Abnormal Critical Fluctuations Revealed by Magnetic Resonance in the Two-Dimensional Ferromagnetic Insulators
Zefang Li, Dong-Hong Xu, Xue Li, Hai-Jun Liao, Xuekui Xi, Yi-Cong Yu, Wenhong Wang
AAbnormal Critical Fluctuations Revealed by Magnetic Resonance in theTwo-Dimensional Ferromagnetic Insulators
Zefang Li,
1, 2
Dong-Hong Xu,
1, 2
Xue Li,
1, 2
Hai-Jun Liao,
1, 3
Xuekui Xi, Yi-Cong Yu,
1, 4, ∗ and Wenhong Wang
1, 3, † Beijing National Laboratory for Condensed Matter Physics,Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China University of Chinese Academy of Sciences, Beijing 100049, China Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,Wuhan Institute of Physics and Mathematics, Innovation Academy for Precision Measurement Science and Technology,Chinese Academy of Sciences, Wuhan 430071, China (Dated: January 8, 2021)Phase transitions and critical phenomena, which are dominated by fluctuations and correlations,are one of the fields replete with physical paradigms and unexpected discoveries. Especially fortwo-dimensional magnetism, the limitation of the Ginzburg criterion leads to enhanced fluctuationsbreaking down the mean-field theory near a critical point. Here, by means of magnetic resonance,we investigate the behavior of critical fluctuations in the two-dimensional ferromagnetic insulatorsCrXTe (X = Si , Ge). After deriving the classical and quantum models of magnetic resonance, wedeem the dramatic anisotropic shift of the measured g factor to originate from fluctuations withanisotropic interactions. The deduction of the g factor behind the fluctuations is consistent with thespin-only state ( g ≈ and 2.039(10) for CrGeTe ). Furthermore, the abnormalenhancement of g shift, supplemented by specific heat and magnetometry measurements, suggeststhat CrSiTe exhibits a more typical two-dimensional nature than CrGeTe and may be closer tothe quantum critical point. Fluctuations and correlations drive abundant phasetransitions and critical phenomena. Regardless of theclassical or quantum regime or of the order parameterand symmetry, they are universal in nature and followstatistical laws [1]. Among them, a particularly fasci-nating aspect of two-dimensional (2D) magnetism asso-ciated with strong intrinsic magnetization fluctuationshas introduced rich physical paradigms: the quantumspin liquid (QSL) state of the Kitaev model, Berezinskii-Kosterlitz-Thouless (BKT) transition of the XY model,Mermin-Wagner theorem of the isotropic Heisenbergmodel, Ising transition, etc. [2]. Herein, recent discover-ies of magnetic van der Waals (vdW) materials providethe ideal platform for exploring intrinsic 2D magnetismdown to the 2D limit and potential opportunities for newspin-related applications [3].Notably, 2D magnetism is particularly susceptible tofluctuations. The Ginzburg criterion indicates that fluc-tuations become much more relevant with decreasing di-mensions, leading to the failure of mean-field theory [4].The Mermin-Wagner theorem recognizes that no long-range order can survive thermal fluctuations at finitetemperature in a 2D system with continuous symme-try [5]. However, by breaking the continuous symmetry,anisotropy in the exchange interaction will open up thespin wave excitation gap to resist thermal agitations ofmagnons at finite temperature. Such notable examples ofmagnetic order in single atomic layers have been discov-ered in CrI [6], CrGeTe [7], Fe GeTe [8] and VSe [9]. ∗ [email protected] † [email protected] Moreover, 2D magnetism is associated with strong intrin-sic competition between quantum fluctuations and ther-mal fluctuations [10]. In the ground state where thermalfluctuations vanish, the quantum fluctuations demandedby Heisenberg’s uncertainty principle will dominate thequantum phase transition (QPT), which is driven bysome nonthermal external parameters such as the mag-netic field, pressure, or chemical doping [11]. At finitetemperature, the energy of a system and the enthalpyof its thermal fluctuations compete, resulting in a clas-sical phase transition (CPT). Although fluctuations playa crucial role in 2D magnetism, most of the theoreti-cal predictions by ab initio methods are based on zerotemperature and ignore fluctuations. Obtaining phaseboundary information of fluctuations and critical pointsthrough experimental detection is very important.Here, we demonstrate magnetization-fluctuation-induced effective g factor anisotropy in the 2D ferromag-netic insulators CrXTe (X = Si , Ge) by means of ferro-magnetic resonance (FMR) and electron paramagneticresonance (ESR). In general, the dominant critical fluc-tuations occur at the critical temperature T c and decayexponentially when deviating from T c . Compared withCrGeTe (CGT), the observation of critical fluctuationswith enhanced intensity and broad temperature rangein CrSiTe (CST) is abnormal, which is associated withthe 2D nature even in the bulk counterparts. Althoughthe critical behavior can be indirectly characterized byneutron scattering [12, 13], magnetic susceptibility mea-surement [14, 15], specific heat measurement [16], nuclearmagnetic resonance [17] and the dynamic magnetoelec-tric coupling technique [18] and directly characterized byreal-time magneto-optical imaging technology [19], accu- a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n rately estimating the temperature dependence of magne-tization fluctuations by means of magnetic resonance isvery exciting. FIG. 1. (a) Crystalline structure of bulk rhombohedralCrXTe (X = Si , Ge) with ABC vdW stacking. (b) Schematicof different coordinate systems from the top view of the abplane: crystallographic axes a, b, and c and FMR coordinateaxes x, y, and z. (c) Magnetic ion Cr surrounded by adistorted octahedral crystal field with an out-of-plane arrowrepresenting its easy magnetization direction. Split d levelsin the e g and t manifolds, with 3 unpaired electrons in the d xy , d yz , and d xz orbitals.
2D vdW CST and CGT are ferromagnetic insulatorsand belong to the family of layered vdW transition metaltrichalcogenides (TMTCs), which are crystallized in the R ¯3 (148) rhombohedral structure. Fig. 1 shows the hon-eycomb ABC layers stacked by a cleavable vdW gap. Ineach layer, the magnetic Cr ions and Ge / Si pairsare located at Te − octahedral sites in a distorted D d local symmetry, with a crystal field splitting the Cr-3 d orbitals and sustaining Cr-Te-Cr ferromagnetic superex-change. As shown in Fig. 1(c), three unpaired electronsof Cr are accommodated in the lower t g triplet or-bital, thus resulting in a quenched orbital moment forspins J = S = 3 / g factor near 2.0023 for freeelectrons. Especially for CST, the strong Coulomb in-teractions from the narrow d bands and the half-filledcondition favor a Mott transition [20]. As further shownin Fig. 1(b), these octahedra arrange in edge-bond-sharing networks in the ab plane and form a magnetichoneycomb lattice. The interplay of spin-orbit couplingand the crystal field currently explains the uniaxial mag-netic anisotropy with an easy axis perpendicular to theab plane [21]. However, controversy about the uncer-tainty among the Kitaev, Ising, Heisenberg and single-ion anisotropy terms remains. CGT has been demon-strated to be well described by the Heisenberg behav-ior with a single-ion anisotropy term, which has beenproven to exhibit ferromagnetic order in the monolayer[7, 14]. In contrast, CST with giant magnetic anisotropyis determined to be consistent with the 2D Ising behav-ior [12, 15], for which the ground state of the monolayer still lacks experimental confirmation. Moreover, in struc-turally related CrI [22, 23], α − RuCl [24] and Na IrO [25], Kitaev anisotropic exchange interactions are foundin competition with Heisenberg interactions, which areassociated with a possibly QSL state. Recently, first-principles-based simulations predicted the possible Ki-taev QSL state in epitaxially strained CST monolayers[26, 27]. After comprehensive consideration, we consideran XXZ Hamiltonian with single-ion anisotropy: H = − X h j,l i ( J S j · S l + Λ S zj S zl ) − X j AS zj S zj − µ B H · g · X j S j . (1)The first term corresponds to the Heisenberg isotropicexchange J and the anisotropic symmetric exchange Λ.The second term is the additional single-ion anisotropyterm, and the last term corresponds to the Zeeman en-ergy. J >
A > /J to infinityrecovers the 2D Ising model, while the isotropic Heisen-berg model is recovered for Λ ≈ A ≈ TABLE I. Physical quantities extracted from magnetometryand specific heat measurements [28].CrSiTe CrGeTe Space group R ¯3(148) R ¯3(148)Critical temp. 34.15 68.15 Derived MT T c (K) 32.50 65.50 Arrott plot32.68 64.90 Specific heatCurie-Weiss temp. 57.16 101.46 H k c Θ (K) 53.86 100.24 H k ab Frustration param. 1.67 1.49 H k cf H k ab Effective mag. a H k cµ eff ( µ B ) 3.97(6) 3.99(6) H k ab Saturation mag. b H k cM s ( µ B / f . u . ) 2.84 3.02 H k ab Mag. entropy c δS (J / mol K) 48.67% 38.85% above T c Critical exponent d β γ δ a Expected value µ eff = g p ( J ( J + 1)) µ B = 3 . µ Bb Expected value M s = gJµ B = 3 . µ B / f . u . c Expected value R ln(2 J + 1) = 11 .
53 J / mol K d CST close to 2D Ising model ( β = 0 . , γ = 1 . β = 0 . , γ = 1 . The presence of critical fluctuations near the criticalpoint can be evidenced by magnetometry and specificheat measurements [28]. As shown in Table I, Curie-Weiss behavior is observed at high temperatures, andan estimate of the effective moment gives µ eff ≈ µ B ,consistent with the spin-only magnetic moment µ eff = g p ( J ( J + 1)) µ B = 3 . µ B for J = S = 3 / g = 2 . FIG. 2. (a) Coplanar waveguide with a rectangular singlecrystal placed parallel to the ab plane. The domain wall isrepresented by a blue solid line, and the angle with the y axisis α . (b) Illustration of three different resonant modes fromthe side view. (c) Frequency- and field-dependent FMR spec-tra (in-plane, β = π/
2) for CrSiTe at 5 K. The solid linesrepresent fitting for the single domain mode (fitted with Eq.2) and multidomain mode (fitted with Eq. 3). (d) Typical mi-crowave transmission at 28 GHz sliced from (c). The resonantpeak of the multidomain mode is formed by the superpositionof a series of peaks with different angles α . low 150 K can be recognized in the M ( T ) curves, result-ing in a higher Weiss temperature Θ than the criticaltemperature T c . The ratio is defined as the frustrationparameter f = | Θ | /T c , which corresponds to short-rangeferromagnetic correlations persisting in the paramagneticstate. Moreover, the magnetic entropy S ( T ) above T c , es-timated from the magnetic specific heat C m ( T ), recoversto a value as large as nearly 48 .
67% (CST) and 38 . M ( H ) curves, which well match the2D Ising model for CST and the tricritical mean-fieldmodel for CGT. The above evidence implies that CSThas a stronger critical fluctuation and an exchange inter-action closer to a 2D nature, but further verification isneeded.On the basis of the correspondence principle, the classi-cal and quantum mechanical descriptions of the magneticresonance are identical. Therefore, concrete expressionsfor the free energy and Hamiltonian are necessary to re-veal the physics behind an observable. Here, we recoverthe resonance solutions of the Larmor equation (classical)and Heisenberg equation of motion (quantum) for a gen-eral case (more details are provided in the supplementarymaterials) [28–30]. We conclude that the anomaly of themeasured effective g factor, as well as the anomaly of therelationship between magnetocrystalline anisotropy K u and saturation magnetization M s , can be explained bythe specificity of fluctuations and correlations, which areunsettled in previously reported FMR measurements ofCrI [22], CrCl [31, 32], and CrGeTe [33–35].One of the important correspondences is the spectro-scopic g factor, which can be determined precisely by gy-romagnetic ratio fitting using FMR and ESR spectra. Asa classical description, shown in Fig. 2(a), a rectangular-shaped single crystal is placed in a coplanar waveguidewhere it is acted upon by an alternating magnetic field H rf . In response to scanning of a strong homogeneousmagnetic field H ext at right angles, resonance absorptionsignals can be detected in the case that ω res = γH eff ,where γ = gµ B / ~ is the gyromagnetic ratio and H eff isthe effective internal field. When resonance occurs, thesaturation magnetization M s induces Larmor precessionalong the effective field direction. In consideration ofthe Zeeman splitting energy, crystallographic anisotropy,demagnetizing field, and Bloch domain structure, the so-lutions to the Larmor equations are given by the Smit-Beljers approach [28, 29]: (cid:18) ω res γ (cid:19) = { H − [ H A − ( N z − N y ) M s ] } { H − ( N y − N x ) M s } , (2)for single-domain mode, ϑ = ϕ = π , H ≥ H A + N y M s . (cid:18) ω res γ (cid:19) = ( H A + N x M s )( H A + M s sin α ) − ( H A + M s sin α − N z M s )( H A + N x M s )( H A + N y M s ) H , (3)for multidomain mode, ϕ = ϕ = π , sin ϑ = sin ϑ = HH A + N y M s , H ≤ H A + N y M s . where H A = 2 K/M s is the anisotropic field, N is thedemagnetization factor, and α is the angle between the domain wall and the external magnetic field. When themagnetic field H ext is applied parallel to the ab plane,we observe three different resonant modes (illustrated inFig. 2(b)), which are plotted as a function of excitationfrequency and applied magnetic field in Fig. 2(c). Thedomain wall resonance peaks excited under weak fieldsare much smaller in amplitude than the FMR peaks.Their dependence on the resonance frequency versus thein-plane field corresponds to the conventional theory forthe Bloch wall model [36]. Remarkably, the multido-main mode has a continuously changing angle α , result-ing in asymmetric peak shapes (Fig. 2(d)). The cross-ing point of the multidomain mode and single-domainmode indicates the saturation field of the domain struc-ture ( H = H A + N y M s ), which exists below the criticaltemperature T c . Moreover, when the magnetic field H ext is applied parallel to the c axis, the resonance frequencycan be determined by the sum of the external field, theequivalent anisotropy field, and the demagnetizing field: ω res γ = H + H A − N z M s , for ϑ = 0 . (4) FIG. 3. (a,b) Temperature dependence of the effective g factorwith the field applied both in-plane and out-of-plane for CSTand CGT. The real g factor is calculated with Eq. 6. (c)Reduced anisotropy constant and magnetization at differenttemperatures shown on the logarithmic scale. The red lineindicates an exponent of 3 for the Callen-Callen power law.(d) Schematic phase diagram showing the paramagnetic andferromagnetic phases. The vertical paths for CST and CGTrepresent the CPT. Herein, we extract the spectroscopic g factor by fittingthe in-plane and out-of-plane FMR and ESR data withthe above equations (more details are provided in thesupplementary materials) [28]. As shown in Fig. 3(a,b),the temperature dependence of the effective g factor forthe in-plane ( H k ab ) and out-of-plane ( H k c ) orien-tations has a contrasting behavior. A downwards (in-plane) or upwards (out-of-plane) shift of the g factoris observed as the temperature increases, with a maxi-mum value at T c . Notably, the deviation of the g fac-tor is beyond an orbital contribution, which is almost completely quenched due to the crystal field [37]. Sucha temperature-dependent shift in the g factor has beenfound in ESR measurements of low-dimensional metal al-loys, metal complexes, or purely organic compounds [38].Based on Nagata’s theory in the ESR case [30, 39–41], westrictly solve a general solution in the FMR case and con-clude that magnetization fluctuations with anisotropicinteractions are responsible for the g shift (more detailsare provided in the supplementary materials) [28]. Tobe more specific, the Hamiltonian is substituted into theprecession motion equation: ~ ω = h [ S − , [ S + , H ]] i h S z i . (5)After calculating the thermodynamic average, we findthat the isotropic Heisenberg term J does not contributeto the g shift, whereas the anisotropic symmetric ex-change Λ and single-ion anisotropy term A do. The ab-solute value of the g shift along the easy axis of mag-netization ∆ g c is twice that along the orthogonal hardplane ∆ g ab . By taking the value-weighted average, wecan obtain the real g factor of the sample: g = 13 × g ab + 23 × g c . (6)As shown in Fig. 3(a,b), the calculated g factor is a con-stant value independent of temperature. After averagingthe g factors over the entire temperature range, we obtainthe averaged g factors as 2.050(10) for CST and 2.039(10)for CGT, which are consistent with the orbital-quenched g factors.In addition, thermal fluctuations lead to an effec-tive reduction in both the saturation magnetization andmagnetocrystalline anisotropy, which can be representedby the Callen-Callen power law based on the single-ionanisotropy model: K u ( T ) K u (0) = (cid:20) M s ( T ) M s (0) (cid:21) l ( l +1) / , (7)where l is the order of spherical harmonics and dependson the symmetry of the crystal. In the case of uniaxialanisotropy for CST and CST, l = 2 and an exponent of3 are expected. However, as shown in Fig. 3(c), CGTshows little agreement with the power law for exponentsof 2.37(2) (FMR) and 2.51(3) (SQUID). In contrast, CSTexhibits nonlinear behavior, which obviously violates thepower law. Hence, the departure from the Callen-Callenpower law suggests that the consideration of thermal fluc-tuations for single-ion anisotropy is incomplete. This isconsistent with the fact that the single-ion anisotropy forCr is sufficiently small due to the weak spin-orbit cou-pling ( ξ L · S ) with quenched orbital angular momentum( L ≈ T is the temper-ature and g is the strength of the ferromagnetic exchangecoupling. On the one hand, the curve of the FM and PMphase boundary corresponds to the critical temperature T c . The CPT occurs by varying the temperature through T c . In the classical critical region, the correlation lengthtends to infinity, and critical fluctuations are dominant.On the other hand, changing g in the ground state willlead to a QPT at the quantum critical point g c , wherethe quantum fluctuations are the strongest. According tothe results of our experiment, the critical temperature ofCST is relatively low, and the fluctuations observed aremuch stronger than those for CGT. Therefore, we canreasonably indicate that CST is closer to the quantumcritical point g c , which is dominated by both classicaland quantum critical behavior. This inference is also sup-ported by a recent report on pressure-induced supercon-ductivity in CST [42]. We believe that doping, pressure,cleavage, and electrical regulation can achieve a QPT inCST, but more experimental verification is needed.In summary, we have combined magnetic resonance,specific heat and magnetometry measurements to inves-tigate the behavior of critical fluctuations in bulk CSTand CGT single crystals. Although fluctuations near thecritical temperature are natural in magnetic materials,the observation of such anisotropic shifts of resonance peaks in low-dimensional systems is unique because ofthe Ginzburg criterion. Despite the structural and elec-tronic similarities, CST and CGT show strong contrastsin critical behavior. Our work implies the presence ofshort-range correlation far above T c and a signally 2Dnature even in bulk counterparts of CST. Although CSTshows a stronger magnetic anisotropy, the absence of fer-romagnetic order in the monolayer should be attributedto the enhanced fluctuations. Last but not least, suchunignorable magnetization fluctuations in 2D magneticmaterials will interact with the spins of scatterers (X-rays, neutron beams, spin currents, etc.) and enhancethe scattering effect. For the application of 2D magneticmaterials in spintronic devices, the influence of magneti-zation fluctuations must be evaluated carefully.This work was supported by the National KeyR&D Program of China (2017YFA0206303 and2017YFA0303202), National Natural Science Foun-dation of China (11974406), and Strategic PriorityResearch Program (B) of the Chinese Academy of Sci-ences (CAS) (XDB33000000). 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Zefang Li,
1, 2
Dong-Hong Xu,
1, 2
Xue Li,
1, 2
Hai-Jun Liao,
1, 3
Xuekui Xi, Yi-Cong Yu,
1, 4, ∗ and Wenhong Wang
1, 3, † Beijing National Laboratory for Condensed Matter Physics,Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China University of Chinese Academy of Sciences, Beijing 100049, China Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,Wuhan Institute of Physics and Mathematics, Innovation Academy for Precision Measurement Science and Technology,Chinese Academy of Sciences, Wuhan 430071, China (Dated: January 8, 2021)
CONTENTS
I. Crystal growth, Magnetometry and Specific heat 1A. Methods 1B. M(T) and M(H) curves 2C. Heat capacity 3D. Arrott plot 4II. Derivation of Ferromagnetic Resonance Model and Experimental Fitting 5A. Single-domain mode 5B. Multi-domain mode 6C. Measured FMR spectra and experimental fitting 7III. Quantum explanation of fluctuations induced g-factor anisotropy 8A. Solution for a general model applicable for FMR and ESR case 8B. Solution for the XXZ model with single ion anisotropy 12References 12
I. CRYSTAL GROWTH, MAGNETOMETRY AND SPECIFIC HEATA. Methods
High quality CrSiTe (CST) and CrGeTe (CGT) single crystals were grown by the self-flux method [1, 2]. Themixture of pure elements (Cr : Si : Te = 1 : 2 : 6 , Cr : Ge : Te = 10 : 13 . .
5) were mounted in an aluminacrucible, which was sealed inside a quartz tube under high vacuum ( < − Pa). Then the quartz tube is heated upto 1100 ◦ C in a tube furnace and slowly cooled down to 700 ◦ C. Excessive molten flux was centrifuged quickly beforesolidification. The final hexagonal flakes were shiny and soft, and were easy to peel off.Energy dispersive X-ray spectroscopy (EDXS, equipped in Hitachi S-4800 microscope) was used for element analy-sis. As well as the X-ray diffraction (XRD, Bruker D8 Anvance) proved the stoichiometric ratio and high quality ofCGT and CST single crystals. We also measured the heat capacity at zero field (PPMS-9T, Quantum Design physi-cal properties measurement system) and characterized the temperature and field dependent magnetization (MPMS,Quantum Design magnetic property measurement system). In consideration of the demagnetization effect, it shouldbe noted that the external applied field has been corrected for the internal magnetic field as H int = H ext − N M ,where N is the demagnetization factor [3] and M is the measured magnetization. ∗ [email protected] † [email protected] a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n FIG. 1. (a), (b) Temperature dependence of the magnetization and its derivative measured in ZFC, FC and FW modes with afield of 100 Oe. The inset shows the enlarged picture of splitting between H k ab and H k c . (c), (d) Inverse of the magneticsusceptibility for FC curve. The black solid lines indicate the linear fits in paramagnetic region with Curie-Weiss law. (e), (f)Magnetization as function of field at ferromagnetic and paramagnetic state respectively. B. M(T) and M(H) curves
The magnetometry results are listed in Fig. 1. Temperature dependent zero field cooled (ZFC), field cooled (FC),and field warmming (FW) with an external field of H = 100 Oe both parallel to c axis and ab plane are shown inFig. 1(a) and Fig. 1(b). The paramagnetic-ferromagnetic (PM-FM) transitions occurs at the Curie temperature(34 .
15 K for CST and 68 .
15 K for CGT) are determined by the derivative of M(T) curve, which are the highest amongpreviously reported values [1, 2, 4–6] and indicating a better sample quality. These two samples both exist a “thermo-hysteresis” between FC and FW curves ( H k c ) below T c . As reported in CrGeTe [7] and Fe GeTe [8], this anomalousbehavior is related to the existence of skyrmions, but has not been confirmed in CrSiTe . And then Fig. 1(c) andFig. 1(d) display the temperature dependent inverse susceptibility H/M under FC, parallel to c axis and ab planerespectively. A linear fit of Curie-Weiss law at PM state yields the Weiss temperature Θ and effective moment µ eff .Above 150 K , the fitted effective magnetic moment is close to the expected value µ eff = g p ( J ( J + 1)) µ B = 3 . µ B .Below 150 K , however, the H/M curves deviate from the linear fit, resulting a higher Weiss temperature Θ thancritical temperature T c . And the ratio of them is defined as frustration parameter f = | Θ | /T c . This suggests thatshort-range magnetic correlations persist in PM state above T c . Fig. 1(e) and Fig. 1(f) show the field dependence ofmagnetization measured at FM and PM state. The M(H) curves splitting at 5K confirm the easy c-axis and largermagnetocrystalline anisotropy in CST than CGT. And the saturation magnetization fitted by the law of approachingsaturation is very close to the expected value M s = gJµ B = 3 . µ B / f . u . . C. Heat capacity
FIG. 2. (a), (b) Temperature dependence of zero-field specific heat for CrSiTe and CrGeTe . The blue line is the fitting oflattice specific heat by Thirring model. The inset represents the magnetic contribution of Cp/T versus T and the integrationof magnetic entropy S after subtracting the lattice contribution. Fig. 2 shows specific heat data at zero field. As expected, the λ -shaped anomaly is observed at PM-FM transition.In order to compare the entropy change associated with magnetism, we subtract the non-magnetic contributions dueto lattice vibrations. For this purpose, the lattice specific heat is estimated with the Thirring model [9]: C lattice = 3 N R ∞ X n =1 b n (cid:18) πTθ D (cid:19) + 1 ! − n . (1)where N is the number of atoms in the unit cell, R is the ideal gas constant, θ D is the Debye temperature. Weuse n up to 4 for fitting and obtain a reasonable accuracy as shown by the blue solid line. After subtracting thelattice contribution, the inset shows the magnetic contribution C p /T versus T and the magnetic entropy obtainedfrom numerical integration. In general, the molecular field results in the second-order phase transition with a jumpin specific heat. However, the fractions of the magnetic entropies gained above T c are 48 .
67% for CST and 38 . J + 1 energy states of N non-interacting magnetic moments consist of W = (2 J + 1) N availablestates. Therefore the corresponding entropy can be calculated by Boltzman’s theory: S = k ln W = N k ln(2 J + 1) = R ln(2 J + 1) = 11 . / mol K for J = S = 3 / . Due to short-range magnetic correlations, the measured entropy changes (3 .
91 J / mol K for CST and 0 .
86 J / mol Kfor CGT) are relatively small. D. Arrott plot
FIG. 3. : (a), (d) Modified Arrott plot of isotherms. The dashed line is the linear fit of isotherm at critical temperature. (b),(e) Temperature dependence of M s and χ . The Tc and critical exponents are obtained from the fitting of Eq. 2 and 3. (c), (f)Isothermal MH at T c . The inset shows the fitting of Eq. 4 and the extracted exponents. The second-order PM-FM phase transition near the critical point can be characterized by a set of critical exponents β , γ , and δ . The spontaneous magnetization M s ( T ) below T c , the inverse initial susceptibility χ − ( T ) above T c andthe measured magnetization M ( H ) at T c follow the power law: M s ( T ) ∝ ( T c − T ) β , T < T c (2) χ − ( T ) ∝ ( T − T c ) γ , T > T c (3) M ∝ H /δ , T = T c (4)It is known that the critical exponents are not independent of each other but follow the scaling relation: δ = 1 + γβ (5)Therefore, we apply a modified Arrott plot by self-consistent iteration to determine the critical exponents and phasetransition temperature T c [10]. As shown in Fig. 3(a) and (b), isotherms plotted in form of M /β versus ( H/M ) γ constitute a set of parallel straight lines, and the isotherm at T c will pass through the origin. The relationship is givenby the following equation: (cid:18) HM (cid:19) /γ = a T − T c T + bM /β (6)where a and b are constants. Linear fitting of the isotherms at high field region gives ( M s ) /β and ( χ − ) /γ as anintercept on M /β and ( H/M ) /γ . According to Eq. 2 and 3, linear fitting of log[ M s ( T )] versus log( T c − T ) andlog[ χ − ( T )] versus log( T − T c ) give new values of β and γ , while free parameter T c is adjusted for best fitting. Andthen we substitute the new values into the Arrott plot and iterate repeatedly to get the optimal solution.In order to test the accuracy of the final solution, we have analyzed the ( M s ) /β and ( χ − ) /γ data by Kouvel-Fisher (KF) plot [11]. As shown in Fig. 3(c) and (d), the values obtained are within the error accuracy. Furthermore,according to Eq. 4, linear fitting of log[ M ( H )] versus log( H ) gives a straight line with slope 1 /δ . And it is noteworthythat these obtained β , γ , and δ are related in the scaling relation of Eq. 5.These obtained critical exponents for CST ( β = 0 . , γ = 1 . β = 0 . , γ = 1 . β = 0 . , γ = 1 . β = 0 . γ = 1 . II. DERIVATION OF FERROMAGNETIC RESONANCE MODEL AND EXPERIMENTAL FITTINGA. Single-domain mode
Considering a regular shaped magnetic sample in the uniform magnetic field, the motion of magnetization vector M follows the Larmor equation: d M dt = − γ [ µ M × H eff ] . (7)The effective magnetizing field H eff is determined by the free energy F per unit volume. In single-domain case, it canbe represented in the sum of Zeeman energy, the demagnetization energy, and magnetocrystalline anisotropy energy: F = − µ M s H sin ϑ sin ϕ + K sin θ + 12 µ M s ( N x sin ϑ cos ϕ + N y sin ϑ sin ϕ + N z cos θ ) . (8)In which K is the magneto-crystalline anisotropy, and N is demagnetization factor. The equilibrium orientation of themagnetization vector M s ( ϑ , ϕ ) are determined by ∂F/∂ϑ = 0 , ∂F/∂ϕ = 0. When the vector M s deviates slightlyfrom the equilibrium position, the total free energy can be Taylor expanded as F ( ϑ + δϑ, ϕ + δϕ ) = F ( ϑ , ϕ ) + ( F ϑϑ δϑ +2 F ϑϕ δϑδϕ + F ϕϕ δϑ ). In this case, the internal effective field are defined as H ϑ = − ( µ M s ) − ∂F/∂ϑ, H ϕ = − ( µ M s sin ϑ ) − ∂F/∂ϕ . Considering that Larmor precession has periodic solution δϑ, δϕ ∝ e iωt , we obtain theequations of motion in the polar coordinate system as the following matrix form: (cid:18) F ϕϑ + iωγ − µ M s sin ϑ F ϕϕ F ϑϑ F ϑϕ − iωγ − µ M s sin ϑ (cid:19) (cid:18) δϑδϕ (cid:19) = (cid:18) (cid:19) . (9)Where F ϑϕ = ∂ F/∂ϑ∂ϕ . The equation of motion has a non-zero solution only when the determinant is zero. Definingthe anisotropic field H A = 2 K/M s , then we can get the resonance frequency as (cid:18) ω res γ (cid:19) = (cid:8) [ H A − ( N z − N y ) M s ] − H (cid:9) H A − ( N y − N x ) M s H A − ( N z − N y ) M s , for sin ϑ = HH A + ( N y − N z ) M s , ϕ = π , H < H A + ( N y − N z ) M s . (10) (cid:18) ω res γ (cid:19) = { H − [ H A − ( N z − N y ) M s ] } { H − ( N y − N x ) M s } , for ϑ = ϕ = π , H > H A + ( N y − N z ) M s . (11) B. Multi-domain mode
For CST and CGT single crystals with hexagonal lattice and uniaxial anisotropy, the Bloch magnetic domainstructure was observed experimentally below the saturation field [7, 14]. It should be pointed out that Smit andBeljers first derived the multi-domain FMR model in BaFe O for a square-shaped sample. But a mistake in thederivation led to an unexplainable item in the result [15]. Here we recover the derivation and generalize to samplesof arbitrary shape.For simplicity, we assume that the magnetic domains of adjacent domains have two kinds of magnetization M ( ϑ , ϕ ) and M ( ϑ , ϕ ). The demagnetization energy of adjacent domains with infinitely thin thickness is equalto 12 (cid:18) ( M − M ) · e n (cid:19) . (12)In considering the multi-domain case, the angle between the domain wall and the external magnetic field is α , andthe expression of free energy is equal to: F = − µ M s H (sin ϑ sin ϕ + sin ϑ sin ϕ ) + 12 K (sin ϑ + sin ϑ )+ 12 µ M s (cid:20) N x ϑ cos ϕ + sin ϑ cos ϕ ) + N y ϑ sin ϕ + sin ϑ sin ϕ ) + N z ϑ + cos ϑ ) + 14 (sin ϑ cos( ϕ − α ) − sin ϑ cos( ϕ − α )) (cid:21) . (13)Similarly, near the equilibrium position M s ( ϑ , ϕ , ϑ , ϕ ), we can get the equation of motion in matrix form: F F − i Ω / F F F + i Ω / F F F F F F F − i Ω / F F F + i Ω / F δϑ δϕ δϑ δϕ = µ M s H/ cos β cos ϑ sin β sin ϑ cos β cos ϑ sin β sin ϑ . (14)Where Ω = ωγ − µ M s sin ϑ , F = ∂ F/ ( ∂ϑ ∂ϕ ), and β is the angle between H rf and H ext . In considering ofthe symmetry, we have F = F , F = F , F = − F = − F = F . For convenience we put ∆ ϑ ± = δϑ ± δϑ ,∆ ϕ ± = δϕ ± δϕ . Then we get F + F − i Ω / i Ω / F + F F − F F − i Ω /
20 0 2 F + i Ω / F − F ∆ ϑ + ∆ ϕ + ∆ ϑ − ∆ ϕ − = µ M s H β sin ϑ cos β cos ϑ . (15)Therefore, the 4 × × H rf ⊥ H ext ( β = π/ (cid:18) ω res γ (cid:19) = ( H A + N x M s )( H A + M s sin α ) − ( H A + M s sin α − N z M s )( H A + N x M s )( H A + N y M s ) H , for sin ϑ = sin ϑ = HH A + N y M s , ϕ = ϕ = π . (16)(b) when H rf k H ext ( β = 0) (cid:18) ω res γ (cid:19) = ( H A + N y M s )( H A + M s cos α ) − H A + M s cos αH A + N y M s H − M s sin α cos α (cid:18) − H ( H A + N y M s ) (cid:19) , for sin ϑ = sin ϑ = HH A + N y M s , ϕ = ϕ = π . (17) C. Measured FMR spectra and experimental fitting
FIG. 4. : (a-m) Frequency and field dependent ferromagnetic resonance spectra for CrSiTe . The color maps show the in-planeresonance spectra, while the white squares are the out-of-plane resonant peaks added afterwards. (n) Optical photograph ofthe single crystal used in the resonance experiment. As shown in Fig. 4 and 5, broadband ferromagnetic resonance experiments were carried out on a home-madecoplanar waveguide (CPW) sample rod, which was adapted to the magnetic field and temperature control system ofPhysical Property Measuring System (PPMS, Quantum Design). FMR spectra was recorded with a vector networkanalyzer (ZVA 40, Rohde & Schwarz) in transmission mode (S12) over a frequency range of 1-40 GHz, in response tothe scanning of magnetic field with the rate of 50 Oe per second. The resonance field was determined by Lorentzianfit to the spectra.When the external magnetic field H ext is applied in ab plane, we use Eq. 11 to fit the data above the saturationfield ( H > H A + N y M s ) and Eq. 17 to fit the data below the saturation field ( H < H A + N y M s ). For H = H A + N y M s ,the two equations 11 and 17 get consistent results: FIG. 5. : (a-h) Frequency and field dependent ferromagnetic resonance spectra for CrGeTe . The color maps show the in-planeresonance spectra, while the white squares are the out-of-plane resonant peaks added afterwards. (i) Optical photograph of thesingle crystal used in the resonance experiment. (cid:18) ω res γ (cid:19) = ( H A + N x M s ) N z M s . (18)In order to determine these two free variables γ , M s , we use equations 16 and 18 as two constraints to fit the dataabove the saturated field. Therefore, we can obtain the spectroscopic splitting factor g from the formula γ = gµ B / ~ ,where µ B = e ~ / (2 m e ) is the Bohr magneton.When the external magnetic field H ext is applied in out-of-plane direction, the internal effective field H eff actingthe spins is equal to the sum of the external field H ext , the equivalent anisotropy field H A = 2 K/M s , and thedemagnetizing field H D = − N z M s , whilst the resonant frequency can be determined from the well-known formula ω res γ = H + H A − N z M s . (19)In the paramagnetic regime, usually the demagnetizing effects can be neglected as the M s is small. But for CSTand CGT, obvious resonance peak shifts causes ω res − H curves to not exceed the zero point. Therefore, the g factorcannot be directly fitted by the formula of paramagnetic resonance ω res = γH . In order to eliminate the influenceof the demagnetizing field, it is reasonable to fit the experimental data with equations 16 and 19. Because when H A = 0, M s = 0, and N x = N y = N z are satisfied with a total paramagnetic state, equations 16 and 19 degenerateinto ω res = γH . Therefore, our fitted g factor has excluded the influence of shape anisotropy. III. QUANTUM EXPLANATION OF FLUCTUATIONS INDUCED G-FACTOR ANISOTROPYA. Solution for a general model applicable for FMR and ESR case
Consider a quasi-Heisenberg magnet consisting of similar spins with a common g -tensor. The Hamiltonian for sucha spin system is expressed as [16] H = − X h j,l i J jl S j · S l − X h j,l i S j · K jl · S l − µ B H · g · X j S j − A X j ( S zj ) . (20)Remark of the Hamiltonian (20): (i) the first term is the usual Heisenberg exchanging coupling; (ii) the second termdescribes the magnetic dipole-dipole interaction, where the K is a tensor and can contain non-diagonal elements; (iii)the third term is the Zeeman energy, g is the generalized g -factor which is also a tensor; (iv) the last term correspondsto the additional single-ion anisotropy [17].The main technique to calculate the g -shift, which was developed by Nagata [18]: ~ ω = h [ S − , [ S + , H ]] i h S z i . (21)Here the ω is the frequency of the precession motion. Notice that in (21) it is demanded that the direction of theeffective magnetic field H is along z -axis (or saying, the axis of the precession motion is the z -axis). Substituting theHamiltonian (20) into the formula (21), the frequency of precession can be evaluated one term by one term. Zeeman term
First we evaluate the Zeeman term H Zeeman = − µ B H · g · P j S j . Notice that this term has spatialtranslational symmetry, thus we only need to evaluate the single site, for this purpose we calculate[ S − , [ S + , S z ]] = ~ X − i Y, [ X + i Y, Z ]] = 2 ~ S z . (22)In the calculation above we take S x = ~ X, S y = ~ Y, S z = ~ Z for convenience. Similarly[ S − , [ S + , S x ]] = ~ ( S x − i S y )[ S − , [ S + , S y ]] = i ~ ( S x − i S y ) . (23)The (23) terms are necessary when the off-diagonal g is take into consideration, although in all the Nagata’s articlesthe g was always a diagonal tensor. For the off-diagonal case, we have g , g xx g xy g xz g yx g yy g yz g zx g zy g zz . (24)The external magnetic field vector is H = [ H x , H y , H z ], then according to (22) and (23), − µ B H · g · S = − µ B ( g xx H x + g yx H y + g zx H z ) S x − µ B ( g xy H x + g yy H y + g zy H z ) S y − µ B ( g xz H x + g yz H y + g zz H z ) S z , (25)then the contribution of the Zeeman term can be evaluated by (21) ~ ω Zeeman = h [ S − , [ S + , − µ B H · g · S ]] i h S z i = − µ B ~ ( g xz H x + g yz H y + g zz H z ) − µ B ~ g xx H x + g yx H y + g zx H z ) + i( g xy H x + g yy H y + g zy H z )] h S x − i S y ih S z i (26)Notice that this term is different from the formula (6) in Nagata’s article [16] with extra terms. Because in theHamiltonian, the Zeeman term is negative defined, we have − µ B H z ~ g zz = µ B | gH | , however, the h S x i , h S y i and h S z i in the extra term are hard to be determined. This difficulty comes from the off-diagonal element in the g tensor. The reason is, according to the original derivation in Nagata’s article [18], the formula (21) was arrived withassumption that “ S + and S − are good normal modes” , which seems to be not true if the g is not diagonal.0However, if the magnetic field was tuned above the saturation field in experiments, the inner effective magneticfield is parallel to the external field, then in this case the formula (25) and (26) can be simplified by H x = H y = 0,and moreover h S x i = h S y i = 0 because the magnetic precession is along the z -direction. In this respect we claim ~ ω Zeeman = − µ B ~ g zz H z . (27) Heisenberg term H Heisenberg = − P h j,l i J jl S j · S l describes the isotropy exchanging interaction between nearneighbour spins, and it is invariant under spatial rotations of the frame. The numerator of the (21) is zero. To proofthis, we evaluate[ H Heisenberg , X i S xi + i S yi ]= − ~ X α,i, h j,l i J jl [ σ αj σ αl , σ xi + i σ yi ] = − ~ X α, h j,l i J jl [ σ αj σ αl , σ xj + i σ yj ] + J jl [ σ αj σ αl , σ xl + i σ yl ]= − ~ X α,β, h j,l i (cid:16) J jl (i (cid:15) αxβ σ βj σ αl + i (cid:15) αxβ σ αj σ βl ) − J jl ( (cid:15) αyβ σ βj σ αl + (cid:15) αyβ σ αj σ βl ) (cid:17) = 0 . (28)In (28) the (cid:15) is the antisymmetry tensor satisfying (cid:15) ijk = ijk ) = ( xyz ) or ( yzx ) or ( zxy ) , − ijk ) = ( xzy ) or ( zyx ) or ( yzx ) , ~ ω Heisenberg = 0 . (30) Dipole-dipole interaction term
The dipole-dipole interaction H dipole = − X h j,l i S j · K jl · S l = − X h j,l i ,α,β K αβjl S αj S βl (31)which was proposed in [16] is the generalization of the cases discussed in [17–19]. Here the K is not necessary diagonal,although the diagonal case (XXZ,XYZ model) is regarded to be quite general to describe the spin-spin interactions.The evaluation of the contribution of this term to the frequency is straight forward: substitute H dipole into (21), aftertedious calculation we arrive at the result ~ ω dipole = X h j,l i ,α,β T αβjl h S αj S βl i h S z i , (32)and the form of the tensor T αβjl can be arranged into matrices T jl = − ~ − K xxjl + 2 K zzjl − K xyjl − K xyjl + i K xxjl − i K yyjl − K xzjl − i K yzjl − K xyjl + i K xxjl − i K yyjl − K yyjl + 2 K zzjl + 2i K xyjl − K yzjl + i K xzjl − K xzjl − i K yzjl − K yzjl + i K xzjl K xxjl + 2 K yyjl − K zzjl (33)Remark: When the external field is above the saturaction filed, the precession of the magnetics are along the z -axisand we could argue h S x S y i = h S y S z i = h S z S x i ≈
0, and under this assumption we can safely drop the off-diagonalterms in (32). Then we have ~ ω dipole = X h j,l i ~ h S z i (cid:16) (2 K zzjl − K xxjl − K yyjl ) h S zj S zl i + ( K xxjl − K zzjl ) h S xj S xl i + ( K yyjl − K zzjl ) h S yj S yl i (cid:17) , (34)1this corresponds with [16]. Notice that in the above derivation we ignore the term K xyjl for the isotropy in the crystalplane. From (34) it can be read that ~ ω k dipole = X h j,l i ~ h S z i (2 K zzjl − K xxjl − K yyjl ) h S zj S zl − S xj S xl i ~ ω ⊥ dipole = X h j,l i ~ h S z i ( K xxjl − K zzjl ) h S zj S zl − S xj S xl i (35) Single-ion anisotropy term
The single-ion anisotropy term H ani = A P j ( S zj ) can be evaluated by[ S − , [ S + , X j ( S zj ) ]] = ( ~ X j [ X j − i Y j , [ X j + i Y j , Z j ]] = X j ~ (cid:0) S zj ) − S xj ) − S yj ) (cid:1) [ S − , [ S + , X j ( S xj ) ]] = ( ~ X j [ X j − i Y j , [ X j + i Y j , X j ]] = X j ~ (cid:0) S xj ) − S zj ) (cid:1) (36)Then for the case that the external magnetic field is along z -axis, we arrive ~ ω k ani = ~ A X j h S zj ) − ( S xj ) − ( S yj ) ih S z i (37)and when the external magnetic field is along x -axis ~ ω ⊥ ani = ~ A X j h ( S xj ) − ( S zj ) ih S z i (38) The thermodynamic average
Consider the finite-temperature case β = k B T in which the mean values of theoperators can be evaluated by the partition function Z = e − β H h S z i = Tr[e − βH − βH f S z ]Tr[e − βH − βH f ] ’ Tr[e − βH (1 − βH f ) S z ]Tr[e − βH ] (39)where H f . = gµ B HS z is the external magnetic filed, and H = H + H f . Then we have h S z i = − βgµ B H hh S z S z ii (40)where “ hh ii ” means average in zero filed. Similarly h S zj S zl i = β g µ B H hh ( S z ) S zj S zl ii = X km β g µ B H hh S zk S zm S zj S zl iih S xj S xl i = β g µ B H hh ( S z ) S xj S xl ii = X km β g µ B H hh S zk S zm S xj S xl ii (41)Then according to formula (35) we arrive at the expression∆ ~ ω k dipole = g k µ B H k B T hh S z S z ii X jlkm ( K zzjl − K xxjl ) hh S zj S zl − S xj S xl ) S zk S zm ii ∆ ~ ω ⊥ dipole = g ⊥ µ B H k B T hh S z S z ii X jlkm ( K zzjl − K xxjl ) hh ( − S zj S zl + S xj S xl ) S zk S zm ii (42)2and from the (42) we can obtain directly that ∆ g k dipole g k dipole = − g ⊥ dipole g ⊥ dipole . (43)The similar argument can also arise for the results (37) and (38), it can be proved∆ g k ani g k ani = − g ⊥ ani g ⊥ ani . (44)Because the g -shift comes from the H dipole and H ani , and the other terms in H do not contribute to the ∆ g , thusaccording to the formulas (43) and (44), and we arrive∆ g k g k = − g ⊥ g ⊥ . (45)We further assume that the g factor is isotropic g k = g ⊥ = g , and obtain ∆ g k = − g ⊥ , which is equivalent to g = 13 × g ab + 23 × g c , (46)where g ab = g + ∆ g ⊥ and g c = g + ∆ g k are measured g factor along the ab plane and c axis. B. Solution for the XXZ model with single ion anisotropy
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