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Makoto Naka, Yukitoshi Motome, [Tsuyoshi Miyazaki](https://orcid.org/0000-0003-3534-4404), Hitoshi Seo

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[Nonrelativistic Piezomagnetic Effect in an Organic Altermagnet](https://mdr.nims.go.jp/datasets/a037af1f-6bf3-4ce2-83e8-5ada58a2cb73)

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Nonrelativistic Piezomagnetic Effect in an Organic AltermagnetarXiv:2505.07327v1  [cond-mat.str-el]  12 May 2025Journal of the Physical Society of Japan LETTERSNonrelativistic Piezomagnetic Effect in an Organic AltermagnetMakoto Naka1 *, Yukitoshi Motome2 †, Tsuyoshi Miyazaki3 ‡, and Hitoshi Seo4 §1School of Science and Engineering, Tokyo Denki University, Ishizaka, Saitama 350-0394, Japan2Department of Applied Physics, The University of Tokyo, Bunkyo, Tokyo 113-8656, Japan3Research Center for Materials Nanoarchitectonics (WPI-MANA), National Institute for Materials Science, Tsukuba,Ibaraki 305-0044, Japan4RIKEN Center for Emergent Matter Science, Wako, Saitama 351-0198, JapanWe theoretically study the piezomagnetic effect on the altermagnetic state in κ-type molecular conductors, focusingon its nonrelativistic mechanism. By introducing shear stress as a monoclinic distortion, we evaluate variations in the ef-fective tight-binding model using first-principles calculations. Using the derived parameters, we investigate the Hubbardmodel and its effective Heisenberg model on the two-dimensional (distorted) κ-type lattice within mean-field approxima-tion. We show that the system exhibits the piezomagnetic effect, i.e., a net magnetization induced at finite temperaturesin the undoped insulating state and both in the ground state and at finite temperatures upon doping. In a real-space pic-ture, this uniform magnetization arises from the ferrimagnetic spin structure due to inequivalent spin sites induced bylattice distortion. Meanwhile, in a momentum-space picture, it stems from the s-wave spin splitting of the electron andmagnon bands, independent of spin-orbit coupling. We find that this nonrelativistic piezomagnetism remains finite, butbecomes smaller in the limit of strong dimerization where the energy gap between the bonding and antibonding orbitalsis infinitely large and the d-wave altermagnetic spin splitting is absent, highlighting the importance of the multi-orbitalnature.Organic conductors, which have been studied as typicalstrongly correlated electron systems,1–3) have recently beenrevealed to exhibit peculiar properties in their antiferromag-netic (AFM) states. Specifically, collinear AFM ordering onthe double herringbone arrangement of molecules, called κ-type, was proposed to show nonrelativistic spin splitting (SS)in the electronic and magnon bands and a spin current gen-eration under electric field or thermal gradient.4) Model cal-culations explicitly demonstrated that the anisotropy of elec-tron hoppings dependent on sublattices is the origin of thesephenomena. Furthermore, this AFM state breaks macroscopictime reversal symmetry and then the anomalous Hall effectcan be expected when the spin-orbit coupling is considered.5)These features—the nonrelativistic SS,6–10) its resultant spincurrent conductivity,4, 11–13) and the anomalous Hall effectsometimes discussed in the finite frequency range5, 14–18)—arein fact the main characteristics of altermagnets,19–21) whichare currently under intensive research.Another physical property of altermagnetism, the piezo-magnetic effect (PME), has been discussed extensively,8, 22–27)and was recently observed in the inorganic altermagnetMnTe.28) The PME is a cross-correlated phenomenon charac-terized by linear coupling between uniform magnetization andmechanical stress in a magnetic crystal; it has been discussedsince early days29, 30) and was observed in AFM ordered CoF2and MnF2.31) These compounds have rutile-type structures,similar to RuO2, an intesively studied altermagnet candidate,and their AFM order breaks macroscopic time-reversal sym-metry. As the microscopic mechanism of this effect, the roleof spin-orbit coupling was investigated.30)On the other hand, the nonrelativistic effect in altermagnets*m-naka@mail.dendai.ac.jp†motome@ap.t.u-tokyo.ac.jp‡MIYAZAKI.Tsuyoshi@nims.go.jp§seo@riken.jpmay also lead the PME. For example, in Refs. 8 and 24, thecoupling bewteen uniform magnetization and a strain field hasbeen pointed out to exist even without the spin-orbit coupling,from the symmetry point of view. Such a nonrelativistic PMEhas been demonstrated in Ref. 22 for a monolayer of V2Se2Oby first-principles calculations. In addition, in Refs. 23 and 25,two-dimensional (2D) Heisenberg models with altermagneticground states were shown to exhibit uniform magnetizationunder strain at finite temperatures (T ), resulting in a ferrimag-netic spin structure. In organic compounds with flexible lat-tices and weak spin–orbit coupling, nonrelativistic contribu-tions to the effect are expected to be particularly pronounced;however, its nature targeting actual materials has not yet beeninvestigated.In this Letter, we study the PME in the organic altermagnetcandidate κ-(BEDT-TTF)2Cu[N(CN)2]Cl by combining first-principles evaluation of strain effects and model calculations.We first derive the effective tight-binding model using first-principles calculations, and then analyze the two-dimensionalHubbard model and the effective Heisenberg model withinmean-field approximation. We show the emergence of a non-relativistic PME at finite T in the undoped insulating state andboth in the ground state and at finite T upon doping, in the pre-sense of the strain. Besides, we find that SS in the unstrainedcase is not essential for the PME and underscore the impor-tance of multi-orbital nature in the BEDT-TTF dimer32) for asizable response. Our results highlight that molecular crystalssensitive to pressure can serve as an ideal platform for explor-ing the PME.The first-principles band calculations were performedbased on a plane-wave density-functional theory within thegeneralized gradient approximation,33) utilizing the QUAN-TUM ESPRESSO code version 7.234) with scalar-relativisticpseudopotentials generated by the projected augmented waveformalism.35) Maximally localized Wannier functions were1https://arxiv.org/abs/2505.07327v1J. Phys. Soc. Jpn. LETTERS-1.0-0.5 0.0 0.5U Z Γ X U Γ(c) (d)U Z Γ X U ΓpqbaAB(a)ab ca1a2p1q1 p2q2b1b2[eV]U-0.4 -0.2  0.0  0.2  0.4-0.4-0.2 0.0 0.2 0.4-0.03-0.02-0.01 0.00 0.01 0.02 0.03-0.4 -0.2  0.0  0.2  0.4(e)-0.4 -0.2  0.0  0.2  0.4kckakckc(f) (g)[eV]XΓ Z(b)ab cBBba2b2cd-waved+s-wave s-waveFig. 1. (Color online) Schematic illustrations of the κ-type lattice structure(a) without and (b) with the monoclinic distortion induced by shear stress.The circles and the ellipses represent the molecular sites and the dimers, re-spectively. The networks of the bonds are shown: four kinds a, b, p, and q inthe undistorted case in (a), and eight kinds {a1, a2}, {b1, b2}, {p1, p2}, and{q1, q2} in the distorted case in (b). Their tight-binding band structures (c)without (tilting angle θ = 0) and (d) with the monoclinic distortion (θ = 2◦).The dashed and solid lines represent those for the paramagnetic (U = 0) andAFM (U = 1 eV) phases, where the up and down spin bands are denoted byred and blue, respectively. The Fermi energies are set to zero. The k-pointpath in the orthorhombic Brillouin zone is shown in (e). We note that thesame symmetry indices are also used in the monoclinic case in (d) for con-venience. The k-space distributions of the SS of the top bands in the AFMground states for (e) θ = 0◦ , (f) θ = 2◦, and (g) the dimer limit for θ = 2◦.generated using the WANNIER90 package36) by setting aWannier center at each BEDT-TTF molecule. The cutoff ener-gies for plane waves and charge densities are set to 40 and 200Ry, respectively, and a Gaussian smearing method was usedwith 2 × 1 × 2 and 4 × 2 × 4 uniform k-point meshes definedin the space group Pnma during the structural optimizationand self-consistent-field calculation, respectively.As for the effect of shear stress, for simplicity, startingfrom the experimental crystal structure at ambient pressure for15 K,37) we tilt the a-axis within the ac-plane, producing anorthorhombic-to-monoclinic distortion, and while fixing thevolume of the unit cell, structurally optimize the internal co-ordinates. This results in the modulation of inter-moleculartransfer integrals as shown in Figs. 1(a) and 1(b), where the2D network at the ambient pressure orthorhombic and that un-der the monoclinic distortion, respectivively, are illustrated.In Table I, we list the transfer integrals by varying the tilt-ing angle θ. The intradimer a, and the interdimer b, p, andq bonds in the orthorhombic case [Fig. 1(a)] are modulatedas {a1, a2}, {b1, b2}, {p1, p2}, and {q1, q2} [Fig. 1(b)], re-spectively. They vary about 8 – 20 % at θ = 2◦, a typicaldegree of distortion in organic crystals under pressure of theorder of GPa.38, 39) The on-site potential energy, ǫi, for the twoBEDT-TTF molecules that becomes crystallographically in-Table I. Tight-binding parameters estimated by the first-principles calcu-lation by varying the tilting of the a-axis with angle θ = 0◦, 1◦, and 2◦, inunit of eV. “bond1” and “bond2” denote the two inequivalent bonds shown inFig. 1(b).orthorhombic (θ = 0◦)monoclinic (θ = 1◦) monoclinic (θ = 2◦)bond1 bond2 bond1 bond2ǫi 4.160 4.165 4.158 4.172 4.161ta -0.193 -0.185 -0.200 -0.178 -0.208tb -0.068 -0.071 -0.065 -0.075 -0.068tp -0.097 -0.106 -0.090 -0.116 -0.082tq 0.050 0.053 0.056 0.056 0.043equivalent for θ , 0 is also evaluated. In Figs. 1(c) and 1(d),the band structures without strain, θ = 0◦, and with distortionof θ = 2◦, are shown. Owing to the monoclinic distortion,the band degeneracy along the Brillouin zone boundaries arelifted.Now we investigate the PME by calculating the magneticstructures under the variation of these parameters. We con-sider the 2D Hubbard model at three-quarter filling, which hasbeen studied intensively for the undistorted case.2, 4, 40) TheHamiltonian is described asHHubb =∑〈i, j〉,sti j(c†isc js + h.c.)+∑i(ǫini + Uni↑ni↓), (1)where cis (c†is) and nis (= c†iscis) are the annihilation (creation)and number operators of an electron at ith molecular site withspin s, respectively, and ni = ni↑ + ni↓. The transfer integralsti j are considered for the pairs 〈i, j〉 along the bonds shownin Figs. 1(a) and 1(b), as mentioned above. U is the on-siteCoulomb repulsion treated within the Hartree-Fock approxi-mation.Without the strain (θ = 0◦), the on-site Coulomb repul-sion leads to an AFM insulating state where the sites on thetwo kinds of dimers A and B show opposite spin directions,40)say, in the z direction, as 〈szA〉 = −〈szB〉. Here 〈szα〉 (α = A, B)represents the expectation value of z-spin moment on dimerα, while the two sites within each dimer have the same value.Since the dimers are connected by glide symmetry, these spinsare compensated resulting in no net uniform magnetization,i.e., mu ≡ 〈szA〉 + 〈szB〉 = 0. In this state, the d-wave altermag-netic SS appears,4) as shown in Figs. 1(c) and 1(e).On the other hand, under monoclinic distortion, the dimersA and B are no longer equivalent, with no symmetry op-eration connecting them. In Figs. 2(a) and 2(b), we showthe T dependence of staggered and uniform magnetization,ms ≡ 〈szA〉−〈szB〉 and mu, respectively, with fixed U = 1 eV, forθ = 0◦ and θ = 2◦. Under the distortion, in the ground state,the staggered component is almost identical to the undistortedcase and there is no net magnetization. Such a state, in whichthe sublattice magnetizations completely cancel each othereven though the sublattices are inequivalent, is referred to as“compensated ferrimagnetism” in recent years.41–43)However, at finite T , mu is induced by the distortion, stabi-lizing a ferrimagnetic state as depicted in the inset of Fig. 2(b).This is the PME. As T is raised from absolute zero, mu in-creases and reaches a maximum in its absolute value, and thendecreases down to zero toward TN. The band structure and SSin the distorted AFM ground state is shown in Figs. 1(d) and1(f), respectively; the d-wave SS is deformed, with an addi-2J. Phys. Soc. Jpn. LETTERS (a) 0.0 0.5 1.0-0.020-0.015-0.010-0.005 0.000 0.00  0.05  0.10  0.15  0.20 (b) AFM orderab c-0.06-0.05-0.04-0.03-0.02-0.01 0.00 (c)[eV] AFM insulator AFM metal-0.07-0.08-0.04 0.00 6.0  6.1  6.2  6.3  6.4  6.5Fig. 2. (Color online) T dependences of (a) the staggered magnetizationms and (b) the net uniform magnetization mu. The undistorted (θ = 0◦), mon-oclinically distorted (θ = 2◦), and its dimer limit (see text) cases are shown.The spin pattern in the ferrimagnetic state is schematically depicted in theinset. (c) T dependences of mu varing the number of electrons in the unit cellne at θ = 2◦. The inset shows the ne dependence of mu in the ground state atθ = 2◦.tional extended s-wave SS being superimposed. The differ-ence in the excitation energies across the AFM gap betweenup- and down-spin electrons, induced by this s-wave splitting,gives rise to the PME at finite T . In contrast, in the groundstate without such thermal excitations, the numbers of up-and down-spin electrons are equal, resulting in a fully com-pensated magnetization.Let us discuss the relationship between SS and the PME,with a focus on the role of dimerization. We artificially tunethe intradimer transfer ingetrals as ta1 → ta1 − ∆ and ta2 →ta2 − ∆; the increase in ∆ leads to the separation of bondingand antibonding orbitals of the dimers and for ∆ = ∞ themodel is reduced to a single-orbital Hubbard model on theanisotropic triangular lattice.40) In this dimer limit, the d-waveSS and the bonding/antibonding orbital degree of freedom arelost both in the orthorhombic and monoclinic cases.4) How-ever, we find that while the uniform component mu becomessmaller as ∆ increase, it converges to a finite value in thedimer limit of ∆→ ∞, as plotted in Figs. 2(a) and 2(b). Inter-estingly, we find that in the dimer limit with monoclinic dis-tortion, only the s-wave component of the spin-splitting sur-vives, as shown in Fig. 1(g). The smaller but finite values ofmu in the dimer limit indicate that even when the d-wave SS—though symmetry-imposed—is negligibly small, the nonrela-tivistic PME remains finite and the multi-orbital nature leadsto the enhancement of the induced magnetization.As shown above, the PME vanishes at T = 0 in the undopedinsulating state. However, when carriers are doped, a finitemagnetization emerge even at T = 0.27) Figure 2(c) shows theelectron doping dependences of the PME at θ = 2◦, where-0.008-0.006-0.004-0.002 0.000 0.000  0.025  0.050  0.075  0.100 (c)[eV] AFM orderAB B A(g)AB B Agroundstate finite TmagnetizationAK1BK2J2’ABab cJ1’J1J2 (a)ikj (b)ab cantibondingbondingi k jK1-0.4 -0.2  0.0  0.2  0.4-0.4-0.2 0.0 0.2 0.4-0.01  0.00  0.01(d)kakc-0.4 -0.2  0.0  0.2  0.4kc-0.02  0.00  0.02 -0.01  0.00  0.01-0.4 -0.2  0.0  0.2  0.4kc(e) (f)d-wave d+s-wave s-waveFig. 3. (Color online) The interdimer exchange interactions in the effec-tive Heisenberg model for κ-type compounds with monoclinic distortion. TheNN exhange interactions, J1 , J2 , J′1, J′2, and the NNN ones, K1, K2, whichmake the system an altermaget, are shown. In the orthorhombic structurewithout the distortion, J ≡ J1 = J2 , J′ ≡ J′1= J′2, and K ≡ K1 = K2 hold.(b) Schematic illustration of a perturbation process leading to the NNN ex-change interaction in the hole picture, where the transfer integrals betweenthe bonding and antibonding orbitals are essential. (c) T dependences of thenet uniform magnetization mu, for the undistorted (θ = 0◦) and monoclini-cally distorted (θ = 2◦) cases, obtained by the mean-field approximation forthe Heisenberg model. The k-space distributions of the magnon SS in theAFM ground states for (d) θ = 0◦ , (e) θ = 2◦, and (f) the dimer limit of (e).(g) A schematic illustration for the mechanism of the nonrelativistic PME(see text).the number of electrons in the unit cell, ne, is increased fromthree-quarter filling (ne = 6.0). Under the doping, the AFMmetallic state is stabilized at low T , and the uniform magneti-zation mu shows the hump below TN as in the undoped case,but it starts to increase again below around kBT/U = 0.03.The magnitude of mu in the ground state increases with anincrease of ne, while reaches a peak at ne ≃ 6.15 and thendecreases as shown in the inset of Fig. 2(c). The low-T in-crease in magnetization originates from the SS of the Fermisurfaces, which induces a difference between the up-spin anddown spin electron densities. The upturn in the T dependenceof mu is governed by the amplitude of the s-wave splitting inthe top band, shown in Figs. 1(f) and 1(g), where the Fermi3J. Phys. Soc. Jpn. LETTERSTable II. Exchange paremeters at U = 1 eV obtained by the perturbationexpansion using the transfer integrals in Table I, for the orthorhombic andmonoclinic structures in unit of meV.orthorhombic (θ = 0◦)monoclinic (θ = 1◦) monoclinic (θ = 2◦)bond1 bond2 bond1 bond2J 60.1 70.2 51.4 82.0 44.2J′ 10.9 11.9 9.98 13.3 9.37K 1.65 2.27 1.22 3.09 0.89energy is located under the electron doping.We can furthermore elaborate on the mechanism of thenonrelativistic PME by deriving the effective 2D Heisenbergmodel through the perturbation expansion with respect toti j/U. We follow the treatment of Ref. 4, which demonstratedthe d-wave SS in the magnon band for the undistorted case.The minimal Heisenberg model describing the altermagneticproperties of κ-type compounds is given by4)HHeis =∑〈i j〉Ji j Si · S j +∑〈i j〉′J′i j Si · S j +∑〈〈i j〉〉Ki j Si · S j , (2)where 〈i j〉 and 〈i j〉′ stand for the nearest-neighbor (NN) bondsof the isosceles triangular lattice, on the legs in two direc-tions and the bases, respectively; 〈〈i j〉〉 is the next-nearest-neighbor (NNN) bonds along one of the leg direction, whichdepends on the sublattices A and B [see Fig. 3(a)]. The NNNexchange interaction originates from the interorbital trans-fer integrals between the neighboring dimers as illustratedin Fig. 3(b), leading to the d-wave spin spliting mentionedabove.4) In the presence of the monoclinic distortion, each ofthese three kinds of bonds splits into two inequivalent ones,as shown in Fig. 3(a). Consequently, the exchange interac-tions are modulated as Ji j = J → {J1, J2}, J′i j= J′ → {J′1, J′2},and Ki j = K → {K1,K2}, where J, J′, and K are the valuesfor the orthorhomic structure, via the splitting of the transferintegrals listed in Table II. Since the monoclinic strain dimin-ishes the glide symmetry existed in the orthorhombic case, thecouplings also lose the glide symmetry.In Fig. 3(c), we show the tempearature dependences of thenet magnetization, mu, for θ = 0◦ and θ = 2◦, together withthe dimer limit at θ = 2◦, calculated within the mean-field ap-proximation as in Ref. 4. Similar to the case for the Hubbardmodel discussed above, mu appears at finite T with similar Tdependence in the undoped case and shows reduction in thedimer limit.The AFM magnon SS obtained by the linear spin-wave the-ory for Eq. (2) are shown in Figs. 3(d)-3(f); the SS is definedby subtracting the energy of down-spin magnon from that ofup-spin magnon. At θ = 2◦, the d-wave magnon splitting atθ = 0◦ is superimposed with the extended s-wave compo-nent induced by the monoclinic distortion. In the dimer limitat θ = 2◦, the d-wave splitting vanishes, leaving only the ex-tended s-wave component. From the analysis of the magnondispersion, we find that the extended s-wave SS induced bythe distortion consists of two components: one proprotionalto the diferrence between the NNN exchange interactionsK1−K2, and the other to J′1− J′2. In the dimer limit, K1 and K2,which are associated with the transfer integrals between thebonding and antibonding orbitals in the neighboring dimers,vanish, whereas J′1and J′2, determined by those between theantibonding orbitals, remain finite. Consequently, the PMEdecreases toward the dimer limit but retains a finite value, asshown in Fig. 3(c). Furthermore, the fact that the SS in thedimer limit becomes purely extended s-wave is also attributedto the vanishing of K, which is responsible for the d-wave SS.These considerations lead us to an intuitive understandingof the T dependence of the nonrelativistic PME [Fig. 3(g)].The couplings between the two sublattices A and B are {J1, J2}while those for A-A and B-B are {J′1,K1} and {J′2,K2}, respec-tively. Owing to the different exchange couplings for the latterintra-sublattice bonds, the finite-T fluctuations become differ-ent between the two sublattices, as was discussed in Ref. 25.In terms of magnons, this can be interpreted as the emergenceof net magnetization due to a difference between the numberof thermally excited up-spin and down-spin magnons as a re-sult of the s-wave SS. We also note that even for K1 = K2 = 0the effect remains finite; considering that Ki j was responsiblefor the d-wave magnon SS in the undistorted case,4) the alter-magnetic SS is not required here, as in the case of the Hubbardmodel discussed above.Finally, we discuss the relevance of our results to experi-ments. As stressed in the introductory part, organic crystalsare tunable by external pressure, and uniaxial pressure hasbeen used to control the properties of correlated electrons.Our results, showing the induced uniform magnetization ofa few percent compared to the staggered component of theAFM order in a moderate range of shear stress, demonstrateorganic systems a promising platform for exploring the PMEin altermagnets. Another issue is the effect of spin-orbit cou-pling, which has conventionally regarded as the microscopicorigin of piezomagnetism. In our case the direction of thenet magnetization is along the sublattice magnetization. InRef. 30, the piezomagnetism owing to the spin-orbit couplingin CoF2 was also predicted to appear also along the sublatticemagnetization. However, the former exhibits a maximum justbelow TN at finite T , while the latter is expected to increaseas T approaches zero. In organic compounds with relativelyweak spin–orbit coupling, the nonrelativistic contribution islikely to dominate over the relativistic one near TN. This sug-gests that detailed analysis of the T dependence could allow aclear distinction between these two mechanisms. Last but notleast, we comment on the effect of magnetic domains. In theκ-type lattice, there are two types of altermagnetic domains,namely {szA.szB} = {↑, ↓} and {↓, ↑}, and their coexistence cancancel out the present PME. However, these domains areaccompanied by weak ferromagnetic moments along the a-axis with opposite directions due to the spin–orbit coupling.5)Therefore, field cooling is sufficient to align the domains andobserve the PME.Acknowledgment We thank T. Aoyama and K. Ohgushi for fruitfuldiscussions. This work is supported by JSPS KAKENHI Grant Numbers,No. JP19K03723, JP19K21860, JP20H04463, JP23H01129, JP23K25826,JP23K03333, JP25H00838, JP25H01247, the GIMRT Program of the Insti-tute for Materials Research, Tohoku University, No. 202212-RDKGE-0062,and No. 202312-RDKGE-0062.1) The Physics of Organic Superconductors and Conductors, edited by A.Lebed, Springer Series in Materials Science (Springer, Berlin, Heidel-berg, 2008).4J. Phys. Soc. Jpn. LETTERS2) H. Seo, C. Hotta, and H. Fukuyama, Chem. Rev. 104, 5005 (2004).3) A. Ardavan, S. Brown, S. Kagoshima, K. Kanoda, K. Kuroki, H. Mori,M. Ogata, S. Uji, and J. Wosnitza, J. Phys. Soc. Jpn. 81, 011004 (2012).4) M. Naka, S. Hayami, H. Kusunose, Y. 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