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Journal of Magnetism and Magnetic Materials
journal homepage:www.elsevier.com/locate/jmmm
Research articles
Spin orders and excitation in frustrated Heisenberg model with four-spin exchange interaction
S.N. Huang
a,1, X.M. Zhang
a,1, W.Z. Zhuo
a, Z.P. Huang
a, D.Y. Chen
a, Z. Fan
a, M. Zeng
a, X.B. Lu
a, X.S. Gao
a, M.H. Qin
a,⁎, J.M. Liu
baInstitute for Advanced Materials, South China Academy of Advanced Optoelectronics and Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, South China Normal University, Guangzhou 510006, China
bLaboratory of Solid State Microstructures and Innovative Center for Advanced Microstructures, Nanjing University, Nanjing 210093, China
A R T I C L E I N F O Keywords:
Iron tellurium Phase diagram Magnetic excitation
A B S T R A C T
In this work, we study the phase diagram of a frustrated Heisenberg model with the additional next nearest neighbor four-spin interaction accounting for the spin–lattice coupling in order to understand the magnetism of iron tellurium. The experimentally identified bicollinear ground state that is always degenerate with the pla- quette state in any Heisenberg-biquadratic models [Glasbrenneret al., Nat. Phys.11, 953 (2015)] and thefirst- order phase transition are well reproduced, strongly confirming the essential role of the four-spin interaction related to the spin–lattice coupling [Bishop, Moreo, and Dagotto, Phys. Rev. Lett.117, 117201 (2016)] in ex- plaining the magnetic properties of iron tellurium. Moreover, based on the linear spin-wave theory, specific magnetic excitation in strained iron tellurium is predicted, appealing for experimental verification.
1. Introduction
In the past a few years, the magnetic orders of iron-based super- conductors have drawn extensive attention since it has been extensively claimed that the superconductivity is probably linked to magnetism [1–4]. One of the major considerations along this line is the scenario that electron pairing mechanism is related to spin fluctuations. This scenario thus stimulates substantial effort in understanding the mag- netic orders and spinfluctuations in these materials. However, it is also well believed that there are generally complicated couplings among the spin, orbit, and lattice degrees of freedoms, prohibiting an exactly theoretical study incorporating all these ingredients. A rational strategy for selecting the essential of the iron-base superconductivity is to seek some simplified effective magnetic models so that the major ingredients regarding the magnetic properties of iron-based superconductors can be considered[5,6].
So far, theoretical models such as the spin-fermion model con- sidering both itinerant electrons and local moments have been suc- cessfully used to describe the magnetic properties including the colli- near antiferromagnetic state (spin configuration is shown inFig. 1(a)) and the in-plane resistivity anisotropy reported in pnictide super- conductors [7–9]. More interestingly, the Heisenberg-biquadratic
models based on the local moments scenery have also demonstrated their great success in explaining complicated physics in these typical correlated electron systems. For example, the effective square-latticeJ1- J2-Kmodel with the nearest neighbor (NN) (J1) and next NN (J2) ex- change couplings and the biquadratic coupling K has been used to successfully reproduce the experimentally reported spin wave and magnetic phases in iron pnictides[10]. Here,J1andJ2couplings at- tribute mainly to the local magnetic exchange and/or superexchange mechanisms, while the biquadratic coupling is mainly of intrinsic electronic origin and/or spin–lattice coupling and could not be ne- glected in any model calculation[11,12]. Furthermore, the third NN exchange interaction J3 (J1-J2-J3-K model) is further considered to study the magnetic orders in iron chalcogenides [13,14]. For suffi- ciently large K and J3, the model has a bicollinear ground state (Fig. 1(b)) which is always degenerate with the plaquette state (Fig. 1(c)). In addition, based on some similar models, the anti- ferroquadrupolar order[15,16]and staggered dimer state (Fig. 1(d)) [17,18]were reported and suggested to relate with the magnetism of FeSe.
Whereas some of the magnetic properties of iron-based super- conductors are qualitatively explained based on the Heisenberg-bi- quadratic models, the origin of the magnetism of iron tellurium FeTe
https://doi.org/10.1016/j.jmmm.2020.166872
Received 18 September 2019; Received in revised form 13 February 2020; Accepted 5 April 2020
⁎Corresponding author.
E-mail address:[email protected](M.H. Qin).
1S Huang and X Zhang contributed equally to this work.
Available online 06 April 2020
0304-8853/ © 2020 Elsevier B.V. All rights reserved.
T
remains to be further clarified. Different from the iron pnictides which normally exhibit the collinear antiferromagnetic order[19], FeTe dis- plays the bicollinear antiferromagnetic state[20]. Experimentally, the first-order antiferromagnetic phase transition accompanied by the monoclinic distortion was reported to occur at temperatureT~ 75 K [21]. On one hand, the monoclinic distortion in iron tellurium generally generates the anisotropy of the next NN exchangeJ2coupling from the view of symmetry and spin–lattice coupling, which has been also re- vealed in thefirst-principles calculations[22]. However, a reasonable interpretation of theJ2anisotropy cannot be obtained in the Heisen- berg-biquadratic model where the bicollinear state and plaquette state are always degenerate[17]. Moreover, the calculated ground state of iron tellurium is the staggered dimer state which is also inconsistent with the well-accepted bicollinear state, strongly demonstrating that the current model should be further modified to better understand the magnetism of iron tellurium[19,23]. Furthermore, the exchange ani- sotropicJ1-J2-J3model exhibits the bicollinear state, while the physical mechanism of the exchange anisotropy is not clear considering the tetragonal symmetry of the system before the structure transition[24].
On the other hand, in a recent work, the spin–lattice coupling was suggested to play an important role in determining the properties of iron tellurium[25]. As an appreciated approximation, the spin–lattice coupling could be reasonably described by higher-order four-spin in- teractions [26–28]. The underlying physics can be understood by looking at some specific systems where the next NN four-spin interac- tion is available and may be essential in explaining the magnetism of FeTe. Indeed, FeTe is a good example if one considers the monoclinic structure of iron tellurium and the orbital polarization symmetry along the next NN bond direction. However, few works on this topic have been reported, as far as we know.
Therefore, in this work, we intend to study aJ1-J2-J3-JL-Kmodel with additional next NN four-spin exchange interactionJLto describe the magnetism of iron tellurium. Specifically, the anisotropy of the next NN exchange interaction can be well interpreted by considering the four-spin interaction, as explained in theSupporting Information sec- tion S1. In addition, the experimentally reported bicollinear ground state and thefirst-order phase transition can be well reproduced by our simulations, further confirming the important role of the spin–lattice coupling. As a comprehensive extension, the magnetic excitation
behaviors with this model will be investigated too, based on the linear spin-wave theory. One will see that such an excitation becomes favored in strained iron tellurium.
2. Model and methods
In this work, theJ1-J2-J3-JL-Kmodel based on the square lattice of FeTe (Fig. 1(a)) under investigation has its Hamiltonian written as:
∑ ∑ ∑
= − + + + + +
H Jn S S· K ( · )S S J ( ·S S )(S ·S )
i δ i j
i δ
i j L
i
i i x y i x i y
, ,
2
n 1 (1)
where the first term accounts for the exchange interactions and Jn
(n= 1, 2, 3) is the coupling between then-th NN Heisenberg spinsiand j=i+δn, the second term accounts for the NN biquadratic interaction along thexorydirections, the third term includes the next NN four- spin exchange couplingJLwith spins at the corners of a square pla- quette[29]. For simplicity,J1and the Boltzmann constant are set to unity over the whole text, which never affects our main conclusion.
Following the earlier works, the ground-state phase diagrams are exactly obtained through comparing the energies of these possible phases, and the explicit expressions of these energies are presented in Supporting Informationsection S2[17]. The magnetic transitions at finite temperature Tare investigated using the Monte Carlo (MC) si- mulations [30], and the simulation details and the definition of the bicollinear order parameter mbi are explained in Supporting Informationsection S3[30–32]. Furthermore, the spin-wave dispersion is investigated based on the linear spin wave theory, and the calculation details are given inSupporting Informationsection S4[24].
3. Results and discussion 3.1. Phase diagrams
We study the phase diagram as favored by the model Hamiltonian, given a set of different next NN four-spin exchange interaction JL. Particular attentions will be paid to the stabilization of the bicollinear phase, the probable ground state in iron tellurium. Here, the phase boundaries in this phase diagram are reasonably calculated through comparing the energies of these possible states, i. e., the typical Neél state (all the NN spins are antiparallel with each other)ENeél, the col- linear stateEcol, the bicollinear stateEbi, the staggered dimer stateEsd, the plaquette stateEpl, and the spiral order with wave vector (q,π)/(q, q)E(q,π)/E(q,q).
Wefirst look at the phase diagram in the absence of the four-spin interaction (JL= 0), and the phase diagram is plotted inFig. 2(a) on the (J2, J3) parameter plane, withK = 0.388 is estimated by the first- principles calculations on FeTe system. In this case, the plaquette state and the bicollinear state are degenerate and favored as long as J3 > 0.5. AsJ3 < 0.5, the phase diagram space is divided into the Neél state, the dimer state, and the collinear state respectively. More- over, the (q,q)/(q,π) spiral order emerges at very small/largeJ2around the boundary between the Neél state/the collinear state and the dimer state. As discussed earlier, for FeTe system, the phase diagram para- meter set is (J1,J2,J3,K) = (1.0, 0.844, 0.481, 0.388), as indicated by the black cross inFig. 2(a). Thus, it is clear that FeTe favors the stag- gered dimer state, inconsistent with experimental report[17]. One may note that these estimated parameters should be modified if a new term is introduced to the effective model. However, the four-spin interaction JLrelated to the spin–lattice coupling is much smaller than the ex- change interactions, which not seriously changes the values of these parameters.
Now one considers the cases ofJL < 0 and the phase diagram at JL=−0.1 is plotted inFig. 2(b) where again black cross marks the position of (J1,J2,J3,K) = (1.0, 0.844, 0.481, 0.388). It is seen that the degeneracy of plaquette state and bicollinear state in the J3 > 0.5 region is broken, favoring the plaquette state rather than the bicollinear Fig. 1.The spin configurations in (a) the collinear state, (b) the bicollinear
state, (c) the plaquette state, and (d) the staggered dimer state. Solid and empty circles represent the up-spins and the down-spins, respectively.
state, implying the essential influence of JL on the phase stability.
Furthermore, the spiral states with wave vector (q,π) and (q,q), located at the two corners, disappear either, resulting in the replacement of the (q,π)/(q,q) spiral state by the collinear/Néel state. This is one view- angle to look into the effect of the four-spin interactionJL.
On the other hand, we look at the other view-angle withJL > 0.
The calculated phase diagrams at JL = 0.1 and 0.2 are plotted in Fig. 2(c) and (d) respectively. In these cases, the phase region at J3 > 0.5 is favorably by the bicollinear state rather than the plaquette state, opposite to the case shown inFig. 2(b). More importantly, the criticalJ3distinguishing the bicollinear state from the staggered dimer state significantly decreases, resulting in the stabilization of the bicol- linear state under the parameters for iron tellurium, as indicated by the black cross in Fig. 2(c). Actually, by comparing the energies of these possible states, one notes that the bicollinear state is stabilized for JL > 0.019 under the parameters for iron tellurium. Thus, the ex- perimentally reported ground state of FeTe is well explained by the consideration of positiveJL, again demonstrating the important role of the spin–lattice coupling in understanding the magnetism of iron tell- urium. With the increase ofJL, the staggered dimer state, the Néel state, and the collinear state are further destabilized and gradually replaced by the bicollinear state, the (q, q) state, and the (q,π) state, respec- tively, as clearly shown inFig. 2(d).
3.2. Bicollinear phase transitions
In experiments, the finite temperature antiferromagnetic phase transition was revealed to be offirst order, which is also captured by the investigated model. In this part, the phase transition from the high temperature paramagnetic state to the bicollinear state is studied using
MC simulations. Specifically, we use Metropolis algorithm [33]and parallel tempering algorithm[34–36]to obtain the equilibrium state, and calculate the order parameters to study the finite temperature transition of the system. Generally, the simulation is performed on a three-dimensional 12 × 12 × 12 lattice with periodic boundary con- ditions, and 150 copies of the system are investigated. Here, the inter- layer NN exchange interactionJz= 1 is introduced considering the character of the real material, noting that the two-dimensional isotropic Heisenberg model has no long range order at anyfinite temperatures, according to the Mermin–Wagner theorem[37]. Typically, the initial 1 × 105 MC steps are discarded for equilibrium consideration and another 3 × 106MC steps are retained for statistic averaging of the simulation. An exchange sampling is taken after every 50 standard MC steps.
The simulated bicollinear order parametermbiand the Binder ratio Bas functions of temperature forJL= 0.25 for the parameter set of (J1, J2,J3,K) = (1.0, 0.844, 0.481, 0.388) are presented inFig. 3(a). When Tis decreased toTN~ 0.528, mbisuddenly increases to a nearly sa- turation value, clearly demonstrating the transition from the para- magnetic state to the bicollinear state occurs atTN~ 0.528. Moreover, the binder ratio shows a sharp peak at the transition temperature, strongly suggesting that the transition is offirst order, consistent with the experimental report[19].
As a matter of fact, the most recent inelastic neutron scattering suggests that the bicollinear state in FeTe is quasidegenerate with the plaquette state, indicating in some extent thatJLmay be very weak in realistic materials. However, the conclusion on the stabilization of the bicollinear ground state and thefirst order transition are hardly af- fected by the value of positiveJL. Specifically, the effect ofJLandJzon the transition temperature TN is also investigated, and the Fig. 2.The calculated phase diagram on the (J2,J3) parameter plane for (a)JL= 0, (b)JL=−0.1, (c)JL= 0.1, and (d)JL= 0.2. The black cross in the phase diagram denotes the calculated parameter set for iron tellurium reproduced from Ref.[17].
corresponding results are summarized in Fig. 3(b) andFig. 3(c).TN
increases with the increasingJLand/orJz, demonstrating that the bi- collinear state can be further stabilized by the enhanced four-spin ex- change and inter-layer exchange interactions.
3.3. Low-energy excitation
Since the ground state andfinite temperature phase transition of iron tellurium are successfully reproduced, the magnetic excitation of the state draws new interest. Following the earlier work, we calculate the magnetic excitations of the bicollinear state using the linear spin- wave theory, and give the calculated results inFig. 4where the spin- wave dispersion along the two directions k1 = kx + ky and k2=kx−kyat the parameter set of (J1,J2,J3,K,JL) = (1.0, 0.844, 0.481, 0.388, 0.1) is presented. As the exchange interaction along thez- axis is not frustrated, we only focus on the in-plane spin-wave disper- sion. Two bands per wave vector and the significant anisotropy of the spin-wave dispersion along the two directions are observed, as also clearly shown in Fig. 5(a) and (b) where presents the spin-wave dis- persion along thek1direction (k2= 0) and thek2direction (k1= 0), respectively.
Moreover, the effect ofJLon the spin-wave dispersion is also in- vestigated, and the calculated spin-wave dispersions for variousJLare
given inFig. 5. It is seen that both the high-ωand the low-ωbranches shift towards the lowωside with the decrease ofJL. Furthermore, the spin-wave velocity (slope at small wave vector of the low-ωbranch) is also decreased. Experimentally, uniaxial stress could be applied along the antiferromagnetic diagonal direction of iron tellurium to reduceJL. In some extent, the spin-wave dispersions predicted here may be available in strained iron tellurium, which deserves to be checked in future experiments.
4. Conclusion
In conclusion, we have studied the phase diagram of the frustrated J1-J2-J3-JL-Kmodel using the energy analysis and Monte Carlo simu- lations in order to understand the magnetism of iron tellurium. The experimentally reported bicollinear ground state andfirst order phase transition have been well reproduced by the consideration of the next NN four-spin exchange interaction. Thus, it is suggested that the four- spin exchange interaction attributing to the spin–lattice coupling may play an essential role in understanding the magnetism of iron tellurium, strongly strengthening the earlier conclusion. Furthermore, the mag- netic excitation is also investigated based on the linear spin-wave theory, and the predictions could be available in strained Iron tell- urium, which deserves to be checked in future experiments.
CRediT authorship contribution statement
S.N. Huang: Conceptualization, Methodology, Writing - original draft.X.M. Zhang:Conceptualization, Methodology, Writing - original Fig. 3.The Monte Carlo simulated (a)mbiandBas function ofTforJL= 0.25, and (b) the critical temperatureTNfor variousJLforJz= 1, and (c)TNfor variousJz forJL= 0.25 under the parameter set of (J1,J2,J3,K) = (1.0, 0.844, 0.481, 0.388).
Fig. 4.The calculated spin-wave dispersionω(k1,k2) for the parameter set of (J1,J2,J3,K) = (1.0, 0.844, 0.481, 0.388) forJL= 0.1 withk1=kx+kyand k2=kx−ky.
Fig. 5.The calculated spin-wave dispersion (a)ω(k1) atk2= 0, and (b)ω(k2) at k1= 0, for variousJL. The high-energy line and low-energy line indicate two branches of spin wave for the specificJL, respectively.
draft.W.Z. Zhuo:Conceptualization.Z.P. Huang:Conceptualization.
D.Y. Chen: Conceptualization. Z. Fan: Conceptualization.M. Zeng:
Conceptualization. X.B. Lu: Conceptualization. X.S. Gao:
Conceptualization.M.H. Qin:Supervision, Writing - review & editing.
J.M. Liu:Supervision, Writing - review & editing.
Declaration of Competing Interest
The authors declare that they have no known competingfinancial interests or personal relationships that could have appeared to influ- ence the work reported in this paper.
Acknowledgements
We sincerely appreciate the insightful discussions with Daoxin Yao.
The work is supported by the Natural Science Foundation of China (No.
51971096 and No. 51721001), and the Science and Technology Planning Project of Guangzhou in China (Grant No. 201904010019), and the Natural Science Foundation of Guangdong Province (Grant No.
2019A1515011028).
Appendix A. Supplementary data
Supplementary data to this article can be found online athttps://
doi.org/10.1016/j.jmmm.2020.166872.
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