Research on multifunctional systems has received increased attention in solid-state physics over the last decade, with a focus on nanosystems with improved storage capacities and ease of conveying information over long distances with less loss. The various concerns are related to both magnetism and ferroelectricity. This was made possible by the discovery of spin [1] [2] , which can be polarised by an external field, resulting in the formation of ME coupling. In recent years, multiferroic materials have gained attention due to their common ferric order (ferroelectric, ferromagnetic, ferroelastic, ferrotoroidic, and antiferromagnetic). These materials can switch polarization, magnetization, and strain between symmetric or opposite values under external fields or mechanical stresses [3] . That is, each field or mechanical stress is thermodynamically conjugated to ferroelectricity, ferromagnetism, and ferroelasticity, respectively [4] - [6] . Furthermore, multiferroic materials have been the subject of much more theoretical [7] [8] and experimental studies [9] - [12] in recent years, with a particular emphasis on how to handle an existing coupling between different ferric orders. Controlling ferroic order enables new technological applications in magnetic storage (stable memories), sensors [13] , spintronics [14] , and nanotechnologies [15] . The multiferroic materials with spin-1/2 chains model under the ME coupling have been theoretically investigated [16] [17] with spin-current mechanism in a model with electric ordering [18] .

Regardless of their origin, magnetism and ferroelectric order coexist in these materials at the same temperature [19] . The partially filled “d” or “f” shell of transition metals, where the interaction between localized moments exhibits magnetic properties, is typically responsible for the microscopic magnetism that exists, while the charge orderings, lone pair, etc., are responsible for the ferroelectric order, where the “d” shell is completely devoid of electrons in contrast to magnetism [20] [21] . Multiferroic materials exhibit remarkably strong couplings between charges, spins, crystal lattice, and orbital degrees of freedom, making them ideal candidates for developing novel devices for a variety of quantum technology applications. Many authors have demonstrated that in ME systems, two transitional phases occur at the same critical temperature, with the coupling between ferroelectricity and ferromagnetism being typically and strongly enhanced at the lower temperature phase transition [22] [23] .

Since the discovery of the giant ME coupling effect in TbMnO₃ (TMO), multiferroic manganite RMnO₃ (where R is a rare-earth element) that hosts an intimately coupled ferroelectrics (FE) and magnetic order has attracted significant attention in the development of new devices with multifunctionality [13] . One of the most common causes of multiferroicity is the inverse DM interaction between adjacent spins. Experimental and theoretical studies on multiferroic BiFeO₃ and RMnO₃ (R = Tb, Dy, and Y) compounds have included the Single Ion Anisotropy (SIA) and spin-orbit coupling interaction in the frustrated spin Hamiltonian [24] - [26] . Recently, research into the TbMnO₃ compound has revealed an intriguing helical magnetic structure for the most likely candidates to host ferroelectricity [26] in contrast to the LaMnO₃ properties, where the single-occupied eg orbitals with antiferro-type ordering imply a ferromagnetic nearest-neighbour (NN) interaction, in the TbMnO₃, a sizeable overlap of the eg orbitals of the next nearest-neighbour (NNN) sites along “b” appears from a large c-axis octahedron rotation [27] , giving rise to a strong NNN antiferromagnetic interaction [14] . The overlap of NNN’s, e.g., orbitals, which define ferromagnetic order well in the “ab” planes, is strongly frustrated, resulting in complex magnetic ordering. Due to the absence of conventional magnetic order [28] [29] , those compounds’ ferroelectricity below about 8K was their primary selling point. This phenomenon has been linked to the emergence of the inverse DM mechanism [18] [20] .

In Refs [26] [30] - [32] , the classical isotropic model with frustrated spin arrangement leading to a helical spin arrangement was investigated using numerical simulation and experiments that demonstrated the interplay of several interactions. In fact, a spin chain that includes an antiferromagnetic NNN interaction as well as a ferromagnetic exchange interaction between NN spins is frustrating [26] [33] . The NNN interaction frustration results in a helical spin structure with a pitch angle determined by the ratio ( $J$ and $J^{\prime }$ represents the NN and NNN exchange couplings, respectively). However, despite the fact that the symmetric and antisymmetric exchange couplings are rigorously determined experimentally or estimated from simulations using the lattice parameters, they may differ under certain conditions. Recent research has shown that rare-earth-metal (R) Mn perovskites, RMnO₃ (R = Tb, Dy, and Y), are well suited to elucidate magnetoelectric excitations because the polarization in the ferroelectric phase is significant and the magnetoelectric coupling is remarkably large [14] .

In this paper, we investigated the simultaneous action of DM and KSEA interactions on the thermodynamic properties and on the ME coupling of a proposed model of RMnO₃ (R = Tb) multiferroic with helical frustrated spin chains, where the NN interactions are ferromagnetic and the NNN antiferromagnetic and driven by an external electric field. The considered action is to toke to improve the behaviour of entropy, heat capacity and the ME coupling factor. The KSEA interactions which actions oppose the symmetric inversion caused by DM interactions are used to control the dynamics of transition phases induced by DM interactions. The observed contributions would be significant in the process of forming and controlling different phases in multiferroics, which is important in spintronics.

This section presents the theoretical approach to determine the thermodynamic properties and the ME coupling factor of the TbMnO₃ multiferroic system model, subjected to an external electric field constraint, while considering the simultaneous action of DM and KSEA interaction [32] [34] .

$H=H_{ex}+H_z+H_{SIA}+H_{Pol}+H_{DM}+H_{KSEA}$(1)

The above Hamiltonian is decomposed as follows:

$H_{ex}=\underset{i}{{\displaystyle \sum }}\left(J_x{S}_i^x{S}_{i+1}^x+J_y{S}_i^y{S}_{i+1}^y\right)+\underset{i}{{\displaystyle \sum }}\left({J^{\prime }}_x{S}_i^x{S}_{i+2}^x+{J^{\prime }}_y{S}_i^y{S}_{i+2}^y\right)$(2)

The system spin-exchanges term, where, $J_x$, ${J^{\prime }}_x$ and $J_y$, ${J^{\prime }}_y$ are the components of the coupling factors in the x- and y-direction respectively,

$H_Z=-\text{Ω}\underset{i}{{\displaystyle \sum }}{S}_i^z$(3)

the Zeeman energy with $\Omega =g{\mu }_{\text{B}}\lambda$ the transverse magnetic field in unit of energy (respectively with the $g$-factor $g=2$ in the present work, ${\mu }_{\text{B}}$ the Bohr magneton and $\lambda$ the magnetic field in Tesla),

$H_{SIA}=-\text{Δ}\underset{i}{{\displaystyle \sum }}{\left(S_i^z\right)}^2+K\underset{i}{{\displaystyle \sum }}\left({\left(S_i^x\right)}^2-{\left(S_i^y\right)}^2\right)$(4)

the contribution of the single-ion anisotropy in the three directions of space given by the wave function of the occupied orbital [23] . This term is governed by both magnetocrystalline constants $\text{Δ}$ and $K$ that render the magnetization along the z-direction.

$H_{POL}=-E\cdot P$(5)

is the interaction part of the external electric field $E\left(0,E,0\right)$ with the induced electric polarization $P={\pi }_{i,i+1}\underset{i}{{\displaystyle \sum }}{e}_x\times \left(S_i\times S_{i+1}\right)$ in the spin-chain. ${\pi }_{i,i+1}$ is the factor inducing the modulation of the polarization term which is calculated from the lattice parameter and expressed in unit energy [35] .

$H_{DM}=D\underset{i}{{\displaystyle \sum }}\left(S_i^y{S}_{i+1}^x-S_i^x{S}_{i+1}^y\right)$(6)

denotes the contribution of the DM interaction (with $D$ the DM parameter taken along the z-direction), due to the antisymmetric exchange interactions of spins in the plane. For large- $S$ limit, the DM favors non-collinear spin spirals and induces spin-current-driven polarization via inverse DM mechanism.

$H_{KSEA}={\text{Γ}}_z\underset{i}{{\displaystyle \sum }}\left(S_i^x{S}_{i+1}^y+S_i^y{S}_{i+1}^x\right)$(7)

is the KSEA interaction term, which in the semiclassical regime limit, adds symmetric anisotropy, modifying the spiral’s pitch and gap, and renormalizes spin-wave velocities and electromagnon energies. For the fact that the KSEA interaction does not have a fixed special point, but depends on the point of intersections in a certain numerical and relatively dense in value range, and sparse in other numerical value ranges, we consider ${\text{Γ}}_z$ such that the symmetric anisotropy KSEA interaction is taken along the z-direction. The sum is over the sites $i\in \left]0,N\right[$; $N$ being the number of spins per unit volume.

The beginning stage for computing the thermodynamic functions is the calculation of the energy spectrum of the proposed model. This can be accomplished from the spin-wave approximation method with the spin operators expressed in terms of Pauli matrices ${\tilde{S}}_i=\hslash 2{\tilde{\sigma }}_i$, by referring to the strongly reduced moment Mn revealed in the neutron-scattering experiment [31] . ${\sigma }_i^{\alpha }\left(\alpha =x,y,z\right)$ denotes the three Pauli matrices at site $i$. This leads to the dispersion relation as free fermions’ energy spectrum

${\omega }_q=\sqrt{{\left(\frac{J_x+J_y}{2}\cos \left(q\right)-\frac{{J^{\prime }}_x+{J^{\prime }}_y}{2}\cos \left(2q\right)+\left(D-{\pi }_{i,i+1}E\right)\sin \left(q\right)-\Omega \right)}^2+{\left({\Gamma }_z\sin \left(q\right)\right)}^2}$

In this paper, depending on the TbMnO₃ material, we deal with isotropic exchange interactions of NN ferromagnetic ( $J_x=J_y=-0.79\text{meV}$) and of NNN antiferromagnetic ( ${J^{\prime }}_x={J^{\prime }}_y=0.62\text{meV}$). The model Hamiltonian in Equation (1) is mapped in the spin-wave approximation to be compared to usual quantum gases which are conveniently treated using the canonical set [3] [36] . Since TbMnO₃ has non-collinear spin order and Mn³⁺ has $S=2\gg -1/2$, quantum fluctuations are weaker, making the spin-wave approximation is used for modelling its magnon spectrum.

$H=\underset{q}{{\displaystyle \sum }}{\omega }_q\left({\eta }_{-q}^{+}{\eta }_q-\frac{1}{2}\right)-\frac{N}{2}\left(\frac{\Delta }{2}+\Omega \right)$(8)

where $\eta$ are Bogoliubov quasiparticles expressed in terms of Fourier transform operator $c_q={\sigma }_i^{-}\left[{\displaystyle {\prod }_{j\le i-1}\left(-{\sigma }_j^z\right)}\right]$ and wave numbers taking in the range $\pi \le q\le -\pi$.

Within the Fermi-Dirac statistical theory, the thermodynamic parameters and the magnetoelectric coupling factor are calculated in the spin-wave approximation using the partition function associated with Hamiltonian Equation (8) given by:

$\begin{array}{c}Z=Tr\exp \left\{-H\cdot \Theta \right\}\\ =\exp \left\{\frac{N\Theta }{2}\left(\frac{\Delta }{2}+\Omega \right)\right\}\underset{q}{{\displaystyle \prod }}\exp \left(\frac{\Theta \cdot {\omega }_q}{2}\right)\underset{q}{{\displaystyle \prod }}\left(1+\exp \left(-\Theta \cdot {\omega }_q\right)\right)\end{array}$(9)

from which the free energy is defined as

$F=-T\ln Z=-T\left(\frac{N\Theta }{2}\left(\frac{\Delta }{2}+\Omega \right)-\frac{\Theta }{2}\underset{q}{{\displaystyle \sum }}{\omega }_q\right)-\frac{1}{\Theta }\underset{q}{{\displaystyle \sum }}\ln \left(1+\exp \left(-\Theta \cdot {\omega }_q\right)\right)$(10)

where $T$ is the absolute temperature, $K_B$ the Boltzman constant and $\text{Θ}=1/K_{B}T$.

From Equation (10), the thermodynamic parameters and the magnetoelectric coupling factor can be evaluated.

This section evaluates the Boltzmann entropy, specific heat capacity, and ME coupling factor in order to demonstrate the influence of the simultaneous KSEA and DM interactions on the thermodynamic properties and on the metamagnetoelectric transition of the system under study. The thermodynamic properties of quantum materials can be found in the canonical distribution using the statistical sum Equation (9) to highlight the influence of the simultaneous effect of DM and KSEA on a helical spin chain system subjected to an external electric field constraint made from TbMnO₃(TMO) multiferroic compound.

The new challenge in solid-state physics is to find new multifunctional materials with a strong ME coupling factor. Following recent research in solid-state materials, multiferroic materials with linear ME coupling have been found to be a candidate of choice [4] [37] . The linear ME coupling in class II multiferroics system is obtained through magnetization or polarization and particularly exhibits the same response of the system under electric or magnetic field excitation with an intriguing survival of the ferroelectric order when the ferromagnetism is dropped, allowing the ferric order separation in such a magnetoelectric system. The significant insight is that, thermodynamically, the coexistence of the magnetic-electric phase in a ferroelectric material with SIA and DM interaction induces a magnetoelectric effect [3] [23] [37] . Why should the contribution of KSEA interaction be juxtaposed to that of DM? The answer to this question is related to the same method used in [36] we also got the same result in Equation (13), plotted respectively in Figures 11-13 ,

$\begin{array}{l}\alpha =\frac{\partial P}{\partial \Omega }=\frac{\partial M}{\partial E}\\ ={\pi }_{I,I+1}\Omega /\lambda \underset{q}{{\displaystyle \sum }}\left(\frac{\sin q}{{\omega }_q}-\frac{{\left(J\cos q-J^{\prime }\cos 2q+\left(D-{\pi }_{I,I+1}E\right)\sin q-\Omega \right)}^2\cdot \sin q}{{\omega }_q^3}\right)\left(\frac{1}{2}-\frac{1}{1+\exp \left(\Theta \cdot {\omega }_q\right)}\right)\\ +\frac{{\pi }_{I,I+1}\Omega \cdot \Theta }{\lambda }\underset{q}{{\displaystyle \sum }}\left(\frac{{\left(J\cos q-J^{\prime }\cos 2q+\left(D-{\pi }_{I,I+1}E\right)\sin q-\Omega \right)}^2\cdot \sin q}{{\omega }_q^2}\left(\frac{\exp \left(\Theta \cdot {\omega }_q\right)}{{\left(1+\exp \left(\Theta \cdot {\omega }_q\right)\right)}^2}\right)\right)\end{array}$ (13)

From the various graphs obtained, we obtain a definite improvement in the magnitude of the ME coupling factor under the simultaneous effect of the two interactions (DM and KSEA). This gives rise to the control of metamagnetic-electric transitions, particularly at low temperatures. Recently, some experimental prospects for the hybrid channel with thermal, magnetic, and classical dephasing parts controlled by static noise, which is used to study the dynamics of the two-qubit Heisenberg spin state as defined by various parameters such as spin-spin, DM, and KSEA interactions have been investigated [38] . Given that their model is fairly similar to ours and that it is experimentally feasible, one can presume that the model and its predictive capabilities are feasible.

The purpose of this study was to investigate the simultaneous action of DM and KSEA interactions used at thermal equilibrium to control RMnO₃ (TbMnO₃) multiferroic compounds. Using a helical frustrated spin chain under the constraint of an external field, a theoretical investigation of ME coupling factor and thermodynamic properties was performed. It emerges from field scans of entropy, heat capacity, and ME coupling a set of interdependent nonlinearities, resulting in metatransitions revealed by a breakdown in the interdependence of magnetic and electric order. The time required for successive and reversed transitions at the expense of an applied external field is maintained by system expenditure in DM and KSEA interactions. This gives rise to manipulating, according to interest, the ME coupling, which is essential for the design of materials with desired functionalities. The magnitude of the thermodynamic properties (entropy and specific heat) decreases, whereas one of the ME coupling factors increases as the applied electric field strength decreases and vice versa. The magnitude change is maintained by the simultaneous effect of DM and KSEA interactions, which favours ME coupling control in TbMnO₃ for designing next-generation spintronics materials.