Nanomagnets. 1. Rotation Behavior of Vector Chirality - American

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Spin Chirality of Cu3 and V3 Nanomagnets. 1. Rotation Behavior of Vector Chirality, Scalar Chirality, and Magnetization in the Rotating Magnetic Field, Magnetochiral Correlations Moisey I. Belinsky* School of Chemistry, Tel-Aviv University, Tel Aviv, Ramat Aviv 69978, Israel S Supporting Information *

ABSTRACT: The rotation behavior of the vector chirality κ, scalar chirality χ, and magnetization M in the rotating magnetic field H1 is considered for the V3 and Cu3 nanomagnets, in which the Dzialoshinsky−Moriya coupling is active. The polar rotation of the field H1 of the given strength H1 results in the energy spectrum characterized by different vector and scalar chiralities in the ground and excited states. The magnetochiral correlations between the vector and scalar chiralities, energy, and magnetization in the rotating field were considered. Under the uniform polar rotation of the field H1, the ground-state chirality vector κI performs sawtooth oscillations and the magnetization vector MI performs the sawtooth oscillating rotation that is accompanied by the correlated transformation of the scalar chirality χI. This demonstrates the magnetochiral effect of the joint rotation behavior and simultaneous frustrations of the spin chiralities and magnetization in the rotating field, which are governed by the correlation between the chiralities and magnetization.

1. INTRODUCTION The concept of chirality plays a fundamental role in chemistry, physics, and biology.1 In spin clusters, molecular magnets, and cluster-based materials, the spin chiralities are important magnetic characteristics. Frustrated spin trimers, the spin chirality of triangular spin systems, molecular and helical magnets, based on chiral spin trimers, have received great attention in nanomagnetism, coordination and bioinorganic chemistry, molecular magnetism, cluster-based materials, and quantum computing.2−6 The spin chirality of many of these systems is connected with the antisymmetric Dzialoshinsky− Moriya (DM)7,8 exchange coupling. The vector spin chirality (κ) and scalar chirality (χ) are of importance for various magnetic systems,4a,b,e,5 in particular, for the triangular clusters with the DM coupling and compounds comprising spin trimers. In the Cu3 and V3 trimers, the DM coupling determines the zero-field splitting, energy spectrum, anisotropy of the Zeeman splitting, magnetism, electron paramagnetic resonance (EPR), inelastic neutron scattering, and magnetic circular dichroism (MCD) spectra.9−38 A large number of trinuclear Cu3 and V3 complexes9,31−38 were characterized by the large Heisenberg (J) and DM exchange (Dz ∼ 5−67 K, 1 K = 0.695 cm−1) parameters and distortions (for review, see ref 33). The spin frustration is important for these DM trimers and for the Heisenberg (J) equilateral Cu3 complexes.4c,39 The V3 (IV)10,11 and Cu3 (II)12,13 nanomagnets in polyoxometalates were described by the intermediate Heisenberg and DM parameters (J ∼ 4−5 K and Dz,⊥ ∼ 0.5 K). The V3 ring of the V15 singlemolecular magnet14,28−30 is characterized by the small J and Dz,⊥ parameters.30 The V3 and Cu3 DM trimers represent the effective model systems for investigation of the spin chiralities, © XXXX American Chemical Society

frustration, and magnetochiral correlations between the energy, chirality, magnetism, polarization, and spin configurations, as well as the dependence of the chiralities on the magnetic fields, temperature, and distortion, with the aim of possible applications of the chirality. The spin chirality is important for understanding the halfstep quantum magnetization observed in the V310 and Cu312 nanomagnets.10,11a,12 In recent years, possible applications of the scalar chirality (χ± = ±1) and vector chirality (κ± = ±1) of the spin trimers in molecular-based devices and quantum computation have attracted significant interest. The vector chirality of the V3 trimers was proposed for manipulation of the spins.10,11a Applications of the scalar chirality of the Cu312a nanomagnet and other (Si = 1/2) trimers in the molecular-based devices and qubits have been proposed.4b,18,19,21,23,24a,40,41 The scalar chirality in the rotating field was proposed23 for qubit encoding for the Cu312a DM trimer. The possibility of qubits entirely based on only the scalar chirality χ± (orbital angular momentum Lz) of trimers has been proposed for the Heisenberg Cu3 trimer40 and for the triple-quantum-dots molecule.41a Possible applications of the chirality of the DM trimers in molecular-based devices require analysis of the properties of their scalar and vector chiralities, in particular, the field dependence and anisotropy of chirality and the magnetochiral correlations between the magnetism and the vector and scalar chiralities in the rotating magnetic field. The vector chirality plays the key role in multiferroics42−44 and in various magnetic materials comprising the DM Received: September 24, 2015

A

DOI: 10.1021/acs.inorgchem.5b02202 Inorg. Chem. XXXX, XXX, XXX−XXX

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Inorganic Chemistry trimers.4a,b,e,5,45−50 The electric polarization (P) and vector chirality in multiferroics44 was investigated43 by rotating of the direction of the magnetic field H of a given strength. The chirality vector κ in multiferroics can rotate43a,c,e by either +90° or −90° (±90° flop of κ) under the 90° rotation of the magnetic field, which demonstrates magnetic control of the vector chirality.43b,e The behavior of the vector chirality κ of the Cu3 and V3 nanomagnets under rotation of the magnetic field H1 of the given strength H1, as well as control of the vector and scalar chiralities of trimers by changing the direction and strength of the magnetic field, has not been considered and is of interest for practical application of the chirality. The chiralities (mainly the scalar chirality) of the trimers Cu312 and V310 with the DM parameters Dz,⊥ ∼ 0.5 K have been proposed for possible applications.4b,10,11a,12,18,19,21,23 The spin chiralities of the Cu3 complexes31−38 with large exchange parameters (J and Dz) and distortions in the rotating field have not been considered. The behavior of the spin chiralities of these trimers with large DM parameters in the rotating magnetic field is of interest in the context of possible applications of the chirality of the trimers with strong DM Dz coupling in molecular devices. The correlation between the scalar and vector chiralities and magnetization mz in the field Hz∥Z has been proposed.20 The dependence of the correlation between the spin chiralities and magnetization on the field rotation and its impact on the rotation behavior and possible application of the chiralities has not been considered. As will be shown, this correlation has the form χ = 2(κ·M) and governs the joint frustrated rotation behavior of the spin chiralities and magnetization in the rotating field. The goals of the paper are the following: (i) to consider the rotation behavior of the vector and scalar chiralities of the V3 and Cu3 trimers under the rotation of the magnetic field H1, H1 < HLC; (ii) to find the rotation behavior of the spin level, the magnetochiral correlations between the spin chiralities, energies, and magnetizations in the ground and excited states in the rotating field; (iii) to consider the gradual and jump rotation behavior of the spin chiralities and magnetization under the field rotation; (iv) to describe the magnetochiral effect of the joint rotation behavior and simultaneous frustration of the spin chiralities and magnetization in the rotating field. The paper is organized as follows. In section 2, the spin Hamiltonian, energy spectrum in the rotating field, and anisotropy of the chiralities are considered. The dependence of the spin chiralities and magnetization on the field rotation is discussed in section 3. The chiralities and magnetizations of the ground and excited states are considered in sections 4 and 5. Magnetochiral correlations in the rotating field are described in section 6. The joint rotation behavior of the vector and scalar chiralities and magnetization is considered in section 7. Sections 8 and 9 are devoted to the main results and conclusion. Temperature and distortion dependences and frustration of the spin chiralities in the rotating magnetic field H1 and the spin chirality in the tilted field Hζ will be considered in paper 2.51

2. ENERGY SPECTRUM AND CHIRALITY OF THE VR3 TRIMER IN THE LONGITUDINAL, TRANSVERSE, AND ROTATING MAGNETIC FIELDS 2.1. Spin Hamiltonian and Vector and Scalar Chiralities of Spin Levels. The energy spectrum, magnetism, and spectroscopy of the equilateral VR3 trimer are described by the spin Hamiltonian. /1 =

⎛

∑ J ⎜⎝SiSj +

3⎞ ⎟ + D ∑ [S S ] + μ g S H z i j z B i i 4⎠ ij

(1)

The HDVV Hamiltonian H0 = J∑(SiSj + 3/4) (the first term) forms the ground 2(S = 1/2) (E0[2(S=1/2)] = 0) and excited S = 3/2 [E0(S=3/2) = 3J/2] states. The axial DM Dz coupling z HDM = Dz ∑ [Si × Sj]z ij

(2) 9

where ij = 12, 23, 31 (the second DM term in eq 1), results in the zero-field splitting (2Δ) of the frustrated state 2(S = 1/2) (2E term9) on the ground (EI,II) and excited (EIII,IV) S = 1/2 Kramers doublets (Figures 1 and 2), where 2Δ = 2dz and dz = Dz√3/2 for (Si = 1/2)3. The anisotropy of the local gi factors was neglected in the third Zeeman coupling term in eq 1. Parts a and b of Figure 1 plot the dependence of the four S = 1/2

Figure 1. Energy spectrum, vector chiralities κ, scalar chiralities χ, and magnetization M of the frustrated 2(S = 1/2) states of the trimer VR3 under polar rotation of the magnetic field H1 of the strength H1: (a) H1 = 1 T; (b) H1 = 0.5 T. J = 4.8 K, Dz− = −0.5 K, and g = 1.96. B


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Inorganic Chemistry

The components Kn, where n = z, x, and y (or Kv), of the operator K of the vector chirality and the operator C of the scalar chirality (or Cz) for the Si = 1/2 trimer have the form4a,b,e,46,47,50a,b K n = (2/ 3 )

∑

[Si × Sj]n (3)

ij = 12,23,31

C = (4/ 3 )(S1·[S2 × S3])

(4)

The matrix elements of the operators Kn and C, calculated on the basis u±(Mz) [eq A1 in the Supporting Information (SI)], which describes the zero-field (H = 0) DM splitting ±|dz| and the Zeeman linear splitting for Hz∥Z (Figure 2), are the following: u+(−)[Mz+(−)]|K z|u+(−)[Mz+(−)] = κzR = +1,

Figure 2. Frustrated Zeeman (S = 1/2) levels of VR3 , characterized by the spin chiralities, in the longitudinal field Hz∥Z [dashed lines, Mz±(Hz)] and in the transverse field Hx∥X (solid lines).

u+(−)[Mz−(+)]|K z|u+(−)[Mz−(+)] = κzL = −1

(5)

u+(−)[Mz−(+)]|Kx{y}|u+(−)[Mz+(−)] = +( −)1{i}

(6)

u+(Mz)|C|u+(Mz) = χ + = +1, u−(Mz)|C|u−(Mz) = χ − = −1

frustrated states EN (N = I−IV) of the equilateral V3 trimer (below V3) on the rotation of the field H1 of the strength H1 = | 0.5T H1| = 1 T (designated as H1T 1 ) and H1 , respectively, H1 performs the counterclockwise (CCW) polar rotation in the range 0 ≤ ϕ ≤ π (Figure 1a; J = 4.8 K,10 Dz− = −0.5 K, and g = 1.9610). The splitting Δn and the avoided levels crossing at ϕ = 90° for the slightly distorted trimer V̅ 3R (below V̅ 3R ) are shown in Figure 1b. The rotation behavior of the spin chiralities of the excited states differ in Figure 1a (H1T 1 , β1 > 1) and Figure 1b (H0.5T 1 , β1 < 1) because this behavior is determined by the competition between the Zeeman energy (gμBH1) and the DM anisotropy energy (2Δ), which is described by the dimensionless strength β1 = gμBH1/2Δ of the rotating field H1. For the achiral (κ = χ = Lz = 0) equilateral Heisenberg trimers VH3 (J ≠ 0 and Dz,⊥ = 0) with the isotropic g factors, the Zeeman splitting (2h1 = gμBH1) between the two doubly degenerate achiral Zeeman states 2Mn− and 2Mn+ (S = 1/2) is conserved during the rotation of the field H1 (H1) direction, as shown by the horizontal linear dark-yellow graphs EHI,II(2Mn−) and EHIII,IV(2Mn+) in Figure 1a,b.

(7)

Equations 3 and 5 show that the DM Hamiltonian (2) can be represented in the form HzDM = dzKz, where Kz is the operator of the axial vector chirality (n = z in eq 3), and as a result, the spin states of the DM trimer [the eigenvalues of the Hamiltonian /1(Hz)] at H = 0, Hz∥Z are characterized15,17,25 by the vector chirality κRz = +1 {κLz = −1} (eq 5). The operators Sz, Cz = C, and Kz commute with /1(Hz), the eigenfunctions u±(Mz) are characterized by the quantum numbers of the vector chirality (κRz , κLz ) and the scalar chiralities χ±: u−R (Mz −) = |κIRz , χI− , MIz − , u+R (Mz +) = |κIIRz , χII+ , MIIz + L + − u+L(Mz −) = |κIII z , χIII , MIIIz , u−L(Mz +) = |κIVLz , χIV − , MIVz +

(8a)

The R [L] superscript in uR+(−) [uL+(−)] (eq 8a) corresponds to the right-handed (RH) κRz = +1 [left-handed (LH) κLz = −1]

Figure 3. (a) Scheme 1: Ground-state chiralities, magnetization, and the longitudinal local spin configuration of VR3 for Hz∥Z (ϕ = 0). Scheme 2: Ground-state magnetization MI and vector chirality κRI and scalar chirality χI for the canted spin configuration for H1(H1=1 T) (ϕ = π/4). (b) R↑ R Coplanar spin configurations and corresponding axial vector chiralities κR↑ Iz (βx), κIIz (βx) (χ = Lz = 0) of the lowest states of V3 for Hx∥X, Hx = 1 T (see the text). C


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Inorganic Chemistry vector chirality, while the subscripts ± designate the corresponding scalar chiralities χ± = ±1. The components MNz and MNx of the magnetization vector MN of the states are defined, following ref 53, as MNα = ⟨SN1 + SN2 + SN3 ⟩α = ⟨SNα⟩, where α = x and z, MN = ∑imNi , with i = 1, 2, 3, and mNi = ⟨SNi ⟩, with N = I, II, III, IV. The DM Dz coupling (2) (HzDM = dzKz) results in the zero-field splitting (±dz) on the Kramers (S = 1/2) doublets characterized by the κRz and κLz vector chiralities:

χI+ = 2κIRz MIz +(Hz −)

for V3R

χI+ = 2κILzMIz −(Hz +),

χI− = 2κILzMIz +(Hz −)

for V3L (9)

C = 2K zSz

(10)

MIz± = ∑miz±, miz± = ±1/6, with i = 1, 2, 3, and miz = ⟨SIiz⟩ is the local spin expectation value. The magnetochiral connections (eq 9) correspond to the correlation (10) (or Kz = 2CSz) between the operators Kz (eq 3), C (eq 4), and Sz. The correlation (9) takes place17 for each frustrated linear Zeeman state Mz±(Hz) in Figure 2, χN = 2κNzMNz. Equation 9 shows that the scalar chirality χ (at Hz∥Z) cannot be considered without the vector chirality κz, which is determined by Dz (see the discussion in Chapter S1.2 in the SI). Equation 9 is consistent with the connection20 χ ∝ (κz12 + κz23 − κz13)m1z. Figure 3a [scheme 1 (ϕ = 0)] shows the axial magnetization − MIz−(Hz+) = −1/2 and the spin chiralities κR↑ Iz = +1 and χI = −1 R of the V3 ground state for Hz∥Z and the corresponding longitudinal spin-parallel configuration (m↓1zm↓2zm↓3z), miz− = −1/6, with i = 1, 2, 3 and miz∥Z. In scheme 2 for H1 tilted by ϕ = π/4, the ground-state vectors MI and κRI and the local magnetization vectors mi of the canted spin-parallel configuration are tilted by ηI = γI < π/4 and ξi = ηI, respectively, and | mi| = 1/6. 2.3. Spin Chirality in the Transverse Field Hx∥X. In R Figures 2 and S1 in the SI, the energies ε I[II] (β x )

E0R [κzR , χ +(−) , Mz +(−)] = dz , E0L[κzL , χ +(−) , Mz −(+)] = −dz

χI− = 2κIRz MIz −(Hz +),

(8b)

The DM Dz− < 0 coupling forms the ground-state doublet (Mz±) of the RH axial vector chirality κRIz = κRIIz = +1 (eq 8) and the excited-state LH κLIIIz = κLIVz = −1 doublet Mz± (Figures 1 and 2). The Dz− < 0 coupling results in the LH ground-state (κLIz,IIz = −1) and RH (κRIIIz,IVz = +1) excited-state Mz± doublets (eq 8). The equilateral DM trimers with the ground-state doublet of the vector chirality κRz [κLz ] (eq 8) will be designated as VR3 ,CuR3 (Dz−) [VL3 ,CuL3 (Dz+)] because the ground-state κ chirality determines the vector chirality of the cluster. The ground-state vector chirality κRz = +1 (κLz = −1) for the DM triangles with Dz− < 0 (Dz+ > 0) coincides with that for the classical coplanar 120° spin configurations of the spin triangles.4b,c,46,47a,b,48 As shown in refs 4b and 19, the scalar chirality χ± correlates with the orbital angular momentum Lz±: Lz± ∝ χ±τz, where τz is the unit axial vector. The net scalar chirality (the orbital angular momentum Lz = χτz) of the degenerate Mz± doublet κRz [κLz ] in 54 doublet + − eq 8 vanishes at zero field, χdoublet Lz(H=0) = 0, (H=0) = χ + χ = 0, doublet and Mz(H=0) = 0. 2.2. Spin Chirality in the Longitudinal Field Hz∥Z. Figures 2 and 3 plot the Zeeman splitting of the frustrated states of VR3 and the spin arrangements for Hz∥Z and Hx∥X, which show large differences in the energies, spin chiralities, and spin configurations (magnetoanisotropy). The energies [EN(Hz))] of the linear Zeeman states Mz±(Hz), shown by dashed lines in Figure 2, are represented also in the dimensionless form εN(βz) = EN/dz, which linearly depends on the dimensionless longitudinal field βz = gμBHz/2Δ (eq A3 in the SI). In the transverse field Hx⊥Z, the 2-fold degenerate nonlinear Zeeman states of VR3 are shown by the solid lines in Figure 2. The low and high fields βn correspond to the small and large Zeeman splitting 2h1 in comparison with the DM splitting 2Δ, LC respectively, n = z and x, and βHF n < βn . The frustrated linear ± R Zeeman levels (ε[Mz (Hz)]) of V3 (Figure 2) are characterized by the vector chiralities κz and scalar chiralities χ; thus, κRIz(Hz±) = +1 and χI−(Hz+) = −1 [χI+(Hz−) = +1] for the ground state MIz−(Hz+) [MIz+(Hz−)], as shown in Figure S1 in the SI. The field-induced inversion of the scalar chirality χI (LIz = χIτz), χI−(Hz+) → χI+(Hz−), correlates with the change of MIz and κIz conservation (see Chapter S1 in the SI). For VL3 (Dz+), the ground-state chiralities are κLIz = −1 and χI±(Hz±) = ±1, for Hz∥Z. Figures 2 and S1 in the SI and eq 8 show that the correlations between the magnetization and chiralities of the ground state of the trimers VR3 and VL3 for Hz+ and Hz− have the form (Hz < HLC)

(= − 1 + βx 2 ) and εLIII[IV](βx) at Hx∥X nonlinearly depend R↑ (βx) (∥Z) correlates with on βx; the axial vector chirality κIz,IIz R↑ R the energy [κIz (βx) = −1/εI (βx)] and with an increase of the → in-plane magnetization MIx,IIx (βx) (∥X). The transformations of the chiralities and magnetizations for Hx∥X are shown in Figure S1 in the SI. As shown in Figures 2 and SI, the lowest Zeeman states εRI,II(βx) of VR3 at Hx are characterized by (i) the fieldR↑ dependent axial RH κR↑ Iz (βx) = κIIz (βx) chirality (Figure S1 in the SI), (ii) zero scalar chirality χ [Lz], χI(βx) = χII(βx) = 0, [Lz(βx) = 0], MIz,IIz(βx) = 0, (iii) the in-plane total spin (S = 1 → /2) magnetizations MIx→(βx) = MIIx (βx), and (iv) the intermediate spin magnetizations 17 M̲ 12I+(S12 → 1, βx ), M̲ 12II−(S12 → 0, βx ), shown in Figure S1 (Chapter C in the SI). Magnetochiral correlations between the energies, chiralities, magnetizations, and spin configurations of VR3 for Hx∥X in Figures 2, 3b, and S1 in the SI are shown in Chapters S1.3, S1.4, and B in the SI. Figure 3b plots the spin configurations of the lowest frustrated states εRI[II](βx) of VR3 at Hx∥X in Figure 2, which have the schematic form of the field-dependent coplanar spin umbrellas (m1pm2pm3p)RI,II in the XY plane, mI,II iz (βx) = 0. These field-dependent coplanar spin arrangements [with χI,II(Hx) = 0 and MIz,IIz(βx) = 0] are characterized by the axial vector R↑ chiralities κR↑ Iz (βx) and κIIz (βx) (∥Z) (Figures 3b and S1 in the SI) and correlated total spin in-plane magnetizations M→ Ix (βx) 14c and M→ IIx(βx) (∥X), whose magnitude increases nonlinearly LF LF LF → from Mx(βx ) ≈ −βx at low-field βx up to Mx (βx) ≈ −0.5 at high-field βxHF directed along the X axis (hard axis of magnetoanisotropy). At low fields (βLF x ), the planar classical 120° spin arrangements IRa and IIRa of the VR3 states in Figure 3b are characterized LF by the maximum vector chirality: κR↑ Iz,IIz(βx ) ≈ 1, χ(βx) = 0, and LF LF Mx(βx ) ∼ −βx (see the discussion in Chapter S1.4 in the SI). Because Hx = 1 T corresponds to ϕ = π/2 in Figure 1a, the D


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Inorganic Chemistry planar spin configurations at Hx = 1 T are of interest. At Hx = 1 T, βbx ≈ 1.51, the coplanar noncollinear spin configurations IRa and IIRb (Figure 3b) of the states εRI[II] (βbx) are characterized by the equal reduced magnetochiral quantities → b ↑ b [ κ̲ IzR,II z(βx ) ≈ 0.552 , χ(βx) = 0, M̲ Ix ,IIx (βx ) ≈ − 0.417 ] and differ by the quantum numbers of the IS S12 magnetization17 M̲ 12I+(βxb , S12 → 1) ≈ +0.834 and M̲ 12II−(βxb , S12 → 0) ≈ −0.834 , which describe different behavior of the intermediate spin S12(βx) (Figure S1 and Chapter C in the SI). The coplanar spin arrangements for Hx = 1 and 0.5 T, the low and high fields are discussed in Chapter S1.4 in the SI. When the high field (βHF x ) suppresses the DM-induced anisotropy, gμBHx ≫ 2Δ, the spin arrangements tend to the achiral (κ = χ = 0) coplanar spin-parallel configurations (−1/3, −1/3, +1/6)SH12x =1 [Ih(S12=1)] and (0, 0, −1/2)SH12x =0) [IIh(S12=0] (mix∥X; Figure 3b), which possess the maximum in-plane → = −0.5 and are characterized by the magnetization MIx,IIx integral Kambe intermediate spin55 S12 = 1 and S12 = 0, as well as the different maximum intermediate spin magnetizations M̲ 12I+(S12 = 1) = +1 and M̲ 12II−(S12 = 0) = −1. The planar spin-collinear configurations (0, 0, −1/2)I(S12=0) and (−1/3, −1/3, +1/6)II(S12=1) of the states, separated by the small zerofield splitting 2δ ≈ 0.08 K, of the slightly distorted isosceles V3 ring of a V15 molecular magnet were observed56 in the lowtemperature 51V NMR experiment. Because the Zeeman states of the equilateral DM trimers at Hx∥X (ϕ = π/2) are degenerate (Figures 1 and 2), the spin arrangements in Figures 3b and S2 in the SI and chiralities were calculated, following refs 17 and 27, for the DM trimer in Figure 1b with the small rhombic distortions ΔJ = J12 − J, ΔJ = −0.05 K, and |ΔJ| ≪ Dz ≪ J. The ΔJ distortion results in the small splitting of the degenerate Zeeman levels of the equilateral triangle at Hx∥X and has a small influence27 on the chiralities, spin arrangements, and magnetization.

Figure 4. Dependences of the scalar chiralities χI and χII, the components κRIn and κRIIn (n = z, x) and the magnitudes κRI and κRII of the chirality vectors κRI and κRII of states I and II of VR3 on the polar rotation of the field H1, H1 = 0.5 T and Dz− = −0.5 K. The continuous graphs ( χ̲ I , χ̲ II ; κ̲ x , κ̲ z , κ̲ I , κ̲ II of κ̲ I and κ̲ II ) of the spin chiralities of the slightly distorted trimer V̅ 3R are also shown.

the Heisenberg achiral equilateral trimer VH3 in the rotating field is shown in Figures S3a and S4 (Chapter S2.1) in the SI. 3.2. Energy, Chirality, and Magnetization in the Rotating Field. The energies εI(β1,ϕ) and εII(β1,ϕ) (eq 11) of the two lowest S = 1/2 states of VR3 in Figure 1b [εI[II](β1,ϕ) = EI,II(H1,ϕ)/dz] and the eigenfunctions25,52 (eq 13) under the polar rotation of the field H1 (β1 < 1) in the ranges 0 ≤ ϕ ≤ π/ 2 (Δa) and π/2 ≤ ϕ ≤ π (Δb) are the following: εI(Δa ) = εII(Δb) = −Π+,

εI(Δb) = εII(Δa ) = −Π−,

εI[II](β1 , π /2) = − 1 + β12 Π± =

3. DEPENDENCE OF THE CHIRALITY AND MAGNETIZATION ON THE FIELD ROTATION 3.1. Change of the Out-of-Plane κz and In-Plane κx Vector Chirality in the Rotating Field. Figure 4 plots the dependences of the scalar chiralities χI and χII, the κz and κx components and the magnitudes (κRI,II = |κRI,II|) of the chirality vectors κRI and κRII of states I and II of VR3 (Figure 1) on the CCW polar rotation of the field H0.5T 1 , β1 ≈ 0.75. These states are characterized by the RH vector chiralities κRI and κRII for H0.5T because κRIz and κRIIz > 0, κR↑ 1 Iz,IIz(ϕ=0,π) = +1 (Figures 4 and 1b). The scalar chiralities χ−(+) and χ+(−) (orbital angular I II ± ± momentum Lz = χ τz) demonstrate the inversion (switching) in the vicinity of the level crossing at ϕ⊥1 : [χ−I (ϕ⊥− 1 ) = −1] → ⊥+ ⊥ ⊥ [χ+I (ϕ⊥+ 1 ) = +1], ϕ1 = ϕ1 ± δϕ, and χI,II(H1x) = 0 at ϕ1 . Parts a−c of Figure S3 [S4] in the SI show the components and magnitudes of the vectors MI, MII, κRI , and κII, as well as the 0.5T scalar chiralities of VR3 in the rotating field H1T 1 [H1 ] (see the discussion in Chapter S2 in the SI). The rotation of the vectors MI and MII does not depend on the sign of Dz. Figures 4 and S3 and S4 in the SI show that the chiralities and magnetizations in the ground and excited states exhibit jump transformations at ϕ⊥1 under the rotation of H0.5T and H1T 1 1 , in accordance with Figure 1a,b. The rotation behavior of the spin chiralities (κLI , κLII, χI, and χII) of the LH trimer VL3 (Dz+ = 0.5 K) for H0.5T is shown 1 in Figure S5 in the SI. The uniform rotation of the vector M0I of

(11)

1 ± 2β1 cos ϕ + β12

(12)

Φ−[+](β1 , ϕ) = A1+[−]u−R[+](Mz−[+]) − A 2+[−]u−L[+](Mz+[−]) (13)

1 [1 + (1 ± β1 cos ϕ)/Π±] , 2 1 [1 − (1 ± β1 cos ϕ)/Π±] A 2± = 2

A1± =

(14)

εIII[IV] = −εII[I], and EIII[IV] = −EII[I]. In the eigenfunctions ΦI[II] = Φ−[+] (eq 13), the functions uR[L] ± (Mz) are shown in eq A1 in the SI, ΦI(Δa) = ΦII(Δb) and ΦII(Δa) = ΦI(Δb). The dependences of the components κIα [κIIα], with α = z and x, of the chirality vector κRI [κRII] (Figure 4) on the rotation of the field H1 (H0.5T 1 ) in the ranges Δa and Δb have the form κIRz (Δa ) = κIIRz(Δb) = (1 + β1 cos ϕ)/Π+, κIRz (Δb) = κIIRz(Δa ) = (1 − β1 cos ϕ)/Π−, κIx(Δa −) = κIIx(Δb−) = β1 sin ϕ/Π+, κIx(Δb−) = κIIx(Δa −) = −β1 sin ϕ/Π−, m κIRz (ϕ1⊥) = κIIRz(ϕ1⊥) = 1/ 1 + β12 = κImin z = κIz

(15) E


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Inorganic Chemistry where Δa− = 0 ≤ ϕ < π/2 and Δb− = π/2 < ϕ ≤ π and

corresponding graphs MIn and MIIn of VR3 when ΔJ → 0. These graphs (M̲ In, M̲ IIn) of V̅ 3R show the in-plane magnetizations ⊥ → ⊥ M̲ I→ x (ϕ1 ) > M̲ IIx (ϕ1 ) (∥X) in Figures S3c and S4 in the SI at ← (ϕ1⊥)] because M̲ Iz ,IIz (ϕ1⊥) = 0 in eq 17, which ϕ1⊥[H1x corresponds4a,b to zero scalar chiralities χI[II](ϕ⊥1 ) = 0.

κN = κNz 2 + κNx 2 . The axial vector chiralities κRIz and κRIIz demonstrate the kink at ϕ⊥1 (Figures 4 and S3a, S4 in the SI). The in-plane chirality κIx (κIIx) exhibits jump change −(+)⊥ 2β1κm Iz at ϕ1 (Figure 4 and eq 15). The axial chirality vectors R↑ ⊥ [ κ̲ Iz (ϕ1 ) < κ̲ IIRz↑(ϕ1⊥)] of states I and II of V̅ 3R are directed parallel to the Z axis at ϕ⊥1 , κIx, IIx(ϕ⊥1 ) = 0 (Figure S3a and eq 15). The scalar chiralities χI and χII (LIz[IIz] = χI[II]τz) of VR3 in the range Δa− and Δb− in Figure 4a read

χI− [χII+ ](Δa −) = −[+]1,

4. SPIN CHIRALITY OF THE GROUND AND EXCITED STATES UNDER THE FIELD ROTATION 4.1. Rotation Behavior of the Chirality Vectors κI and κII in the Rotating Field. Figures 5 and S6 in the SI plot the

χI+ [χII− ](Δb−) = +[−]1 (16)

The scalar chiralities change the sign at ϕ⊥1 , with χI, II(ϕ⊥1 ) = 0. For V̅ 3R , the scalar chiralities vanish for the nondegenerate states at ϕ⊥1 in eq 16, χI, II(ϕ⊥1 ) = 0, which corresponds to MIz,IIz(ϕ⊥1 ) = 0 in Figure 3b. The continuous graphs κ̲ IRn , κ̲ IIRn , κ̲ IR , κ̲ IIR , with n = z and x, and χ̲ I , χ̲ II in Figure 4 with R κ̲ IRx ,IIx(ϕ1⊥) = 0 (κ I,II ∥Z) and χ̲ I,II (ϕ1⊥) = 0 show the

chiralities of V̅ 3R , which are the results of the small gap Δn(ΔJ) at ϕ⊥1 (avoided level crossing) in Figure 1b. The continuous graphs κ̲ IRn , κ̲ IIRn ( χ̲ I , χ̲ II ) of V̅ 3R in Figures 4 and S3a,b in the SI coincide with the corresponding graphs κRIn, κRIIn (χI, χII) of VR3 in a wide range outside the vicinity of ϕ⊥1 and tend to these graphs when ΔJ → 0. The continuous graph χ̲ I of the nondegenerate ground state of V̅ 3R demonstrates fast inversions (switching) of the scalar chirality (Lz) in the vicinity o f ϕ 1⊥ i n d u c e d b y t h e fi e l d r o t a t i o n χ̲ I− (ϕ1⊥−) ⇄ [ χ̲ I (ϕ1⊥) = 0], [ χ̲ I (ϕ1⊥) = 0] ⇄ χ̲ I+ (ϕ1⊥+)),

Figure 5. Dependence of the canting (rotation) angles γI and ηI of the ground-state chirality vector κRI and magnetization vector MI of VR3 , respectively, on the rotation of the field H1; γI′, ηI′ − H1 = 0.5 T, γI, ηI − H1 = 1 T; γI*, ηI* − H1 = 2 T (see the text). The continuous graphs γI̅ and ηI̅ show the rotation behavior of the vectors κRI and MI, respectively, of the slightly distorted trimer V̅ 3R . The uniform rotation θ0M = ϕ of the vector M0I of the Heisenberg trimer VH3 is also shown.

and χ̲ I− (ϕ1⊥−) ⇄ χ̲ I+ (ϕ1⊥+) (Figures 4 and S3b in the SI). The components MIα (MIIα) (α = x, z) of the 2D magnetization vector MI (MII) in the XZ plane in the ranges Δa and Δb (Figure S4 in the SI) have the form

dependence of the canting (rotation) angle γI of the groundstate chirality vector κRI on the rotation of the field H1. The canting (rotation) angles γI and γII of the 2D vectors κI and κII within the XZ plane in the rotating field are defined by the equation tan γN = κNx/κNz, where N = I and II. Using eq 15 and Figures 4 and S3a in the SI, one obtains the dependence of the canting angle γI [γII] of the 2D vector κRI [κRII] in the XZ plane − − on the rotation of the field (H0.5T 1 ) in the ranges Δa and Δb in the forms

MIz(Δa −) = MIIz(Δb−) = −(1 + β1 cos ϕ)/2Π+, MIIz(Δa −) = MIz(Δb−) = (1 − β1 cos ϕ)/2Π−, MIx(Δa ) = MIIx(Δb) = −β1 sin ϕ/2Π+, MIIx(Δa ) = MIx(Δb) = −β1 sin ϕ/2Π−, MIx(ϕ1⊥) = MIIx(ϕ1⊥) = −β1/2 1 + β12 = −β1κImz /2 (17)

γI[II](Δa −) = −[+] arctan{β1 sin ϕ/(1 + [−]β1 cos ϕ)},

MN = MNz + MNx and MIIIα[IVα] = −MIIα[Iα], where N = I−IV. The out-of-plane magnetization MIz (MIIz) demonstrates ⊥ the sharp change +(−)κm Iz/2 at ϕ1 (eq 17; see Figures S3c and S4 and the discussion in Chapter S2.1 in the SI). The in-plane magnetizations MIx and MIIx demonstrate the kink at ϕ⊥1 (eq 17). The resulting reduced in-plane magnetization vectors ⊥ → ⊥ ⊥ M→ Ix (ϕ1 ) = MIIx(ϕ1 ) (∥X) of the degenerated doublet at ϕ1 are 14c ← ⊥ directed against the in-plane field H1x(ϕ1 ). Equations 15−17 for the crossing branches I and II in Figure 1 result in the correlation κN = 2χNMN (N = I, II) between the chiralities and magnetization. The continuous graphs M̲ In(M̲ IIn), n = z, x, of the vectors M̲ I , M̲ II of V̅ 3R (Figures S3c and S4 in the SI) coincide with the corresponding components MIn (MIIn) of VR3 in a wide range outside the avoided level crossing at ϕ⊥1 and tend toward the 2

2

γI[II](Δb−) = −[+] arctan{[β1 sin ϕ /(1 − [+]β1 cos ϕ)}

(18)

In the denominators (1 ± β1 cos ϕ) in eq 18, the dimensionless term β0z = 1 corresponds to the dimensionless DM anisotropy β0z (in comparison with β1 cos ϕ) and determines the hindered rotation [the difference of the κRI rotation from the uniform rotation θ0M of the M0I vector of VH3 (tan θ0M = tan ϕ)]. Equation 18 corresponds to the crossing of the branches I and II at ϕ⊥1 in Figure 1a. The DM-induced zero-field splitting 2Δ (=2dz = Dz 3 ) between the κR and κL doublets (eq 8) [the energy of the DM magnetoanisotropy] can be estimated, for convenience, by the corresponding axial effective field Hz0 = Dz 3 /gμB. For VR3 (Dz = −0.5 K), eq 18 in the form F


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Inorganic Chemistry of tan γI(Δa−) = h1 sin ϕ/(dz + h1 cos ϕ) = H1 sin ϕ/(H0z + H1 cos ϕ) shows that the DM Dz anisotropy term dz (≈0.433 K) [H0z ≈ 0.662 T] in the denominator corresponds to h1 ≈ 0.653 K [H1 = 1 T]. For the canting angle ϕ = π/2, H1∥X, one has tan γI = β1 = h1/dz = H1/H0z . This axial magnetoanisotropy of the DM (Dz) origin determines the nonuniform rotation behavior of the κ and M vectors in Figures 5−8. For the

κRII perform the opposite oscillations γI′ and γII′ in states I and II with respect to the Z axis within the upper vertical half-plane 1 (ZX )+ (κRIz,IIz > 0; Figures 7b and 8b). 2 For VL3 (Figure S5 in the SI), the LH chirality vector κLI performs the nonuniform oscillations γLI within the lower half1 plane 2 (ZX )− in the rotating field H1, where tan γLI = κLIx/κLIz and γLI = γRI . 4.2. Gradual and Sharp Rotations of the Ground-State Chirality Vector κRI in the Rotating Field. Dependence of the canting angle γI (eq 18) of the ground-state chirality vector κRI of VR3 on the polar rotation of the H1 direction in the range 0 0.5T ≤ ϕ ≤ π is shown in Figure 5 for H1T 1 (γI), H1 (γI′), and 2T H1 (γI*), with H1 < HLC, and in Figure S6 in the SI (H11T). In Figure 5, under the CCW 90° rotation of H1 in the range Δa, the vector κRI performs the gradual CCW hindered rotation γg1(ϕ) up to the maximal left tilt and then the reverse jump CW rotation (flop) − γ̲ 2s by the equal angle, which leads to the total nonuniform oscillation +γ1m − γ̲ 2s = 0 (Figure S6 in the SI). The gradual CCW hindered rotation γg1(ϕ) < ϕ of the vector κRI in the range Δa− occurs up to the maximal left canting ⊥− ⊥− m +γmax 1 (ϕ1 ) at ϕ1 , as shown in Figures 5 and S6 in the SI (γ1 max 1T = γ1 ≈ 56.5° for H1 ). Under the uniform Δb(ϕ) = 90° CCW H1 rotation in the range Δb, the ground-state vector κRI also performs the nonuniform oscillation − γ̲ 3s + γ4max = 0, which includes the jump CW rotation − γ̲ 3s (right flop, γ̲ 3s = γ̲ 2s) and the gradual CCW rotation γg4(ϕ) (see the discussion in Chapter S3.1 in the SI). As a result, under the CCW polar rotation of the field H1 (H1 < HLC) by Δ(ϕ) = π in the Δa + Δb range, the ground-state chirality vector κRI of VR3 performs the complete nonuniform ΓγIκ oscillation (eq 19 and Figure 5), which includes the gradual CCW and jump CW (reverse) rotations:

Figure 6. Sawtooth oscillations γI of the chirality vector κRI and the sawtooth oscillating rotation ηI of the magnetization vector MI under the complete (2π) CCW polar rotation of the field H1(H1T 1 ). The sawtooth oscillations (ηI − θ0M) of the vector MI with respect to the uniform rotation θ0M are also shown. The inset shows the χI− and χI+ plateaus and inversions (switching), χI− ⇄ χI+ at ϕ⊥1 and ϕ⊥2 of the scalar chirality χI± in the rotating field. The continuous γI̅ , ηI̅ , and χ̲ I graphs of V̅ 3R are also shown.

Γ γIκ = γ1g(ϕ) − γ̲ 2s − γ̲ 3s + γ4g(ϕ)

coplanar spin configurations at ϕ⊥1 [H1x(ϕ⊥1 )∥X] with χI,II(ϕ⊥1 ) = 0 (Figure 3b), the reduced out-of-plane vector chiralities ↑ ⊥ ⊥ ( κ̲ IRz ,II z(ϕ1 )|| Z ) correspond to the canting angles γI(ϕ1 ) = 0 ⊥ ⊥ and γII(ϕ1 ) = 0 at ϕ1 in eq 18. The canting angles γI′ and γII′ in Figure 7b (eq 18, H0.5T 1 ) show that under the H0.5T rotation, the chirality vectors κRI and 1

(19)

0.5T The continuous graphs γI̅(ϕ)(H1T 1 ) and γI̅ ′(ϕ)(H1 ) in Figures 5−7 and S6 in the SI demonstrate the nonuniform continuous oscillations of the vector κ̲ IR of V̅ 3R with respect to the Z axis, which include the fast CW rotations of κ̲ IR in the

Figure 7. Dependence of the canting (rotation) angles γI, γII and ηI, ηII of the chirality vectors κI, κII and magnetization vectors MI, MII in the ground and excited states of VR3 , respectively, on the rotation of the field H1: (a) H1 = 1 T; (b) H1 = 0.5 T (see the text and Figure 8a,b). G


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Inorganic Chemistry

Figure 8. Correlated rotation behavior of the chirality vectors κI and κII and magnetization vectors MI and MII of the EI and EII states of VR3 for the 0.5T − ± ± R given canting angles ϕn of the rotating field H1: (a) H1T 1 , β1 ≈ 1.51; (b) H1 , β1 ≈ 0.75, and Dz = −0.5 K. The scalar chiralities χI and χII of V3 , as 0 H well as the magnetization vector MI of V3 , are also shown.

nondegenerate ground state in the vicinity of ϕ⊥1 (Figure S7), where γI̅(ϕ1⊥) = γI̅′(ϕ1⊥) = 0 and κ̲ IR (ϕ1⊥)||Z ( κ̲ IIR (ϕ1⊥)||Z ).

XZ plane under the H1 rotation in the range 0 ≤ ϕ ≤ π for 1T 2T R − H0.5T 1 , H1 , and H1 , respectively, for V3 (Dz = −0.5 K). The nonuniform rotation ηI of the vector MI in the rotating field includes the gradual and sharp rotations (Figures 5−8). Under the CCW Δa(ϕ) = 90° polar rotation of H1 in the range Δa, the vector MI performs the CCW polar rotation ηIa = π/2, which includes the gradual CCW hindered rotation (ηg1 = γg1 < ϕ) up s to ϕ⊥+ 1 and the jump CCW rotation η̲ 2 (flop), Figures 5 and S6 in the SI, the total CCW nonuniform rotation is ηIa = η1m(ϕ) + η̲ 2s = π /2 . Large jump CCW rotations (flops) η̲ 2s of MI are shown in the graphs ηI′ (ηI) [ηI*] in Figure 5; η̲ 2s ≈ 53° (33.5°) [19°] because ηm I ≈ 37° (56.5°) 1T 2T [71°] for H0.5T (H ) [H ]. During the CCW Δb(ϕ) = 90° 1 1 1 rotation of H1 in the range Δb, MI performs the CCW nonuniform rotation ηIb = π/2 (Figure 5), which also includes the jump CCW rotation η̲ 3s and the gradual CCW rotation ηg4; the nonuniform CCW rotation in the range Δ b is ηIb = η̲ 3s + η4max = π /2 (see Figure S6 and the discussion in Chapter S3.2 in the SI). R ⊥− ⊥− The tilted antiparallel vectors [MI(ϕ⊥− 1 ), κI (ϕ1 )] at ϕ1 R ⊥+ and the parallel stacked together vectors [MI(ϕ⊥+ ), κ (ϕ )] at 1 I 1 ϕ⊥+ of the ground state (T = 0 K), shown in the schemes 1 1 K and 2, respectively, in Figures S6 in the SI and 8, demonstrate large CWW flop ( η̲ 2s + η̲ 3s) of the magnetization vector MI

The continuous curves γI̅ and γI̅′ of V̅ 3R coincide with the corresponding graphs γI and γI′ of the equilateral VR3 in the large gradual ranges outside the vicinity of ϕ⊥1 and tend toward γI and γI′, respectively, when ΔJ → 0, which also demonstrates the scheme of the nonuniform oscillation of the vector κRI of VR3 , V̅ 3R in Figures 5 and 6 and eq 19. 4.3. Maximal Gradual Canting of the Chirality Vector κRI under the Field Rotation. Under the polar rotation of the field H1 in the range Δa−, the chirality vector κRI of the ground state (TK = 0 K) of VR3 performs the gradual CCW hindered rotation (γg1 < ϕ) up to the maximum left canting angle ⊥− ⊥− γm 1 (ϕ1 ) at ϕ1 γIm(β1) ≈ arctan β1

(20)

γ 1m

This angle can be significantly smaller than the ⊥− corresponding angle ϕ⊥− (γm 1 1 < ϕ1 ; Figure 5). The reverse R jump CW κI rotation is also large in Figure 5, − γ̲ 2s = −γ1max . ⊥+ The maximum κRI canting at ϕ⊥+ 1 is γI (ϕ1 ) = −arctan β1 (eq s 18), which defines the jump rotation − γ̲ 3 (= − γ̲ 2s). Under the ⊥+ small polar rotation ϕ⊥− 1 → ϕ1 of H1, the total jump rotation s s R (flop) of κI is − γ̲ I = −( γ̲ 2 + γ̲ 3s) in Figure 5. An increase of the strength H1 (β1) results in an increase of the maximal m canting angle γm 1 of the gradual rotation in Figure 5: γ1 ≈ 37° 0.5T 1T (56.5°) [71°] for γI′, H1 , β1 ≈ 0.75 (γI, H1 , β1 ≈ 1.51) [γI*, − H2T 1 , β1 ≈ 3.02] for Dz = −0.5 K. This confirms eq 20 and shows that the axial DM-induced anisotropy determines the 0 canting angle γm 1 because β1 = h1/dz = H1/Hz .

and the simultaneous CW (reverse) flop −( γ̲ 2s + γ̲ 3s) of the ⊥+ chirality vector κRI under the small CCW rotation (ϕ⊥− 1 → ϕ1 ) ⊥ of the field H1 in the vicinity of ϕ1 (level crossing) that represents the frustration of the vectors MI and κRI in the rotating field (Figures 5 and 6). In summary, under the uniform CCW polar rotation of the field H1 (H1 < HLC) in the range Δa + Δb, the nonuniform CCW rotation of the ground-state magnetization vector MI reads

5. NONUNIFORM ROTATION OF THE MAGNETIZATION VECTOR MI IN THE ROTATING FIELD 5.1. Gradual and Sharp Rotations of the Magnetization Vector MI. The canting (rotation) ηI of the groundstate magnetization vector MI is shown in Figure 5 (tan ηI = MIx/MIz; see Figures S3c in the SI (H1T 1 ), Figure S4 in the SI (H0.5T 1 ), and eq 17). In Figure 5, the graphs ηI′, ηI, and ηI* show the polar nonuniform rotations of the 2D vector MI within the

η Γ IM = η1m(ϕ) + η̲ 2s + η̲ 3s + η4m(ϕ) = π

(21)

2T In Figure 5, the graphs ηI′ (H0.5T 1 ) and ηI* (H1 ) also include the gradual and jump CCW rotations. Figure 5 shows that an increase of β1 increases ηI in the range Δa−: ηI′ < ηI < ηI* < θ0M

H


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Inorganic Chemistry = ϕ. In the range Δb−, an increase of β1 decreases ηI, θ0M = ϕ < ηI* < ηI < ηI′. Thus, an increase of β1 tends the graph ηI of the nonuniform MI rotation toward to the graph θ0M of the uniform M0I rotation in the same rotating field H1. The nonuniform rotation ηI of MI depends on β1(H1), whereas the uniform rotation θ0M = ϕ of M0I does not depend on H1 (Figure 5). 5.2. Angle Dependence of the Rotation of the Magnetization Vector MI. Using eq 17, one obtains the angular dependence of the rotation of the vector MI in the 0 ≤ ϕ ≤ π range:

to the Z axis with the triangle sawtooth oscillations along the θ0M linear graph in Figure 6. This sawtooth graph ηI describes the gradual CCW MI rotations ηgI (ϕ) altered by the CCW flops η̲ 2s + η̲ 3s (jump rotations η̲ Is ) at ϕ⊥1 = π/2 and ϕ⊥2 = 3π/2, when H1∥X (Figure 6). This uniform polar 2π rotation of H1 (H1T 1 ) results also in the two simultaneous complete oscillations (with the period π) of the chirality vector κRI , shown by the sawtooth graph γI in Figure 6, which describes the gradual CCW rotations γI(ϕ) altered by the jump CW κRI rotations − γ̲ Is at ϕ⊥1 and ϕ⊥2 , − γ̲ Is = −( γ̲ 2s + γ̲ 3s), during the passage of the field H1 direction through the X axis in the triangle plane (the hard plane of magnetoanisotropy). The sawtooth γI oscillations of the vector κRI occur between the maximum canting angles +γm 1 and − γ̲ 3s (amplitudes of the sawtooth oscillation) with respect to the Z axis within the upper half-plane 1/2(XZ)+ (Figures 5 and 6). The rotation dependence of the scalar chirality χI is described by the plateaus χI− = −1 and χI+ = +1 with the inversions (switching) χI− → χI+ at ϕ⊥1 and χI+ → χI− at ϕ⊥2 , as shown in the inset in Figure 6. The continuous curve χ̲ I in the inset

ηI(ϕ , Δa −) = arctan[h1 sin ϕ/(dz + h1 cos ϕ)] = arctan[β1 sin ϕ/(1 + β1 cos ϕ)] ηI(π /2) = η1g − m(ϕ1⊥−) + η̲ 2s = π /2

(22a) (22b)

ηI(ϕ , Δb−) = π − arctan[h1 sin ϕ/(dz − h1 cos ϕ)] = π − arctan[β1 sin ϕ/(1 − β1 cos ϕ)]

(22c)

η̲ 3s = η̲ 2s. As discussed for the κRI canting γI (eq 18), the axial Dz DM-induced magnetoanisotropy [the term dz (β0z = 1) in the denominator dz ± h1 cos ϕ (1 ± β1 cos ϕ) in eq 22] determines the deviation ηI < θ0M (ηI > θ0M) of the nonuniform MI rotation ηI from the uniform rotation θ0M in the range Δa− (Δb−). For state II, the rotation behavior of the vector MII in Figure 7b reads ηII(ϕ , Δa −) = −arctan[β1 sin ϕ /(1 − β1 cos ϕ)]

(23a)

ηII(π /2) = −η1IIg − m(ϕ1⊥−) − η̲ 2IIs = −π /2

(23b)

shows the change of the scalar chirality of V̅ 3R . The graph (ηI − θ0M) in Figure 6 depicts the two complete sawtooth oscillations (with the period π) of the vector MI within the XZ plane, which were performed with respect to the uniform rotation θ0M linear graph during the complete field rotation. This shows that, under field rotation by Δϕ = 2π, the vector MI performs the complete CCW sawtooth oscillating rotation ηI = 2π, which consists of the uniform rotation θ0M and the simultaneous sawtooth oscillations ηI − θ0M (Figure 6). An increase of β1 leads to the reduction of the amplitude η̲ Is (Figure 5) of the sawtooth oscillations (ηI − θ0M) of MI in Figure 6 [ηI → θ0M when βHF 1 → βLC] but does not influence the uniform rotation θ0M (Figures 5 and 6 and S6 in the SI). This means the reduction of the Dz DM-induced effect of the sawtooth MI oscillations ηI − θ0M (Figure 6) in the non-uniform ηI rotation by an increase of the rotating field strength H1. The sawtooth graphs γI, ηI − θ0M (and ηI) with the opposite jump rotations − γ̲ Is and + η̲ Is of the vectors κRI and MI at ϕ⊥1 and ϕ⊥2 , correlated with the χI inversions, demonstrate the joint frustrations of these quantities of the ground state (TK = 0 K) of the DM trimer in the rotating field. The correlated rotation behavior and frustrations of the magnetochiral quantities κRI , χI, and MI in the rotating field in Figure 6 is governed by the connection χI = 2(κI·MI) (eq 25). Figure 6 with the sawtooth oscillations γI and ηI − θ0M of the κIR and MI vectors, respectively, and with the sawtooth oscillating rotation ηI of MI, which are correlated with the inversions and plateaus of the scalar chirality χI, show the magnetochiral effect of the joint rotation behavior and joint frustrations of the vector and scalar chiralities and magnetization of the Cu3 and V3 DM nanomagnets in the rotating field. This frustrated magnetochiral behavior takes place for the Cu3 and V3 DM trimers under the CCW and CW polar rotations of HI with respect to the Z axis within the XZ plane or any vertical plane perpendicular to the triangle XY plane (the axial symmetry). Sharp changes of the magnetization, such as the jumps and plateaus (TK = 0 K) for a varying magnetic field, are characteristic for the frustrated cluster systems.4d

ηII(ϕ , Δb−) = −π + arctan[β1 sin ϕ /(1 + β1 cos ϕ)] (23c)

β1 < 1. Dependence ηII (ηII′) of the nonuniform rotation of MII on the field rotation is shown in Figure 7b, β1 ≈ 0.75. The rotations ηI (eq 22a) and ηII (eq 23a) of the vectors MI and MII in the Δa− range are consistent with that in refs 25 and 52 and with the αk rotations23 of the local spin vectors ⟨si⟩ of Cu3 within the XZ plane under the field rotation in the range Δa−. ⊥− The maximum angle ηm 1 (ϕ1 ) of the CCW gradual canting of ⊥− the vector MI at ϕ1 has the form

η1m(ϕ1⊥−) ≈ arctan β1

(24)

= π /2 − The canting η1m is determined by the dimensionless field β1 = h1/dz = H1/H0z , as shown in Figures m ⊥− 5, S6 in the SI, and 7b, where ηm 1 = γ1 < ϕ1 . The maximum ⊥+ m ⊥+ canting at ϕ1 is η1 (ϕ1 ) ≈ π − arctan β1, which defines the jump CCW rotation η̲ 3s . Under the small CCW polar field H1 ⊥+ ⊥ rotation ϕ⊥− 1 → ϕ1 in the vicinity of ϕ1 , the total jump CCW s s rotation of the vector MI is η̲ I = ( η̲ 2 + η̲ 3s). 5.3. Simultaneous Rotation Behavior of the MI and κRI Vectors and the Scalar Chirality χI under the Complete Field Rotation. Figure 6 plots the simultaneous rotation behaviors ηI and γIof the ground-state magnetization MI and chirality κRI vectors, respectively, as well as the change of the scalar chirality χI, under the complete (2π) CCW polar rotation of the field H1T 1 . Under this complete polar field H1 rotation, the magnetization vector MI performs the complete nonuniform CCW rotation ηI = 2π within the XZ plane, which is described by the sawtooth graph ηI canted by π/4 with respect η̲ 2s

η1m .

I


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Inorganic Chemistry The continuous graphs γI̅ and ηI̅ in Figure 6 show the continuous very slightly smoothed sawtooth oscillation γI̅ and rotation ηI̅ of the ground-state chirality κ̲ IR and magnetization M̲ I vectors, respectively, of V̅ 3R , and the correlated continuous scalar chirality χ̲ I is shown in the inset. The continuous rotation behavior and frustrations of these quantities

κ̲ IR ,

vector, the Dz magnitude is 1.5 K. The Dz parameter can also be m m found, using the maximum canting angle γm I = ηI (tan γI ≈ β1), ⊥− ⊥ in the vicinity ϕ1 of ϕ1 for the given strength H1, Dz = gμBH1/ √3(tan ηm I ).

6. MAGNETOCHIRAL CORRELATIONS IN THE ROTATING FIELD Equations 15−18 and Figures 4 and S3−S6 in the SI show that the correlations between the vector and scalar chiralities and magnetization in the ground states of VR3 and VL3 in the rotating field H1 (H1 < HLC) have the form of the scalar product of the chirality κR[L] and magnetization MI vectors: I

χ̲ I ,

and M̲ I of the ground state of the slightly distorted trimer V̅ 3R are governed by the correlation χ̲ I = 2( κ̲ IR ·M̲ I) (see the discussion in Chapter S2.2 in the SI). 5.4. Gradual and Sharp Rotations of the Vectors MI and κRI during the 90° Rotation of the Field H1. The ranges of the gradual and sharp rotations of the vectors MI and κRI of the ground state (TK = 0 K) depend on the dimensionless strength β1 of the rotating field H1 (Figure 5). Thus, (i) for the trimer VR3n (Dz = −2.5 K), the polar Δaϕ = 90° rotation of the LF field H1T 1 (β1 ≈ 0.3) in the range Δa results first in the small gradual rotation ηgI ≈ 16° (ηmI ≈ 17°) of the vector MI under the field rotation by Δϕ = 80° (89°) and then in the corresponding large jump CCW rotation η̲ 2s ≈ 73° (flop) of MI during the small field rotation δϕ = 10° (1°) up to Δa (90°), the total rotation of the vector MI is ηIa = η1m + η̲ 2s = 90°;

χI = 2(κIR ·MI),

χI = 2(κIL ·MI)

(25)

The sawtooth oscillations (rotation) behavior and frustration of the vectors κRI and MI, connected with the plateaus and inversions of χI in the rotating field in Figures 5−8, are governed by the magnetochiral correlation (25). The correlation (25) explains the joint nonuniform rotation of the antiparallel [parallel] vectors MI and κRI in the range Δa− [Δb−] (Figures 5−8). Equation 25 in the form χ̲ I = 2( κ̲ IR[L]·M̲ I) describes the correlation for the magnetochiral quantities of the slightly distorted V̅ 3R trimer, which demonstrate the continuous transformations of the rotation angles and magnitudes (Figures 4−8 and S3−S7 in the SI). The magnetochiral correlation (eq 25) is the vector generalization of eq 9 (Hz∥Z) for all directions of the rotating field H1, including the azimuthal (in-plane) field rotation. For each frustrated state of the DM trimers in the rotating field, the chirality−magnetization correlations have the form χN = 2(κN·MN), where MN = ∑imNi , with N = I−IV. Equation 25 shows that the scalar chirality cannot be considered without the vector chirality and cannot be decoupled from the magnetization. Equation 25 is consistent with the connection20 χjkl ∼ ⟨[Sj × Sk]⟩⟨Sl⟩. The magnetochiral correlations (eq 25) correspond to the connection (eq 26) between the vector operators of the chirality K (eq 3) and magnetization S and the operator C (eq 4) of the scalar chirality

⊥ ⊥ M̲ I→ x ,IIx (ϕ1 ) ≈ − 0.14 is the in-plane magnetization at ϕ1 for LF β1 (Figures 5 and S6 in the SI). This scheme of the fast large flop MI is realized also for V̅ 3R . The chirality vector κRI performs the small simultaneous sawtooth oscillation with respect to the R↑ ⊥ Z axis with γm 1 ≈ 17° and κIz,IIz(ϕ1 ) ≈ 0.96 (see the discussion in Chapter S3.3 in the SI). (ii) For VR3m (Dz = −0.1 K), the CCW polar rotation of the HF same field H1T 1 (β1 ≈ 7.5) in the range Δa by the angle Δϕ = 80° (89°) results in the gradual MI rotation up to the large canting angle ηgI ≈ 73° (ηm 1 ≈ 81°; eq 22), with the following CCW small flop η̲ 2s ≈ 9° ( η̲ 2s ≈ π /2 − η1m ) during the field rotation up to Δa(90°) [M̲ Ix→,IIx (ϕ1⊥) ≈ −0.496 at ϕ⊥1 and ⊥ κR↑ Iz,IIz(ϕ1 ) ≈ 0.13]. An increase of β1 increases the gradual rotation and decreases the jump rotation of the vector MI. (iii) Under the polar 90° rotation of H1, the chirality vector κRI of VR3 , which exhibits the sawtooth oscillation γI with respect to the Z axis with the maximum canting γm I < π/2 (Figures 5 and 6), cannot perform the 90° flop, in contrast to multiferroics.43a,c,e The chirality under the in-plane (azimuthal) field rotation is considered in Chapter S4 in the SI. In summary, the DM trimers demonstrate various types of rotation behavior of the vectors MI and κI in the rotating field, depending on β1. 5.5. Application of the Gradual Rotation of the Vectors MI and κRI in the Rotating Field. Usually, the magnitude Dz of the DM parameters of the Cu3 and V3 DM trimers were found in the experiments on the magnetic susceptibility and spin heat capacity, as well as EPR, MCD, and inelastic neutron scattering spectroscopy.9−13,31−38 The Dz induced hindered gradual rotation of the vectors MI and κRI under the field H1 rotation in the range Δa− (eqs 18, 19, 22, and 24 and Figure 5) allows one to propose the method of measurement of the Dz magnitude in the rotating field. Thus, e.g., according to eqs 18 and 22, if the CCW rotation Δϕ = 60° of H1T 1 with respect to the Z axis in the range Δa results in the total CCW hindered polar rotation ηI = γI ≈ 19° of the MI (κI)

C = 2(Κ ·S)

(26)

7. JOINT ROTATION BEHAVIOR OF THE MAGNETOCHIRAL QUANTITIES κI, MI, AND χI AND κII, MII, AND χII Figure 7a plots the comparison of the canting (rotation) angles γI [ηI] and γII [ηII] of the vectors κRI [MI] and κII [MII] of states I and II in the rotating field H1T 1 . The rotation behavior of the vectors κII and MII (γII′ and ηII′; Figure 7b; β1 < 1) significantly differs from that (γII, ηII) in Figure 7a (β1 > 1). The continuous graphs γII̅ , ηII̅ and γII̅ ′, ηII̅ ′ show the continuous fast oscillation (rotation) behavior of the vectors κII and MII of the nondegenerate state II of V̅ 3R at the avoided level crossing at ϕ⊥1 (Figure 8b) in comparison with the corresponding rotations (γI̅ , ηI̅ and γI̅′, ηI̅ ′) of the vectors κI and MI in the ground state (Figures 7 and 8). It should be noted that, because the vectors MI, κI and MII, κII are nonparallel to the rotating magnetic field H1 (Figures 5−8), the nonzero torques on these vectors (τM = [M × H1] and τκ = [κ × H1]) are directed perpendicular to the XZ plane and determine the dynamics (precessions) of the vectors, which are not considered here. J


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Inorganic Chemistry

sawtooth oscillating graph ηI describes the gradual CCW rotations of the vector MI altered by the jump CCW rotations at ϕ⊥1 and ϕ⊥2 . The vector MI performs the sawtooth oscillations (ηI − θ0M) with respect to the uniform rotation graph θ0M with the jump rotations at ϕ⊥1 and ϕ⊥2 , which shows that the CCW polar sawtooth oscillating rotation ηI of MI in the rotating field consists of the CCW uniform rotation θ0M and the sawtooth oscillations (ηI − θ0M; Figure 6). With increasing H1(β1), the DM-induced sawtooth oscillations (ηI − θ0M) of the vector MI are reduced, and, as a result, the nonuniform rotation ηI of the vector MI tends to the uniform rotation θ0M independent of H1. The sawtooth oscillations γI and ηI − θ0M of the vectors κRI and MI and the ηI rotation of the vector MI, correlated with the plateaus χ±I = ±1 and inversions of the scalar chirality χI, show the magnetochiral effect of the joint rotation behavior and joint frustration of the spin chiralities and magnetization in the rotating field. Figures 5−8 show control of the vector and scalar chiralities by the change of the direction and strength of the field under the CCW or CW polar H1 rotation with respect to the Z axis (axial symmetry). The joint rotation behavior and simultaneous frustrations of the quantities χI, κI, and MI in the rotating field are governed by the magnetochiral correlation χI = 2(κI·MI). This rotation behavior, frustration, and anisotropy of the spin chiralities are determined by the Dz DM-induced axial magnetoanisotropy. The gradual hindered rotation of the vectors MI and κRI in the rotating field allows one to find the DM parameter Dz. The small trimer distortions result in continuous transformations of the spin chiralities in the rotating field.

In Figure 8a, the simultaneous rotation behavior of the chirality vectors κI and κII, as well as the magnetization vectors MI and MII, of the states EI,II of VR3 (Figure 1a) in the rotating field H1T 1 is plotted schematically, in accordance with Figure 7a. Figure 8a demonstrates directly how the ground-state chirality vector κRI performs the sawtooth oscillation γI with respect to the Z axis within the upper half-plane 1/2(XZ)+, as shown above, while the vector κII performs the complete (2π) nonuniform rotation γII (Figure 7a) under the H1 rotation by Δϕ = π. The nonuniform rotation behavior of the magnetochiral quantities κ, M, and χ demonstrates essential differences in the ground and excited states in Figures 7a, 8a, and 1a (H1T 1 ) and Figures 7b, 8b, and 1b (H0.5T ). Large jump rotations of the 1 vectors κRI and MI under the small field rotation in the vicinity of ϕ⊥1 discussed above are shown in Figure 8a. The vectors κII and MII also perform simultaneous opposite rotations at ϕ⊥1 . Figure 8b (H0.5T 1 ) demonstrates schematically that the RH chirality vectors κRI and κRII, as well as the magnetization vectors MI andMII, perform the opposite sawtooth oscillations and sawtooth oscillating rotations, respectively, in the rotating field (Figure 7b) that correlate with the corresponding scalar chiralities χI and χII (eq 25). The gap Δn(ϕ⊥1 ) = ERII − ERI at ϕ⊥1 for V̅ 3R in Figure 8b results in the continuous opposite more smoothed oscillation γI̅′ [γII̅ ′] of the vector κ̲ IR [ κ̲ IIR ] and rotation ηI̅ ′ [ηII̅ ′] of MI [MII] in Figure 7b. Figure S8 in the SI plots the rotation behavior of the spin chiralities κLI,II and χI,II and the magnetization MI,II of the LH trimer VL3 (Dz+) for H0.5T 1 .

8. MAIN RESULTS The DM V3 and Cu3 trimers are chiral nanomagnets, where the spin chirality and magnetism are correlated. The polar rotation of the field H1 of the given strength H1 results in the energy spectrum of the equilateral DM VR3 trimer (Figure 1), which demonstrates the nonlinear rotation dependence and the degeneracy at the level crossing and is characterized by the different rotation behaviors of the spin chiralities in the ground and excited states. The spin chiralities depend on the dimensionless strength β1 of the rotating magnetic field H1, which represents the competition between the Zeeman energy and the DM magnetoanisotropy energy, β1 = gμBH1/Dz√3. Frustrated states and spin configurations of the DM trimers are characterized by the anisotropic vector and scalar chiralities (Figures 1−4). The magnetochiral correlations between the energy, vector and scalar chiralities, orbital angular momentum, magnetization, and spin configurations of the RH and LH V3 and Cu3 trimers in the rotating field were found. The groundstate magnetization MI and chirality κRI vectors demonstrate the gradual and jump rotations (flops) during the uniform polar rotation of the H1 field (Figures 5−8). The ranges of the gradual and jump rotations of the vectors MI and κRI depend on β1. Under the complete (2π) CCW polar rotation of the field H1, the ground-state chirality vector κRI of VR3 performs the sawtooth oscillations with respect to the Z axis within the upper half-plane 1/2(XZ)+, which are presented by the sawtooth graph γI in Figures 5−7. This graph γI describes the gradual CCW κRI rotations up to maximal left canting, which are altered by the jump CW rotations at ϕ⊥1 and ϕ⊥2 (Figure 6). Under the complete H1 rotation, the magnetization vector MI performs the simultaneous complete CCW polar nonuniform rotation ηI = 2π within the XZ plane, shown by the sawtooth graph ηI canted by π/4 with respect to the Z axis in Figure 6. This

9. CONCLUSION To summarize, we have investigated the magnetic field dependence of the spin chiralities and magnetization of the V3 and Cu3 chiral nanomagnets. The rotation behavior of the vector and scalar chiralities, orbital angular momentum, and magnetization in the rotating magnetic field H1 of the given strength H1 was considered. Under the uniform polar rotation of the field H1, the ground-state chirality vector performs sawtooth oscillations, which depend on the rotating field strength β1, while the magnetization vector performs the sawtooth oscillating rotation, correlated with the simultaneous change of the scalar chirality. This represents the magnetochiral effect of the joint rotation behavior and simultaneous frustration of the spin chiralities and magnetization in the rotating field. The correlated frustrated rotation behavior of the magnetization and chiralities is determined by the DM-induced axial magnetoanisotropy of the V3 and Cu3 chiral nanomagnets. The rotation behavior of the spin chiralities in the rotating field, including the sawtooth oscillations of the chirality vector and inversions (switching) and plateaus of the scalar chirality, as well as the magnetochiral correlations, provides control of the vector and scalar chiralities by the strength and direction of the rotating magnetic field, which is relevant for the potential applications of the spin chiralities of the frustrated V3 and Cu3 chiral nanomagnets in molecular-based devices.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.inorgchem.5b02202. K


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Additional considerations and details related to the rotation behavior of the spin chiralities and magnetization of the V3 and Cu3 DM trimers in the rotating magnetic field, anisotropy of spin chiralities and spin configurations in the longitudinal and transverse magnetic fields, and intermediate spin magnetization (PDF)

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*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

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