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)MASCIOCCHI G ET AL. 2022. Control of magnetoelastic coupling in Ni/Fe multilayers using He+ ion irradiation. Appl Phys Lett 121: 182401., Chen et al. (2023)CHEN C, LI Y & ZHANG J 2023. Characteristics of magnon-phonon coupling in magnetic insulator based on the Boltzmann equation. AIP Advances 13: 025221., Luo et al. (2023)LUO J ET AL. 2023. Evidence for Topological Magnon-Phonon Hybridization in a 2D Antiferromagnet down to the Monolayer Limit. Nano Lett 23: 2023-2030., Gries et al. (2022)GRIES L, JONAK M, ELGHANDOUR A, DEY K & KLINGELER R 2022. Role of magnetoelastic coupling and magnetic anisotropy in MnTiO3. Phys Rev B 106: 174425., Cong et al. (2022)CONG A, LIU J, XUE W, LIU H, LIU Y & SHEN K 2022. Exchange-mediated magnon-phonon scattering in monolayer CrI3. Phys Rev B 106: 214-424., Challali et al. (2023)CHALLALI R, SAIT S, BOURAHLA B & FERRAH L 2023. Localized Surface Magnon Modes in Cubic Ferromagnetic Lattices. SPIN 13: 2350001., Sun et al. (2022)SUN Y-J, LAI J-M, PANG S-M, LIU X-L, TAN P-H & ZHANG J 2022. Magneto-Raman Study of Magnon?Phonon Coupling in Two-Dimensional Ising Antiferromagnetic FePS3. Phys Chem Lett 13: 1533-1539., Holanda et al. (2018)HOLANDA J, MAIOR DS & AZEVEDO A 2018. Detecting the phonon spin in magnon-Phonon conversion experiments. Nature Phys 14: 500-506., Xiong et al. (2017)XIONG Z, DATTA T, STIWINTER K & YAO D-X 2017. Magnon-phonon coupling effects on the indirect K-edge resonant inelastic x-ray scattering spectrum of a two-dimensional Heisenberg antiferromagnet. Phys Rev B 96: 144436., Li et al. (2020)LI J, SIMENSEN HT, REITZ D, SUN Q, YUAN W, LI C, TSERKOVNYAK Y, BRATAAS A & SHI J. 2020. Observation of Magnon Polarons in a Uniaxial Antiferromagnetic Insulator. Phys Rev Lett 125: 217201., Hayashi & Ando (2018)HAYASHI H & ANDO K 2018. Spin Pumping Driven by Magnon Polarons. Phys Rev Lett 121: 237202., Weiler et al. (2012)WEILER M, HUEBL H, GOERG FS, CZESCHKA FD, GROSS R & GOENNENWEIN STB. 2012. Spin Pumping with Coherent Elastic Waves. Phys Rev Lett 108: 176601., Zhang et al. (2019, 2020), Sasaki et al. (2021)SASAKI R, NII Y & ONOSE Y. 2021. Magnetization control by angular momentum transfer from surface acoustic wave to ferromagnetic spin moments. Nat Commun 12: 2599., Mingran et al. (2020)MINGRAN X, KEI Y, JORGE P, KORBINIAN B, BIVAS R, KATSUYA M, HIROMASA T, DIRK G, SADAMICHI M & YOSHICHIKA O. 2020. Nonreciprocal surface acoustic wave propagation via magneto-rotation coupling. Science Advances 6: 32., Morais et al. (2021)MORAIS D, DE MOURA FABF & DIAS WS. 2021. Magnon-polaron formation in XXZ quantum Heisenberg chains. Phys Rev B 103: 195445., Sales et al. (2018, 2023). In Chen et al. (2023)CHEN C, LI Y & ZHANG J 2023. Characteristics of magnon-phonon coupling in magnetic insulator based on the Boltzmann equation. AIP Advances 13: 025221., 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)ZHANG X, ZHANG Y, OKAMOTO S & XIAO D. 2019. Thermal Hall Effect Induced by Magnon-Phonon Interactions. Phys Rev Lett 123: 167202; demonstrates that the magnon-phonon coupling controls the thermal Hall effect on a ferromagnetic square lattice featuring Dzyaloshinskii-Moriya interaction. Li et al. (2020)LI J, SIMENSEN HT, REITZ D, SUN Q, YUAN W, LI C, TSERKOVNYAK Y, BRATAAS A & SHI J. 2020. Observation of Magnon Polarons in a Uniaxial Antiferromagnetic Insulator. Phys Rev Lett 125: 217201. reported the existence of collective antiferromagnetic magnon-phonon pair formation in an insulator . The results in ref. Hayashi & Ando (2018)HAYASHI H & ANDO K 2018. Spin Pumping Driven by Magnon Polarons. Phys Rev Lett 121: 237202. indicate that the magnon-phonon coupling could amplify spin pumping in a Pt/YIG bi-layer film. Mingran et al. (2020)MINGRAN X, KEI Y, JORGE P, KORBINIAN B, BIVAS R, KATSUYA M, HIROMASA T, DIRK G, SADAMICHI M & YOSHICHIKA O. 2020. Nonreciprocal surface acoustic wave propagation via magneto-rotation coupling. Science Advances 6: 32. 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)SALES MO, NETO AR & DE MOURA FABF. 2018. Spin-wave dynamics in nonlinear chains with spin-lattice interactions. Phys Rev E 98: 062136 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)MORAIS D, DE MOURA FABF & DIAS WS. 2021. Magnon-polaron formation in XXZ quantum Heisenberg chains. Phys Rev B 103: 195445. 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)SALES MO, NETO AR & DE MOURA FABF. 2018. Spin-wave dynamics in nonlinear chains with spin-lattice interactions. Phys Rev E 98: 062136, 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 spin 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)EVANGELOU SN & KATSANOS DE 1992. Super-diffusion in random chains with correlated disorder. Phys Lett A 164: 456-464., de Moura et al. (2002)DE MOURA FABF, COUTINHO-FILHO MD, RAPOSO EP & LYRA ML 2002. Delocalization and spin-wave dynamics in ferromagnetic chains with long-range correlated random exchange. Phys Rev B 66: 014-418.:
The interaction between spins and is given by . is the displacement of spin 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 ). 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 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)HENNIG D, CHETVERIKOV A, VELARDE MG & EBELING W 2007. Electron capture and transport mediated by lattice solitons. Phys Rev E 76: 046602., Ikeda et al. (2007)IKEDA K, DOI Y, FENG BF & KAWAHARA T 2007. Chaotic breathers of two types in a two-dimensional Morse lattice with an on-site harmonic potential. Physica D: Nonlinear Phenomena 225: 184-196., de Lima & de Cavalho (2012), Carrillo et al. (2013)CARRILLO JA, MARTIN S & PANFEROV V 2013. A new interaction potential for swarming models. Physica D: Nonlinear Phenomena 260: 112-126.:
Here, represents the particle moment at site . We emphasize that we are using the dimensionless representation considered in ref. Hennig et al. (2007)HENNIG D, CHETVERIKOV A, VELARDE MG & EBELING W 2007. Electron capture and transport mediated by lattice solitons. Phys Rev E 76: 046602.. The time is scaled as , with 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)HENNIG D, CHETVERIKOV A, VELARDE MG & EBELING W 2007. Electron capture and transport mediated by lattice solitons. Phys Rev E 76: 046602.. The magnon dynamics is represented by the time-dependent Schrödinger equation for () defined as Sales et al. (2018)SALES MO, NETO AR & DE MOURA FABF. 2018. Spin-wave dynamics in nonlinear chains with spin-lattice interactions. Phys Rev E 98: 062136:
To clarify, we want to point out that the previous equations were written considering a ferromagnetic ground state denoted as and a set of kets represented by . Therefore, the value corresponds to the wave function amplitude associated with the spin deviation at position . By utilizing the Hamilton formalism, we have derived the equations governing the dynamics of the lattice:
It’s important to note that we changed the time scale in the previous equation by rescaling to , with rrepresenting the frequency of oscillations around the minimum of the Morse potential Hennig et al. (2007)HENNIG D, CHETVERIKOV A, VELARDE MG & EBELING W 2007. Electron capture and transport mediated by lattice solitons. Phys Rev E 76: 046602.. 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)HENNIG D, CHETVERIKOV A, VELARDE MG & EBELING W 2007. Electron capture and transport mediated by lattice solitons. Phys Rev E 76: 046602., Davydov (1991)DAVYDOV AS 1991. Solitons in Molecular Systems. 2nd ed., Reidel, Dordrecht., Scott (1992)SCOTT AC. 1992. Davydov’s soliton. Phys Rep 217: 1-67.. To put it simply, this framework involves a factor that multiplies the spin equation Hennig et al. (2007), Davydov (1991)DAVYDOV AS 1991. Solitons in Molecular Systems. 2nd ed., Reidel, Dordrecht., Scott (1992)SCOTT AC. 1992. Davydov’s soliton. Phys Rep 217: 1-67.. In our work, we will use, which is in alignment with previous research Hennig et al. (2007)HENNIG D, CHETVERIKOV A, VELARDE MG & EBELING W 2007. Electron capture and transport mediated by lattice solitons. Phys Rev E 76: 046602., Davydov (1991)DAVYDOV AS 1991. Solitons in Molecular Systems. 2nd ed., Reidel, Dordrecht., Scott (1992)SCOTT AC. 1992. Davydov’s soliton. Phys Rep 217: 1-67., Korotin et al. (2015)KOROTIN DMM, MAZURENKO VV, ANISIMOV VI & STRELTSOV SV 2015. Calculation of exchange constants of the Heisenberg model in plane-wave-based methods using the Green’s function approach. Phys Rev B 91: 224405., Satija et al. (1980)SATIJA SK, AXE JD, SHIRANE G, YOSHIZAWA H & HIRAKAWA K. 1980. Neutron scattering study of spin waves in one-dimensional antiferromagnet KCuF3. Phys Rev B 21: 2001., Hutchings et al. (1979)HUTCHINGS MT, MILNE JM & IKEDA H 1979. Spin wave energy dispersion in KCuF3: a nearly one-dimensional spin-1/2 antiferromagnet. Journal of Physics C: Solid State Physics 12: L739., Kadota et al. (1967)KADOTA S, YAMADA I, YONEYAMA S & HIRAKAWA K 1967. Formation of One-Dimensional Antiferromagnet in KCuF3 with the Perovskite Structure. Journal of the Physical Society of Japan 23: 751-756.. The value of 𝜏 will be adjustable, but in previous works, it was typically chosen to be around 10 Hennig et al. (2007)HENNIG D, CHETVERIKOV A, VELARDE MG & EBELING W 2007. Electron capture and transport mediated by lattice solitons. Phys Rev E 76: 046602., Ranciaro-Neto & de Moura (2016), Sales et al. (2018)SALES MO, NETO AR & DE MOURA FABF. 2018. Spin-wave dynamics in nonlinear chains with spin-lattice interactions. Phys Rev E 98: 062136 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 , , and , with as a normalization constant. We will use a Taylor procedure de Moura (2011)DE MOURA FABF 2011. Dynamics of one-electron in a one-dimensional systems with an aperiodic hopping distribution. Int J M Phys C 22: 63-69. to solve the set of equations 3, and a standard second-order Verlet’s like procedure Allen & Tildesley (1987)ALLEN MP & TILDESLEY TJ 1987. Computer Simulation of Liquids. Oxford University Press, p. 71-80., 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 defined as Sales et al. (2018)SALES MO, NETO AR & DE MOURA FABF. 2018. Spin-wave dynamics in nonlinear chains with spin-lattice interactions. Phys Rev E 98: 062136:
The lattice properties can be analyzed using the mean position of the lattice deformation defined as:
We want to emphasize that and represent the mean position of the spin-wave excitation and the lattice deformation, respectively. These measurements generally are in units of lattice spacing (). Using these quantities, we can obtain the magnon and the lattice deformation velocities and using fittings of the curves and . 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 and 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 ; this quantity represents a generalized probability that deformation around site occurs. This is obtained by normalizing , that is . We will plot where (i.e., represents the center of the chain). In fig. 1 we plot our results for , , , and . We can see that independent of the value of , 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.
The lattice deformation versus and where represents the center of the chain. Calculations were done for , , , and
Results and Discussion
Our findings on the velocities and in relation to are presented below. We obtained and through the linear fitting of the curves and . Our calculations of and 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 . The linear fitting was conducted using the last 20% of the complete time interval, roughly within the time interval . 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 . 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 : , and . Here, is a normalization constant, is a tunable parameter, and the and are larger than zero. We varied the parameter within the interval . We considering initially , , and several values of . We show our results in figs. 2(a-d). We emphasize that the curves indicate the velocities (black solid line) and (red dotted line) versus for several values of . To construct these curves, we calculate the dynamics of the spin and the lattice for long times for several values of and . We calculate the and curves versus using a linear fitting. We can see that is roughly independent of . On another side, spin propagation strongly depends on the spin-lattice interaction parameter . Let us clarify this important matter in simpler terms. The lattice’s deformation is governed by eq. (4). We can see that when is small, the nonlinear Morse terms, i.e., the first two terms, dominate over the terms that depend directly on and the wave functions. Therefore, the soliton velocity remains roughly constant; however, when 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 in both the diagonal (first term) and the off-diagonal (last two terms). As such, it was indeed expected that the value of would influence the magnon’s velocity. By analyzing all curves for several of we have considered, there is a matching of the magnon’s and lattice’s velocity () for a specific value of . This result suggests that for this particular value of , 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 has increased, this value of in which the velocities are the same become smaller. To comprehend this phenomenon, we need to emphasize that when 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 .
Magnon and lattice deformation velocities [(black solid line) and (red dotted line)] versus . We have considered , , and up to .
We also calculate the long-time mean distance between the magnon and the lattice deformation. The distance is defined as . We emphasize that 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)) versus where represents the maximum of the distance between the magnon and the lattice position. We can observe that for the same value of in which that , we can see that , 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 in which is in good agreement with the critical value found using the velocity curves versus (see fig. ). Therefore all measures of , , and are topological quantities that characterize the propagation of the magnon and the lattice deformation. Our calculations numerically demonstrate that for some specific values of , 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.
The normalized distante between the Magnon and the lattice deformation () versus . Calculations done for , , and up to .
In fig. 4, we collect the critical value of versus . We stress that for 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 with is a direct consequence of the role played by at the off-diagonal terms at eq. (3). As the 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 and change the values of and respectively to and ; we kept the same range of values of . We can observe that the results are qualitatively the same obtained in figs. and i.e.: as the value of is increased, the magnon-lattice pair formation is obtained for a specific value of . We also obtained that as is increased decreases (see fig. 7). In figures 8 and 9, we show our results considering and , and we kept the same range of values of and . The results obtained are similar to those in the previous figures. It appears that the magnon-lattice pair only exists when equals the critical value . This critical value decreases as increases, as shown in figure 10
Magnon and lattice deformation velocities [(black solid line) and (red dotted line)] versus computed using , , and up to .
The normalized distante between the Magnon and the lattice deformation () versus computed for , , and up to .
The velocities [(black solid line) and (red dotted line)] versus computed using , , and up to .
The normalized distante between the Magnon and the lattice deformation () versus computed for , , and up to .
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 . 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 . By computing and , 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 , our calculations indicated that and . In ref. , the possibility of magnon-soliton propagation in nonlinear lattices was demonstrated considering localized initial states (i.e., ). 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 . The parameter 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 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 . 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 , which allowed us to observe the signatures of magnon-soliton pairs. We discovered that within the range of values between 1 and 10, a magnon-soliton pair exists when equals a certain value, denoted as . However, this critical value depends on the value of . For instance, the results we obtained for and (refer to fig. 11(a)) revealed that when is low (less than 3), the critical value remains stable at around , while for values greater than 3, the critical value decreases by approximately half. This decrease in with increasing 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 and , the results remain qualitatively the same. The velocity of the magnon-soliton pair () is also dependent on the initial velocity . In fig. 11(b), we observe that for small , . For , . It is essential to note that the results shown in fig. 11 do not depend on the values of , , and . Generally, as velocity increases, the solitonic mode becomes faster, resulting in a faster magnon-soliton pair.
a) The critical value versus ;b) The magnon-soliton velocity versus . Calculations were done for and .
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 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 (). 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 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
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Publication in this collection
22 Dec 2023 -
Date of issue
2023
History
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Received
12 Apr 2023 -
Accepted
31 Aug 2023