y+1}{\overrightarrow{S}}_y{\overrightarrow{S}}_{y+1}\}.$$ (1)

The interaction between spins $y$ and $y+1$ is given by $J_{y,y+1}=J{e}^{-\alpha (X_{y+1}-X_y)}$. $X_y$ is the displacement of spin $y$ from it equilibrium position. We emphasize again that we are dealing with a one-dimensional geometry. Therefore, without lattice vibrations, all spins are in equally spaced positions (in equilibrium, the distance between nearest-neighbor spins is the lattice spacing, an adimensional parameter $l_s=1$). However, our model will consider that the spins can move around their equilibrium position. The spatial movement of the spins produces variations in the value of the spin-spin interaction. Our model will consider that these variations follow this exponential dependence shown earlier. The parameter $\alpha$ characterizes this exponential dependence within this formalism, thus controlling the effective spin-lattice interaction. The lattice dynamics here will be governed by a Morse potential represented by the classical Hamiltonian Hennig et al. (2007), Ikeda et al. (2007), de Lima & de Cavalho (2012), Carrillo et al. (2013):

$$\begin{array}{rl}{H}_L& =\sum _{y=1}^N\frac{P_y^2}{2}\\ & +\sum _{y=1}^N\{1-exp[-(X_y-X_{y-1})]{\}}^2.\end{array}$$ (2)

Here, $P_y$ represents the particle moment at site $y$. We emphasize that we are using the dimensionless representation considered in ref. Hennig et al. (2007). The time is scaled as $t\to \omega t$, with $\omega$ representing the frequency of oscillations around the minimum of the Morse potential. The energy scale is measured in units of the depth of the Morse potential Hennig et al. (2007). The magnon dynamics is represented by the time-dependent Schrödinger equation for ($\hslash =1$) defined as Sales et al. (2018):

$$\begin{array}{rl}i\frac{d{u}_y(t)}{dt}& =\frac{\tau }{2}\{[exp(-\alpha (X_{y+1}-X_y))\\ & +exp(-\alpha (X_y-X_{y-1}))]u_y(t)\\ & -exp(-\alpha (X_y-X_{y-1}))u_{y-1}(t)\\ & -exp(-\alpha (X_{y+1}-X_y))u_{y+1}(t)\}.\end{array}$$ (3)

To clarify, we want to point out that the previous equations were written considering a ferromagnetic ground state denoted as $|0⟩$ and a set of kets represented by $|y⟩=S_y^{+}|0⟩$. Therefore, the $u_y\left(t\right)$ value corresponds to the wave function amplitude associated with the spin deviation at position $y$. By utilizing the Hamilton formalism, we have derived the equations governing the dynamics of the lattice:

$$\begin{array}{rl}\frac{d^2{X}_y}{d{t}^2}& =\{1-exp[-(X_{y+1}-X_y)]\}exp[-(X_{y+1}-X_y)]\\ & -\{1-exp[-(X_y-X_{y-1})]\}exp[-(X_y-X_{y-1})]\\ & +\frac{J\alpha }{2}[e^{-\alpha (X_y-X_{y-1})}(u_y^{\ast }{u}_y+u_{y-1}^{\ast }{u}_{y-1})\\ & -e^{-\alpha (X_{y+1}-X_y)}(u_y^{\ast }{u}_y+u_{y+1}^{\ast }{u}_{y+1})\\ & +e^{-\alpha (X_{y+1}-X_y)}(u_{y+1}^{\ast }{u}_y+u_y^{\ast }{u}_{y+1})\\ & amp;-& amp;e^{-\alpha (X_y-X_{y-1})}(u_{y-1}^{\ast }{u}_y+u_y^{\ast }{u}_{y-1})].\end{array}$$ (4)

It’s important to note that we changed the time scale in the previous equation by rescaling $t$ to $\omega t$, with $\omega$ rrepresenting the frequency of oscillations around the minimum of the Morse potential Hennig et al. (2007). This step is necessary to account for the difference in timescale between electron dynamics (which is faster) and lattice vibrations (which is slower) Hennig et al. (2007), Davydov (1991), Scott (1992). To put it simply, this framework involves a factor $\tau =J/\left(\hslash \omega \right)$ that multiplies the spin equation Hennig et al. (2007), Davydov (1991), Scott (1992). In our work, we will use$J=0.1$, which is in alignment with previous research Hennig et al. (2007), Davydov (1991), Scott (1992), Korotin et al. (2015), Satija et al. (1980), Hutchings et al. (1979), Kadota et al. (1967). The value of 𝜏 will be adjustable, but in previous works, it was typically chosen to be around 10 Hennig et al. (2007), Ranciaro-Neto & de Moura (2016), Sales et al. (2018) due to potential differences in time scales between quantum and classical propagation. However, we will explore the effects of varying 𝜏 around this value. Our initial conditions will be $u_y(t=0)=A{e}^{-{(y-N/2)}^2/\left(4{\sigma }_S^2\right)}$, $X_y(t=0)=0$, and $P_y(t=0)=e^{-{(y-N/2)}^2/\left(4{\sigma }_L^2\right)}$, with $A$ as a normalization constant. We will use a Taylor procedure de Moura (2011) to solve the set of equations 3, and a standard second-order Verlet’s like procedure Allen & Tildesley (1987), da Silva et al. (2019) to solve the lattice dynamics. Our analysis will focus on magnon propagation and lattice deformation dynamics along the chain, which can be observed using the quantity $n_S$ defined as Sales et al. (2018):

$$\begin{array}{c}\hfill n_S=\sum _y(y-N/2){\left|u_y\left(t\right)\right|}^2.\end{array}$$

The lattice properties can be analyzed using the mean position of the lattice deformation defined as:

$$\begin{array}{c}\hfill n_L=\frac{\sum _y(y-N/2){[1-e^{-(X_y-X_{y-1})}]}^2}{\sum _y{[1-e^{-(X_y-X_{y-1})}]}^2}.\end{array}$$

We want to emphasize that $n_S$ and $n_L$ represent the mean position of the spin-wave excitation and the lattice deformation, respectively. These measurements generally are in units of lattice spacing ($l_s=1$). Using these quantities, we can obtain the magnon and the lattice deformation velocities $V_S$ and $V_L$ using fittings of the curves $n_S\times t$ and $n_L\times t$. We stress that here we will use a methodology similar to that was used in the previous literature . We will follow the propagation of the magnon and the lattice deformation to describe the existence (or not) of magnon-lattice coupled movement. Generally, stable dynamics with $n_S\approx n_L$ and $V_S\approx V_L$ indicate the presence of magnon-lattice pair formation. The nonlinear Morse chain considered here contains a solitonic mode propagation along the chain. We can see this solitonic mode by calculating the lattice deformation $Z_y$; this quantity represents a generalized probability that deformation around site $y$ occurs. This is obtained by normalizing $B_y={(1-e^{[-X_y+X_{y-1}]})}^2$, that is $Z_y=B_y/{\sum }_y\left(B_y\right)$. We will plot $Z_n\times t\times n$ where $n=y-N/2$ (i.e., $n=0$ represents the center of the chain). In fig. 1 we plot our results for $\alpha =0,1,2,3$, $v_0=1$, ${\sigma }_S=0.5$, ${\sigma }_L=0.5$ and $\tau =10$. We can see that independent of the value of $\alpha$, the lattice deformation exhibits a stable solitonic mode propagating along the chain. Therefore, the main focus of our work is investigating the existence of a possible magnon-soliton pair formation and its dependence on all tunable parameters.

Figure 1
The lattice deformation $Z_n$ versus $t$ and $n$ where $n=y-N/2=0$ represents the center of the chain. Calculations were done for $v_0=1$, ${\sigma }_S=0.5$, ${\sigma }_L=0.5$, $\tau =10$ and $\alpha =0,1,2,3$

Results and Discussion

Our findings on the velocities $V_S$ and $V_L$ in relation to $\alpha$ are presented below. We obtained $V_S$ and $V_L$ through the linear fitting of the curves $n_S\times t$ and $n_L\times t$. Our calculations of $n_S$ and $n_L$ suggest that both quantities exhibit long-term linear behavior, consistent with the solitonic dynamics found in references . We performed these calculations using a time limit of $t_{max}\approx {10}^4$. The linear fitting was conducted using the last 20% of the complete time interval, roughly within the time interval $[8000,10000]$. We used a Taylor expansion up to the tenth order to solve the quantum equations and a second-order Verlet-like method to solve the classical equations. We performed our numerical procedure using a time step of $\Delta t=0.001$. It is important to emphasize that this method is faster than the Runge-Kutta formalism for this type of problem. The initial condition was given by : $u_y(t=0)=A{e}^{-{(y-N/2)}^2/\left(4{\sigma }_S^2\right)}$, $X_y(t=0)=0$ and $P_y(t=0)=v_0{e}^{-{(y-N/2)}^2/\left(4{\sigma }_L^2\right)}$. Here, $A$ is a normalization constant, $v_0$ is a tunable parameter, and the ${\sigma }_L$ and ${\sigma }_S$ are larger than zero. We varied the parameter $\tau$ within the interval $[1,15]$. We considering initially $v_0=1$, ${\sigma }_S=0.5$, ${\sigma }_L=0.5$ and several values of $\tau$. We show our results in figs. 2(a-d). We emphasize that the curves indicate the velocities $V_S$ (black solid line) and $V_L$ (red dotted line) versus $\alpha$ for several values of $\tau$. To construct these curves, we calculate the dynamics of the spin and the lattice for long times for several values of $\alpha$ and $\tau$. We calculate the $V_S$ and $V_L$ curves versus $\alpha$ using a linear fitting. We can see that $V_L$ is roughly independent of $\alpha$. On another side, spin propagation strongly depends on the spin-lattice interaction parameter $\alpha$. Let us clarify this important matter in simpler terms. The lattice’s deformation is governed by eq. (4). We can see that when $\alpha$ is small, the nonlinear Morse terms, i.e., the first two terms, dominate over the terms that depend directly on $\alpha$ and the wave functions. Therefore, the soliton velocity remains roughly constant; however, when $\alpha$ increases, the final terms become more significant and have a greater impact on the soliton propagation, causing a slight increase in velocity. On a different note, the behavior of spin dynamics is dictated by equation (3), which shows a significant dependence on the magnitude of $\alpha$ in both the diagonal (first term) and the off-diagonal (last two terms). As such, it was indeed expected that the value of $\alpha$ would influence the magnon’s velocity. By analyzing all curves for several of $\tau$ we have considered, there is a matching of the magnon’s and lattice’s velocity ($V_S\approx V_L$) for a specific value of $\alpha$. This result suggests that for this particular value of $\alpha$, the magnon and the lattice deformation travel at the same velocity. We stress that it is the first indication that magnon and lattice may move in a kind of "correlated propagation" (like a magnon-lattice pair formation). We can also see that as the parameter $\tau$ has increased, this value of $\alpha$ in which the velocities are the same become smaller. To comprehend this phenomenon, we need to emphasize that when $\tau$ increases, the off-diagonal terms in the Schrödinger equation become more effective. This results in a stronger coupling with the lattice deformation even for smaller values of $\alpha$.

Figure 2
Magnon and lattice deformation velocities [$V_S$(black solid line) and $V_L$(red dotted line)] versus $\alpha$. We have considered $v_0=1$, ${\sigma }_S=0.5$, ${\sigma }_L=0.5$ and $\tau =2.5$ up to $15$.

We also calculate the long-time mean distance between the magnon and the lattice deformation. The distance is defined as $D=|n_L(t\to \infty )-n_S(t\to \infty )|$. We emphasize that $D$ represents also an measurement of the possible existence of the magnon-soliton pair state. In general, bound states exhibit a smaller value of intrinsic internal distances. used this kind of measure to detect the existence of electron-electron bound states in the low-dimensional two-electron Hubbard model. We emphasize that we will plot (see figs. 3(a-d)) $D/D_{max}$ versus $\alpha$ where $D_{max}$ represents the maximum of the distance between the magnon and the lattice position. We can observe that for the same value of $\alpha$ in which that $V_L\approx V_S$, we can see that $D/D_{max}\approx 0$, i.e., the magnon and the lattice position are close, thus suggesting the existence of magnon-lattice pair formation. We can see that the critical value of $\alpha$ in which $D/D_{max}\approx 0$ is in good agreement with the critical value found using the velocity curves versus $\alpha$ (see fig. ). Therefore all measures of $D/D_{max}$, $V_S$, and $V_L$ are topological quantities that characterize the propagation of the magnon and the lattice deformation. Our calculations numerically demonstrate that for some specific values of $\alpha ={\alpha }_c$, the distance D is small, and the magnon and the lattice deformation travel at the same velocity. This result strongly indicates a magnon-soliton pair formation for these special situations.

Figure 3
The normalized distante between the Magnon and the lattice deformation ($D/D_{max}$) versus $\alpha$. Calculations done for $v_0=1$, ${\sigma }_S=0.5$, ${\sigma }_L=0.5$ and $\tau =2.5$ up to $15$.

In fig. 4, we collect the critical value of $\alpha$ versus $\tau$. We stress that for $\alpha ={\alpha }_c$ the system exhibits a magnon-lattice pair formation, i.e., the magnetic excitation moves along with the lattice vibration and at the same velocity. We emphasize again that the decreasing of ${\alpha }_c$ with $\tau$ is a direct consequence of the role played by $\tau$ at the off-diagonal terms at eq. (3). As the $\tau$ is increased, the effective off-diagonal term also increases. Increasing the effective spin-spin interaction makes coupling between the spin and the lattice deformations easier. In figures 5 and 6 we consider again $v_0=1$ and change the values of ${\sigma }_S$ and ${\sigma }_L$ respectively to $0.5$ and $1$; we kept the same range of values of $\tau$. We can observe that the results are qualitatively the same obtained in figs. and i.e.: as the value of $\tau$ is increased, the magnon-lattice pair formation is obtained for a specific value of $\alpha ={\alpha }_c$. We also obtained that as $\tau$ is increased ${\alpha }_c$ decreases (see fig. 7). In figures 8 and 9, we show our results considering ${\sigma }_S=1$ and ${\sigma }_L=1$, and we kept the same range of values of $\tau$ and $v_0$. The results obtained are similar to those in the previous figures. It appears that the magnon-lattice pair only exists when $\alpha$ equals the critical value ${\alpha }_c$. This critical value decreases as $\tau$ increases, as shown in figure 10

Figure 4
The critical value ${\alpha }_c$ versus $\tau$ for ${\sigma }_S={\sigma }_L=0.5$.

Figure 5
Magnon and lattice deformation velocities [$V_S$(black solid line) and $V_L$(red dotted line)] versus $\alpha$ computed using $v_0=1$, ${\sigma }_S=0.5$, ${\sigma }_L=1$ and $\tau =2.5$ up to $15$.

Figure 6
The normalized distante between the Magnon and the lattice deformation ($D/D_{max}$) versus $\alpha$ computed for $v_0=1$, ${\sigma }_S=0.5$, ${\sigma }_L=1$ and $\tau =2.5$ up to $15$.

Figure 7
The critical value ${\alpha }_c$ versus $\tau$ for ${\sigma }_S=0.5$ and ${\sigma }_L=1$.

Figure 8
The velocities [$V_S$(black solid line) and $V_L$(red dotted line)] versus $\alpha$ computed using $v_0=1$, ${\sigma }_S=1$, ${\sigma }_L=1$ and $\tau =2.5$ up to $15$.

Figure 9
The normalized distante between the Magnon and the lattice deformation ($D/D_{max}$) versus $\alpha$ computed for $v_0=1$, ${\sigma }_S=1$, ${\sigma }_L=1$ and $\tau =2.5$ up to $15$.

Figure 10
The critical value ${\alpha }_c$ versus $\tau$ for ${\sigma }_S={\sigma }_L=1$.

To summarize the previous results, an initial vibrational Gaussian velocity pulse was introduced into the lattice, and a finite amount of the initial energy propagated along the lattice through the nonlinear solitonic mode. This behavior was observed by tracking the lattice position over time, with our calculations indicating that a finite fraction of the initial energy remained trapped in a finite region of the lattice. This localized pulse could travel along the lattice with a constant velocity of $V_L$. Additionally, the quantum equation was initialized using a Gaussian initial magnon wave packet, and our calculations showed that the dominant wave packet exhibited a solitonic profile with a position given by $n_S\left(t\right)\approx V_{S}t$. By computing $V_S$ and $V_L$, we numerically demonstrated that depending on the value of magnon-lattice coupling, we could obtain a good indication of magnon-soliton pair formation. For certain values of $\alpha ={\alpha }_c$, our calculations indicated that $n_S\approx n_L$ and $V_S\approx V_L$. In ref. , the possibility of magnon-soliton propagation in nonlinear lattices was demonstrated considering localized initial states (i.e., ${\sigma }_L={\sigma }_S=0$). However, we have discovered that broad initial conditions can also lead to magnon-soliton propagation. The critical value for the occurrence of magnon-solitons depends on the width of the initial conditions and the value of $\tau$. The parameter $\tau$ measures the time scale difference between the magnon and lattice deformation and acts as the intensity of the effective spin-spin interaction within the quantum equations. This spin-spin interaction is also the key to the magnon-lattice interaction. As $\tau$ increases, the spin-lattice terms become stronger, making it easier to promote magnon-soliton pair formation.

Before we finish our work, we need to examine how our results vary with the value of $v_0$. Specifically, we want to see how the formation of magnon-soliton pairs is affected by the strength of the initial impulse. To do this, we conducted many numerical experiments with different values of $v_0$, which allowed us to observe the signatures of magnon-soliton pairs. We discovered that within the range of $v_0$ values between 1 and 10, a magnon-soliton pair exists when $\alpha$ equals a certain value, denoted as ${\alpha }_c$. However, this critical value depends on the value of $v_0$. For instance, the results we obtained for ${\sigma }_S={\sigma }_L=1$ and $\tau =10$ (refer to fig. 11(a)) revealed that when $v_0$ is low (less than 3), the critical value remains stable at around ${\alpha }_c=1.205$, while for $v_0$ values greater than 3, the critical value decreases by approximately half. This decrease in ${\alpha }_c$ with increasing $v_0$ may seem counterintuitive, but we believe it is mainly due to the intensity of the solitonic mode. As the initial impulse grows, the soliton gains more intensity, resulting in an increased spin-lattice interaction that favors the magnon-soliton pairing. Our calculations show that regardless of the values of ${\sigma }_L$ and ${\sigma }_S$, the results remain qualitatively the same. The velocity of the magnon-soliton pair ($V_{SL}$) is also dependent on the initial velocity $v_0$. In fig. 11(b), we observe that for small $v_0$, $V_{SL}\propto v_0$. For $v_0>3$, $V_{SL}\propto v_0^{0.55\left(5\right)}$. It is essential to note that the results shown in fig. 11 do not depend on the values of ${\sigma }_L$, ${\sigma }_S$, and $\tau$. Generally, as velocity increases, the solitonic mode becomes faster, resulting in a faster magnon-soliton pair.

Figure 11
a) The critical value ${\alpha }_c$ versus $v_0$;b) The magnon-soliton velocity $V_{SL}$ versus $v_0$. Calculations were done for ${\sigma }_L={\sigma }_S=1$ and $\tau =10$.

Summary and conclusions

Our research delves into the behavior of a single magnon state in a nonlinear Morse chain, considering the magnon-lattice coupling through the Heisenberg spin-spin term that directly depends on the spin positions. We begin with a Gaussian wave packet for the magnon state and a Gaussian impulse packet for the lattice. The velocity intensity and width of these initial Gaussian pulses are adjustable parameters in our model. We also vary the time scales between the magnon and lattice dynamics. We provide a detailed numerical analysis of how magnon-soliton pairs propagate and their dependence on these parameters. Our findings reveal that magnon-soliton propagations are attainable for specific values of the magnon-lattice interaction (called ${\alpha }_c$ in our model) and that this critical value is highly reliant on the width of the initial Gaussian pulses. Our numerical calculations indicate that increasing the velocity of the initial Gaussian pulse decreases the critical value of spin-lattice interaction (${\alpha }_c$). Furthermore, as the time difference between the magnon and lattice dynamics increases, the intensity of magnon-lattice coupling needed to promote pair formations decreases. Overall, our study underscores the importance of the initial conditions and the specifics of the magnon/lattice dynamics in the existence of magnon-soliton pairs in nonlinear chains. We demonstrate that a time difference of $\tau \ge 10$ yields a more reliable existence of magnon-lattice coupling, consistent with previous research. Our work is intended to inspire further research in this area.

ACKNOWLEDGMENTS

This work was partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) and Financiadora de Estudos e Projetos (FINEP), CNPq-Rede Nanobioestruturas, as well as FAPEAL (Alagoas State Agency).

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Publication Dates

• Publication in this collection
  22 Dec 2023
• Date of issue
  2023

History

• Received
  12 Apr 2023
• Accepted
  31 Aug 2023