Nanomagnets. 2. Frustration, Temperature, and Distort - American

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Spin Chirality of Cu3 and V3 Nanomagnets. 2. Frustration, Temperature, and Distortion Dependence of Spin Chiralities and Magnetization in the Rotating and Tilted Magnetic Fields Moisey I. Belinsky* School of Chemistry, Tel-Aviv University, Tel Aviv, Ramat Aviv 69978, Israel S Supporting Information *

ABSTRACT: We consider the frustration, magnetochiral correlations, temperature and distortion dependences of the vector and scalar chiralities, magnetization, and orbital angular momentum of the Cu3 and V3 nanomagnets in the rotating magnetic field, as well as the spin chiralities and frustration in the tilted magnetic field, the joint frustrated rotation behavior of the correlated spin chiralities and magnetization. Spin chiralities and magnetization demonstrate strong frustration in the rotating and tilted magnetic fields. An increase of the temperature and trimer distortions results in the reduction of the chiralities and frustration. The equilateral and distorted clusters with large Dzialoshinsky−Moriya (DM) parameters are characterized by the large spin chirality. An increase of the strength of the tilted magnetic field Hζ leads to the inhomogeneous polar rotation of the chirality and magnetization vectors, which depends on the temperature.

1. INTRODUCTION Spin chirality, magnetochiral correlations between chirality, magnetism and spin configurations, spin frustration, field manipulation and applications of chirality attract significant interest in molecular magnetism and magnetism of the chiraland cluster-based compounds.1−8 The exchange trimers V3 and Cu3,9−11,12−19 in which the Dzialoshinsky−Moriya (DM)20,21 coupling is active, demonstrate the DM-induced anisotropy of their magnetic and spectroscopic characteristics.9−11,12−19,22−34 Possible applications of the spin triangle chirality in molecularbased devices and quantum computation have been proposed for the V310a and Cu311a DM trimers2b,10a,b,11a,b,31−33 and other spin trimers.35,36 The properties of spin chiralities in the magnetic field and temperature dependence of the spin chirality are important for application of the chirality of spin trimers. The temperature dependences of the net vector (κ) and scalar (χ) chiralities of the DM trimers in the rotating magnetic field have not been considered. As will be shown, an increase of the temperature strongly and differently influences the vector and scalar chiralities of the DM trimers that should be taken into account when considering the chirality application. The spin chiralities of the Cu3 complexes9,12−19 with large axial DM parameters (Dz) have not been considered and are of interest for possible applications. A large number of Cu3 complexes9,12−19 with large DM parameters (for review, see ref 14), the V310 and Cu311 nanomagnets, and the V3 ring of a V15 single-molecular magnet (SMM)22−24 (for review, see ref 24a) are distorted. The influence of the trimer distortions on the spin chiralities of the DM trimers in the rotating field H1 should be considered because the chiralities of a large number of distorted DM trimers are of interest for possible applications. © XXXX American Chemical Society

Correlations between quantum magnetism and molecular chirality have been observed in molecular-based chiral magnets,37a a chiral trinuclear SMM,37b and chiral single-chain magnets,37c formed by the helically stacked heterometallic triangles and other chiral SMMs and chains.37d−h In this context, magnetochiral correlations between the vector and scalar spin chiralities, energy, magnetism, and spin configurations in the V3 and Cu3 nanomagnets in the rotating and tilted magnetic fields are of interest. Geometric frustrations in the spin trimers,5 tetramers, polynuclear clusters, and spin rings6,37−45 and various magnetic systems7,46,47 have attracted significant interest (for review, see ref 6). Magnetic molecules, which exhibit geometric frustration, demonstrate such interesting phenomena as jumps and plateaus of the magnetization (at TK = 0 K) for a varying magnetic field.6 Various types of frustrations were found in molecular magnetism.6,42 In a quantum-mechanical description of the frustration in discrete trimers,5,43 frustration is the result of the degeneracy of the ground state of the equilateral Heisenberg achiral trimer5 CuH3 at zero magnetic field. The states of the V3 and Cu3 DM trimers are characterized by the spin chirality. Consideration of the joint frustration of the correlated vector and scalar chiralities and magnetization of the equilateral DM trimers in the rotating and tilted magnetic fields, the temperature and distortion dependences of the frustration of the spin chiralities are of interest for the field control and possible applications of the spin chiralities of the trimers. Received: September 24, 2015

A

DOI: 10.1021/acs.inorgchem.5b02204 Inorg. Chem. XXXX, XXX, XXX−XXX

Article

Inorganic Chemistry

where MNz and MNx are the components of the magnetization vector of the state EN(S,H) for the field H(H,ϕ), N = I−IV for the frustrated 2(S = 1/2) states and N = V−VIII for the achiral S = 3/2 states, and ZN = ∑N exp[−EN(S,H,ϕ)/kBTK] is the partition function. The temperature dependences of the components κα (α = x and z) of the chirality vector κ of the trimer and the scalar chirality χ in the magnetic field are described by the equations

The vector chirality in multiferroics was investigated in a sweeping tilted field Hζ; the specific vector chirality behavior and manipulation of the electric polarization and chirality in the slanted fields were observed.48−50 For the DM trimers, an increase of the strength Hζ of the tilted field Hζ leads to the nonlinear Zeeman levels.9,51 The behavior of the vector and scalar chiralities and magnetization of the nonlinear Zeeman levels of these trimers, the magnetochiral correlations and frustration, as well as the temperature dependence of the spin chiralities, under an increase of the strength of the tilted field Hζ, are of interest for possible application of the chiralities of the V3 and Cu3 DM trimers in the tilted magnetic field. In paper 1 (P1),30 the rotation behavior of the spin chiralities and magnetization in the ground and excited states, the magnetochiral effect of the joint rotation behavior and frustration of the spin chiralities κI, χI, and magnetization MI of the ground state of the DM trimers, and the chirality− magnetization correlations were considered. The goals of the present paper are the following: (i) using the results of P1, to consider the temperature dependence of the net vector chirality, magnetization, and scalar chirality of the V3 and Cu3 trimers in the rotating field; (ii) to describe the magnetochiral correlations between the spin chiralities, magnetization, and spin configurations in the rotating field for the equilateral and distorted DM trimers; (iii) to consider the joint frustration of the vector and scalar chiralities, orbital angular momentum, magnetization, and spin arrangements of the DM trimers and its temperature and deformation dependences in the rotating and tilted fields; (iv) to consider the deformation dependence of the spin chiralities in the rotating field and the spin chiralities of the Cu3 complexes with large DM parameters and distortions; (vi) to describe the dependence of the spin chiralities and magnetization of the DM trimer upon an increase of the strength Hζ of the tilted field and its temperature dependence.

κα(H , ϕ , TK ) = (1/ZN ) ∑ κNα(H , ϕ) exp[−EN (S , H , ϕ) N

/kBTK ]

χ (H , ϕ , TK ) = (1/ZN ) ∑ χN (H , ϕ) exp[−EN (S , H , ϕ) N

/kBTK ]

(3)

where κNα(H,ϕ) is the α component of the chirality vector κN and χN(H,ϕ) is the scalar chirality of the frustrated state N (Figure 130); κ = χ = 0 for S = 3/2. Figures 1 and 2 and S1−S3 in the Supporting Information (SI) depict the rotation behavior (for H1T 1 ) of the net vector chirality κR, scalar chirality χ, and magnetization vector M of the equilateral trimer VR3 (below VR3 ) at low temperature TK = 0.1 K, Dz− = −0.5 K. The energy spectra in Figures 1 and 830 determine the temperature dependence of the spin chiralities of the system, as shown in Figures 1−3. At TK = 0.1 K, the two lowest levels, εI and εII, with the level crossing at ϕ⊥1 = 90° (Figure 130) mainly determine the magnetochiral quantities κR, χ, and M of VR3 with the zero-field-splitting Dz√3 ( 0 in Figure S1, which shows that the cluster vector chirality oscillates in the upper half-plane (1/2)(XZ)+ and is determined mainly by the ground-state vector chirality κRI . For VL3 (Dz+), the cluster chirality vector κL [left-handed (LH)] with κLz < 0 oscillates in the low half-plane (1/2)(XZ)−. A continuous graph of the scalar chirality χ at TK = 0.1 K shows fast transformation of χ in the vicinity of ϕ⊥1 , where the cluster scalar chirality, as well as the cluster orbital angular momentum Lz = χτz, vanishes; χ(ϕ⊥1 ) = Lz(ϕ⊥1 ) = 0 (Figure S1). The variable magnitude M of the vector M (Mz and Mx) in the rotating field (Figure S2) demonstrates the decrease of the length M of the vector M in a wide range of rotation with the minimum at ϕ⊥1 , where the reduced cluster magnetization vector is directed parallel to the X axis. Mx(ϕ⊥1 )∥X and Mz(ϕ⊥1 ) = 0 correspond to χ(ϕ⊥1 ) = 0. In Figure 1a, the continuous graphs γκ and ηM show the rotation behavior of the net chirality κR and magnetization M vectors of VR3 , respectively, under the polar uniform rotation of 1T 1T the field H1T 1 , γκ = γκ(H1 ,ϕ,TK), ηM = ηM(H1 ,ϕ,TK), and TK = 0.1 K. The rotation angles γκ and ηM are defined by tan γκ = κRx / κRz and tan ηM = Mx/Mz, respectively. The sawtooth oscillation γI and sawtooth oscillating rotation ηI of the ground-state vectors κRI and MI for H1T 1 (see Figures 5 and 6 in P1) are also shown in Figure 1a. The graphs γκ′ and ηM′ in Figure 1a correspond to H0.5T and TK = 0.1 K. The graphs γκ* and ηM* 1 for the temperature TK* = 0.01 K (H1T 1 ) are very close to the ground-state sawtooth graphs γI and ηI (TK = 0 K) in P1, respectively, with the flop at ϕ⊥1 because only the ground state

2. SPIN CHIRALITIES AND MAGNETIZATION IN THE ROTATING FIELD AT LOW TEMPERATURE 2.1. Smoothed Sawtooth Oscillation of the Chirality Vector κR at Low Temperature. The Hamiltonian of the Cu3 and V3 DM trimers, energy spectra, ground-state vector and scalar chiralities in the magnetic field were considered in P1.30 The spin chirality (κ and χ) of the DM trimers is determined by the ground-state vector chirality κRI for Dz− (κLI for Dz+) and the corresponding scalar chirality χI. The rotation behavior of the two-dimensional (2D) net chirality vector [κR = κR(β1,TK)] and net magnetization vector [M = M(β1,TK)] within the XZ plane, as well as the net scalar chirality χ = χ(β1,TK) of the system at the temperature TK, were calculated, taking into account the temperature population ρN(TK) of the frustrated excited states and different rotation behavior of the chiralities in the ground and excited states in the rotating field H1, H1 < HLC in P1;30 here TK designates the temperature because T is tesla (the rotating field strength H1) in the figures. The components Mα (α = x and z) of the 2D net magnetization vector M as a function of the field and temperature TK are described by the standard expression8a Mα(H , ϕ , TK ) = (1/ZN ) ∑ MNα(H , ϕ) N

exp[−EN (S , H , ϕ)/kBTK ]

(2)

(1) B


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Inorganic Chemistry

Figure 2. Smoothed sawtooth oscillations γκ of the chirality vector κR and smoothed sawtooth oscillating rotation ηM of the magnetization vector M of VR3 under the complete (2π) rotation of the field H1, H1 = 1 T, TK = 0.1 K. The graph ηM − θ0M shows the smoothed sawtooth oscillation of the vector M with respect to the uniform rotation θ0M = ϕ; see the text. The inset shows the smoothed plateaus and inversions of the scalar chirality χ± and the simultaneous change of the magnitudes κR and M of the vectors κR and M, shown in the scheme 1.

Figure 1. (a) Smoothed sawtooth oscillation γκ of the chirality vector κR and smoothed sawtooth polar rotation ηM of the magnetization vector M of VR3 at the temperature TK under uniform polar rotation of the field H1, for H1 = 1 and 0.5 T and TK = 0.1 K (for γκ* and ηM*, H1 = 1 T and TK* = 0.01 K); see the text. (b) Comparison of the smoothed sawtooth rotations (γκ and ηM) of the vectors κR and M at TK = 0.1 K with the pure sawtooth rotations (γI and ηI) of the groundstate vectors κRI and MI. The scheme 1 (2) corresponds to the range Δa (Δb).

is mainly thermally populated at TK*. The uniform rotation θ0M = ϕ of the vector M0I of the trimer VH3 is shown in Figure 1. The graph γκ of the continuous oscillations of the chirality vector κR at TK = 0.1 K in Figure 1a,b shows that the cluster vector κR rotates slower than the ground-state vector κRI in a wide range of the field rotation, γκ(ϕ) < γI(ϕ) (Figure S3). At the same time, the graph ηM of the continuous CCW rotation of the vector M in Figures 1a,b show that the net cluster vector M rotates faster at low-temperature TK than the ground-state vector MI in the range Δ2 and slower in Δ3 because ηM > ηI in the scheme 1, while ηM < ηI in the scheme 2 in Figure 1b (see the discussion in Chapter S1 and Figure S3). As a result, the rotating vectors κR and M are nonparallel in a large range in the vicinity of the level crossing at ϕ⊥1 = π/2 even at lowtemperature TK = 0.1 K (Figure 1b) because γκ < ηM < ϕ in this range. The rotating vectors κR and M are nonparallel to the rotating field H1, including the angle ϕ⊥1 = π/2, where ηM(ϕ⊥1 ) = π/2, γκ(ϕ⊥1 ) = 0, and M∥H1 and M⊥κR (Figure 1a,b). In the range Δ1, the vectors κR and M rotate together being parallel because γκ = ηM coincides with the graph γI = ηI of the hindered

Figure 3. Scalar chirality χTK and magnitude κTK of the chirality vector κRTK of VR3 in the rotating field H1 (H1 = 0.5 T) for the temperatures TK = 0, 0.1, 05, 1, 2, and 5 K and Dz = −0.5 K. The inset shows the smoothed sawtooth oscillations ±γTK of the vector κRTK in the rotating field as the temperature increases.

rotation. In Figure S3, the schemes 1 and 2 explain this rotation behavior γκ and ηM of the vectors κR and M, respectively. The vector κR performs the continuous smoothed sawtooth oscillation γκ during the counterclockwise (CCW) rotation of H1T 1 (Figure 1a,b and S3 and S1). In the range Δa = 90°, this oscillation of κR includes the continuous slow left rotation and fast right (reverse) rotation. The slow CCW rotation (γslow 1α ) takes place under the large field rotation Δϕ up to the maximal left inclination of κR [+γs_max (ϕm)] at ϕm < ϕ⊥1 , designated as γm κ κ (ϕm) (Figure 1b), m which is smaller than the maximal left canting (γmax 1 = γI ) of the ⊥− m m ground-state vector κI at ϕ1 , γκ < γI (Figure 1a,b). Then, fast R clockwise (CW) reverse rotation (−γfast 2̲ β ) takes place up to κ ∥Z C


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Inorganic Chemistry [γκ(ϕ⊥1 ) = 0] at ϕ⊥1 under small field rotation Δϕ′ = π/2 − ϕm, and the maximal angle of the slow left rotation is equal in magnitude to the angle of the fast opposite rotation, γs_max − 1α γf_max = 0 (Figure 1b). The slower rotation γκ takes place when 2̲ β large field rotation Δϕ = ϕm leads to γmax κ (ϕm) < ϕm, while Δϕ′ < γmax corresponds to the faster rotation γκ. The change of the κ length κR of the smoothed sawtooth oscillating vector κR during the H1 rotation is shown in Figure S2. The times of the slow left and fast reverse right angle rotations in the smoothed sawtooth oscillation γκ of the vector κR can be estimated using ϕm on an example of the uniform polar Δa = 90° rotation of the field H1T 1 with the angular velocity49 ω0H = 0.5°/s in this range Δa, which occurs within the time t1 = 180 s. In this case, the time ts1 of the slower CCW rotation (γslow 1α ) under large field rotation Δϕ = ϕm up to the left max maximum tilt [+γmax κ (ϕm)] at ϕm (γκ (ϕm) ≈ 39° < ϕm ≈71°) s is t1 ≈ 142 s (Figure 1b), while the time tf1 of the following f faster reverse (CW) rotation (−γfast 2̲ β ) is t1 ≈ 38 s, for Δϕ′ ≈ 19° max s f < γκ , where t1 + t1 = t1. Under the consequent uniform rotation of H1T 1 in the Δb range, the graph γκ in Figure 1a and the scheme 2 in Figure 1b show the smoothed sawtooth oscillation of the vector κR, which f f includes the CW rotation (−γfast 3̲ β ) within the time t2 = t1 up to the maximal right fast tilt [−γf_max (ϕ )] at ϕ (Figure 1b) and ̲m ̲m 3̲ β s then the slower CCW rotation (γslow ) within the time t = ts1 4α 2 ↑R s f from the maximal right tilt up to κz (π)∥Z, where t2 + t2 = t and m fast |γm 3̲ β| = |γ2̲ β|. The total smoothed sawtooth oscillation is −γ3̲ β + γslow = 0. 4α As a result, the uniform polar rotation of the field (H1T 1 ) in the range 0 ≤ ϕ ≤ π leads to the complete continuous smoothed sawtooth oscillation of the chirality vector κR of VR3 at low temperature, shown in Figures 1a,b and S3, which has the form Γ̅ γκ = γ1sα(ϕ) − γ̲ 2fβ(ϕ) − γ̲ 3fβ(ϕ) + γ4sα(ϕ)

As a result, under the uniform polar CCW rotation of the field H1T 1 in the range 0 ≤ ϕ ≤ π, the vector M of the variable magnitude M performs the continuous smoothed sawtooth CCW rotation (Figure 1a−c), which can be presented in the schematic form Γ̅ ηM(ϕ , TK ) = ηMs (ϕ) + η̲ Mf+(ϕ) + η̲ Mf−(ϕ) + ηMs (ϕ) = π (5)

This sawtooth continuous rotation (eq 5) of the vector M includes the continuous slower CCW and faster CCW rotations with acceleration and deceleration (see Figure S7 in P1). Continuous smoothed sawtooth graphs γκ′ and ηM′ (H0.5T 1 ) and γκ and ηM (H1T 1 ) in Figure 1a show different rotation behavior of the chirality (κR) and magnetization (M) vectors in the wide range of field rotation. The graph γκ′ shows the smaller magnitude of the maximal canting (γm κ ′) of the vector 1T κR in comparison with γm κ (ϕm) for H1 (Figure 1a). The smoothed sawtooth graph ηM′ (H0.5T 1 ) in Figure 1a shows larger deviation from the θ0M = ϕ linear graph than that in the case of the graph ηM (H1T 1 ): β1′ < β1 corresponds to the larger range of the fast sawtooth rotation of the vector M. A decrease of the rotating field strength β1 increases the nonuniform character of the ηM′ rotation. 2.3. Rotation Behavior of the Vectors κR and M under Complete Field Rotation. Figure 2 shows that under complete (2π) uniform polar rotation of the field H1 (H1T 1 ), the chirality vector κR of VR3 at TK = 0.1 K performs two complete continuous oscillations (±γmax κ ) with respect to the Z axis within the upper half-plane (1/2)(XZ)+ (scheme 1 in Figure 2). The ground-state graphs γI, ηI, and ηI − θ0M of the pure sawtooth oscillations (rotation) at TK = 0 K are also shown for comparison (Figure 6 in P1). These continuous oscillations γκ of the vector κR are represented by the smooth sawtooth continuous graph γκ (solid red) in Figure 2 in comparison with the sawtooth graph γI (dashed magenta), which describes the sawtooth oscillations of the ground-state chirality vector κRI (TK = 0 K) discussed in P1. The continuous sawtooth curve γκ shows the fast continuous CW rotations of κR in the vicinity of the level crossing instead of the CW jumps s s R ⊥ ⊥ −γm I̲ = −(γ2̲ + γ3̲ ) of the vector κI at ϕ1 and ϕ2 in Figure 6 in P1; the slow rotations γκ and γI coincide in the large range of field rotation. Figure 2 depicts different intervals of the slower CCW and faster CW continuous rotations of κR with acceleration and deceleration, which depend on the temperature TK and the rotating field strength β1 (Figure 1a,b). The amplitude ±γmax of the oscillations of the vector κR with the κ̲ period π (Figure 2) is smaller in comparison with the amplitude R (−γm I̲ ) of the sawtooth oscillations of the ground-state vector κI ⊥ ⊥ at ϕ1 and ϕ2 (Figure 2), which shows the temperature-induced reduction of the maximum angle amplitude ±γmax of the κ smoothed sawtooth oscillations γκ of the net chirality vector κR of the system. In Figure 2, under the complete rotation of H1 (H1T 1 ), the magnetization vector M performs the complete continuous oscillating rotation ηM = 2π, shown by the continuous smoothed sawtooth oscillating graph ηM (solid blue). The sawtooth continuous graph ηM in Figure 2 shows the smoothed oscillations (with the period π) with respect to the linear graph θ0M = ϕ, inclined at π/4 with respect to the Z axis. A comparison of the smoothed sawtooth oscillating graph ηM of the vector M at low temperature with the sawtooth graph ηI (dash-dotted purple) of the oscillating rotation of the ground-state (TK = 0 K) vector MI shows that an increase of the temperature leads to

(4)

The smoothed sawtooth oscillation (eq 4 and Figure 1a) of the vector κR of the variable κR length, which includes the slower CCW and faster CW rotations, occurs with respect to the Z axis within the upper half-plane (1/2)(XZ)+ with maximal angle canting (amplitude) ± γmax κ . 2.2. Sawtooth Oscillating Rotation of the Magnetization Vector M under Field Rotation. The magnetization vector M of the trimer VR3 at low temperature (TK = 0.1 K) performs the continuous sawtooth polar CCW rotation ηM within the XZ plane under CCW polar rotation of the field H1, simultaneously with the smoothed sawtooth oscillation of the vector κR (Figures 1 and 2). In the range Δa, the vector M of magnitude M ≈ 0.5 (S = 1/2) performs continuous polar CCW slow rotation (ηslow M ) with constant angular velocity (see Figure S7 in the SI for P1), and then the vector M of the decreasing length M (Figure S2) performs fast continuous CCW rotation f+ (η̲fast ̲ (ϕ). As a result, the continuous CCW M ), designated as ηM sawtooth rotation ηM = π/2 in this range consists of slow and fast CCW rotations, ηsM + ηf+ ̲ = π/2 (Figure 1a). In the M subsequent range Δb, the continuous smoothed sawtooth rotation ηM = π/2 of M also includes fast and slow CCW rotations: first, the vector M of the increasing length M (Figure f− S2) performs fast CCW rotation (ηfast ̲ ) designated as (η̲M ), and M then the vector M (M ≈ 0.5) performs slow CCW rotation (ηslow M ) with constant angular velocity. The smoothed sawtooth continuous CCW rotation of the vector M has the form ηf− ̲ + M ηsM = π/2 in this range (Figures 1a−c and S3). D


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Inorganic Chemistry fast continuous rotations of M in the vicinity of ϕ⊥1 and ϕ⊥2 instead of the jumps η̲sI = η̲s2 + η̲s3 of the vector MI at ϕ⊥1 and ϕ⊥2 in the graph ηI. The slow rotations ηM and ηI coincide in the large ranges of field rotation. The graphs ηM − θ0M (solid green) and ηI − θ0M (dash-dot-dotted violet) in Figure 2 show the smoothed sawtooth and pure sawtooth oscillations, which are performed by the vector M (of the variable length M) and the ground-state vector MI, respectively, with respect to the uniform rotation θ0M of the vector M0I of VH3 , during the complete ϕ rotation of the field H1. The graphs ηM − θ0M and θ0M show that, under the complete uniform CCW polar H1 rotation (H1T 1 ), the vector M performs the complicated CCW sawtooth oscillating rotation ηM − θ0M (Figure 6), which consists of the pure uniform rotation θ0M of M (with the period 2π) and the simultaneous smoothed oscillations ηM − θ0M of the sawtooth shape (with the period π) with respect to θ0M. An increase of the temperature does not influence the uniform rotation θ0M but reduces the angle amplitude [±(ηM − θ0M)max] of the continuous sawtooth periodic oscillations, which leads to the rotation ηM being closer to the uniform rotation θ0M. The inset in Figure 2 depicts the simultaneous variation of the lengths κR and M of the sawtooth oscillating vector κR and rotating vector M, respectively, with rotating field H1. Figures 1 and 2 show that the DM trimers VR3 and CuR3 (VL3 and CuL3 ) can act as molecular devices that produce the R L sawtooth oscillations (±γmax κ ) of the chirality vector κTK (κTK) with respect to the Z axis in the upper (1/2)(XZ)+ [lower (1/ 2)(XZ)−] half-plane and the sawtoothed oscillating rotations of the vector M, which accompany the continuous inversions of the scalar chirality, when the applied magnetic field H1 (H1) performs continuous uniform polar rotation. 2.4. Scalar Chirality in the Rotating Field at Low Temperature. Variation of the net scalar chirality χ (orbital angular momentum, Lz = χτz) of VR3 is shown in the inset in Figure 2 (TK = 0.1 K) and in Figure S1. The graph χ (open olive triangles) demonstrates the continuous frustration of the scalar chirality shown in the smoothed plateaus and inversions of the scalar chirality in the rotating field. The graph χ shows also the continuous fast inversions (switching) of the magnitude and sign of χ (Lz) by field rotation in the vicinity of the angles ϕ⊥1 and ϕ⊥2 at TK = 0.1 K [χ− → (χ = 0) → χ+ and χ+ → (χ = 0) → χ−]. The scalar chirality χ (Lz) vanishes at ϕ⊥1,2, χ(ϕ⊥1,2) = Lz(ϕ⊥1,2) = 0, when H1∥X. Control of the scalar chirality by the rotating field at low temperature, shown in Figure 2, provides the possibility of fast continuous inversions (switching) of the scalar chirality (Lz) by the H1 rotation in the vicinity of ϕ⊥1,2: χ− = −1 ⇄ χ(ϕ⊥1 ) = 0, χ(ϕ⊥1 ) = 0 ⇄ χ+ = +1, χ− ⇄ χ(ϕ⊥1 ) = 0 ⇄ χ+, and χ− = −1 ⇄ χ+ = +1. The inversion or switching of the molecular chirality (helicity) is of interest for various areas of chemistry,52 such as supramolecular, molecular, organic and coordination chemistry, and chiral nanotechnology.52e Thus, for example, the molecular sensors, switches, and motors based on inversion of the helicity (chirality) were proposed52d for application. The continuous molecular chirality in transition-metal chemistry was discussed in ref 52g. 2.5. Chirality−Magnetization Correlation under an Increase of the Temperature. The continuous nonlinear change of the scalar chirality χ± of VR3 in the rotating field (Figure 2, inset) is correlated with the smoothed sawtooth oscillations of the chirality vector κR (of the variable length) and with the smoothed sawtooth oscillating rotation of the magnetization vector M of the system at TK = 0.1 K (Figure 2).

For the VR3 and VL3 trimers in the rotating field H1 (H1 < HLC) at the temperature TK, the temperature-dependent rotation behavior of the magnetization M, scalar chirality χ, and corresponding vectors κR and κL in Figures 1−6 and S1−S3, S5, S6 is governed by the correlations χT = 2(κTRK ·M TK), K

χT = 2(κTLK ·M TK) K

(6)

The magnetochiral correlation (eq 6) for the cluster quantities χ, κ, and M is realized for all RH- and LH-frustrated DM trimers during the CCW and CW uniform rotation of the field H1 of a given strength under an increase of the temperature TK. The temperature-dependent rotation behavior of the scalar chirality χ (Lz), vector chirality κ, and magnetization M of the DM trimers is correlated and cannot be considered independently in the chirality applications.

3. TEMPERATURE DEPENDENCE OF SPIN CHIRALITIES IN THE ROTATING FIELD 3.1. Chirality of the DM Trimer V 3R with the Intermediate DM Parameter. The temperature dependence of the spin chirality is different for the trimers with intermediate and large DM parameters. At very low-temperature TK*, the chirality of VR3 is determined by the ground-state chiralities κI (κI,γI) and χI. Figure 3 demonstrates the variation of the scalar chirality χTK [orbital angular momentum Lz(TK)] of the trimer VR3 (Dz− = −0.5 K; 2Δ ≈ 0.9 K) in the rotating field H0.5T for 1 the given temperatures TK = 0, 0.1, 05, 1, 2, and 5 K. Significant reduction of the scalar chirality χTK [Lz(TK)] takes place when TK increases. At relatively high temperature for this trimer in the rotating field of the given strength (H0.5T 1 ), when all frustrated states are thermally populated, the scalar chirality χTK vanishes because of the opposite chiralities within the doublets. This is shown in the total reduction of the scalar chirality at TK = 5 K (Figure 3). For the given temperature TK and H0.5T 1 , the maximal − [+] scalar chirality χTK takes place for the longitudinal fields H↑1z(ϕ = 0) [H↓1z(ϕ = π)] in Figure 3 because the Zeeman splitting is maximal for H1z∥Z (Figure 1b).30 Figure 3 shows that the scalar chirality of the DM trimers with the intermediate DM parameters (Dz ∼ 0.5 K) can be observed (and used in applications) at low temperatures and only in the presence of an external magnetic field (preferably, longitudinal); χTK(H = 0) = 0. Figure 3 also depicts the reduction of the magnitude κTK of the oscillating chirality vector κRTK of VR3 in the rotating field H0.5T at elevated temperature TK. 1 With increasing TK, an increase of the temperature population L L of the excited doublet (εIII and εIV ) of V3R, which is characterized by the LH vector chirality κL (κL↓ z < 0), leads to the partial compensation of the RH vector chirality κRI,II (κR↑ z > 0) of the ground-state doublet (εRI and εRII; Figure 1b30). Because the zero-field DM splitting is relatively small for this DM trimer (2Δ < 1 K), an increase of TK up to TK = 5 K leads to the strong reduction of the magnitude κTK; κ5 K ≪ 1 (Figure 3). The reduced magnitudes κTK do not depend on the field rotation; κ1 K ≈ 0.41 and κ2 K ≈ 0.21. The reduction of the length κTK of the vector κRTK with increasing TK is smaller than the reduction of the corresponding maximal scalar chirality χTK (Figure 3). The inset in Figure 3 depicts the graphs γTK of the smoothed sawtooth oscillations ±γκ of the chirality vector κRTK E


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Inorganic Chemistry in the rotating field for the temperatures TK = 0.1, 05, and 1 K. An increase of the temperature TK results in the strong R reduction of the sawtooth oscillations (±γm TK) of the vector κTK and the shift of the maximum. Thus, the vector κR1 K (κR2 K) m m performs the small oscillations ± γm TK with γ1 K ≈ 1° (γ2 K ≈ m 0.3°), whereas γ0.1 K ≈ 22°. 3.2. Chirality of the DM Trimer Cu3 with the Large DM Parameter in the Rotating Field. A large number of trinuclear complexes Cu312−19 were characterized by the large axial DM parameters (Dz ∼ 5−67 K14). Large DM intervals (2Δ) between the ground εI,II(ϕ) and excited εIII,IV(ϕ) doublets with S = 1/2 in these trimers allow the investigation and use of their chiralities even at relatively higher temperatures (Figure 4). Parts a and b of Figure 4 show the rotation behavior of the chiralities of the trimer CuR3 (Dz− = 5 K; J = 100 K) for TK = 0, 0.1, 05, 1, 2, 5, 10, 20, and 50 K, H1 = 2 T, and β1 ≈ 0.3. An increase of TK results in the reduction of the magnitude κTK

from κ0.1 K ≈ 1 to κ10 K ≈ 0.40 (κ20 K ≈ 0.19) for TK = 10 K (20 K) (Figure 4a). Beginning from TK = 3 K, the reduced magnitudes κTK do not depend on the field rotation (Figure 4a). An increase of TK results also in the reduction of the smoothed R sawtooth oscillations ±γm TK of the vector κTK (Figure 4a), as shown in Figure 4b; thus, γm 10 K ≈ 0.6° for TK = 10 K in comparison with γmax 1 K ≈ 7° for TK = 1 K. An increase of TK leads to the reduction of the scalar chirality χTK [Lz(TK)]. The scalar chirality χTK of this trimer is significant up to TK = 5 K for H1 = 2 T. Completely reduced scalar chirality χ10 K (χ0.1 K ≈ −1) is much smaller than the magnitude κ10 K for TK = 10 K. In Figure 4b, the graphs ηTK show that, with increasing TK, the smoothed sawtooth oscillating rotation ηTK of the vector MTK of this trimer tends toward the linear graph of the uniform rotation θ0M. An increase of TK reduces the smoothed sawtooth oscillations (ηTK − θ0M) of the vector MTK with respect to the uniform rotation θ0M (Figure 2). Thus, the maximal deviation of the vector MTK from θ0M is reduced from |η0.1 K − θ0M|max ≈ 49° for TK = 0.1 K to |η5 K − θ0M|max ≈ 4.6° for TK = 5 K in Figure 4b. For β1 ≈ 0.3, the amplitude of the MTK smoothed sawtooth oscillations (ηTK − θ0M) is significantly larger that for the κRTK 0 m oscillations ±γm TK (at the same TK, |η1 K − θM|max ≈ 21° and γ1 K ≈ 7° for TK = 1 K; Figure 4b). Figure 5 depicts the temperature dependence of the magnitudes κz(Hz)a,b,c of the axial RH vector chirality

Figure 5. Temperature dependence of the scalar chirality χ(TK,Hz)n and magnitude κz(TK,Hz)n of the axial vector chirality at Hz = 0.5 and 1 T for the DM trimers and n = a, b, and c: (a) CuR3 (Daz = 5 K and J = 100 K); (b) VR3 (Dbz = 0.5 K and J = 4.8 K); (c) VR3 (Dcz = −0.1 K and J = 2.5 K).

κRz (TK,Hz) (∥Z) and the negative scalar chirality χ(Hz)a,b,c in the longitudinal magnetic field Hz+∥Z (when the magnitude of the scalar chirality is maximum; Figures 3 and 4) of the given strength H0.5T and H1T z z in the temperature range 0 ≤ TK ≤ 10 K for the following trimers: (a) Cua3 (Daz = −5 K and J = 100 K), (b) Vb3 (Dbz = −0.5 K and J = 4.8 K), and (c) Vc3 (Dcz = −0.1 K and J = 2.5 K). The magnitude κz(H0.5T z )a of the vector chirality significantly exceeds κz(Hz0.5T)b, which, in turn, exceeds κz(H0.5T z )c, at all temperatures. The scalar chiralities χ(Hz)a,b,c are reduced with increasing TK more rapidly than the

Figure 4. (a) Dependence of the scalar chirality χTK and magnitude κTK of the vector chirality κRTK of CuR3 on the field rotation (H1 = 2 T) for the given temperatures TK, Dz = −5 K, and J = 100 K; see the text. (b) Dependence of the smoothed sawtooth oscillations γTK and the smoothed sawtooth rotation ηTK of the chirality vector κRTK and the magnetization vector MTK of CuR3 , respectively, on an increase of the temperature TK in the rotating field. F


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Inorganic Chemistry corresponding vector chiralities κz(Hz)a,b,c, in accordance with 0.5T Figure 4. The scalar chirality χ(H0.5T z )a exceeds χ(Hz )b, which, 0.5T in turn, exceeds χ(Hz )c at all temperatures. The temperature 1T dependences χ(H0.5T z )a and χ(Hz )a show significant increases of the scalar chirality under an increase of Hz, while the vector 1T chiralities κz(H0.5T z )a and κz(Hz )a of the same DM trimer coincide. This consideration and comparison of the chiralities for Hx∥X and Hz∥Z (not shown) demonstrate some preference of the chiralities of the Cu3 trimers with the large Dz parameters in possible applications. Joint rotation behavior of the magnetochiral quantities κTK, χTK [Lz(TK)], and MTK of the DM trimers in the rotating field at temperatures TK in Figures 1−5 is governed by the correlation equation (6). Figures 1−5 and S1−S3 show control of the vector and scalar chiralities of the equilateral DM trimers by variation of the temperature TK, the strength H1, and the ϕ angle of the rotating field H1. Figures 3−5 demonstrate that the temperature-dependent vector chirality κTK and scalar chirality χTK of the trinuclear Cu3 DM complexes with the large exchange parameters (Dz and J) are nonzero at higher temperatures, in comparison with the chiralities of the trimers with the intermediate and small DM parameters Dz (Figures 1−5). This shows some advantage of the chirality of the Cu3 trimers9,12−19 with the large Dz parameters over the chiralities of the trimers with the intermediate and small Dz parameters in the possible applications of the spin chiralities (orbital angular momentum Lz). As shown in P1, the chirality and magnetization, as well as the dimensionless energy (εN) in the rotating field, depend on the dimensionless magnetic field β1 = gμBHz/Dz√3. This shows that, with increasing temperature, the magnetism and chirality of the DM trimer depend on the effective dimensionless temperature T̲ = Tk/dz in eqs 1−3.

Figure 6. Scheme of frustration of the ground-state chirality vector κRI , magnetization vector MI, and scalar chirality χ±I (orbital angular ⊥ momentum L±Iz) of VR3 under the small polar rotations (ϕ⊥± 1 = ϕ1 ± δϕ) of the field H1 with respect to the orientation H1x(ϕ⊥1 )∥X (1), H1 = 1 T, and Dz− = −0.5 K.

vectors κRI and MI are shown. In the scheme 3, the stacked together vectors κRI and MI are tilted by the angles −γs̲ (=−γs3̲ ) and +η̲s (=+ηs3̲ ), respectively, with respect to the Z and X axes. These schemes are shown in comparison with the scheme 1 for ⊥ ⊥ the transverse field β← 1x(ϕ1 )∥X at ϕ1 , in which the degenerate ground state of the equilateral trimer is characterized by the ⊥ reduced axial vector chirality κ̲R↑ Iz,IIz[β1x(ϕ1 )] (∥Z), zero scalar chirality, χI,II(β1x) = LIz,IIz(β1x) = 0, and the in-plane (XY) ⊥ reduced magnetization M→ Ix,IIx[β1x(ϕ1 )] (∥X). In order to avoid ⊥ the problem of degeneracy at ϕ1 , the scheme 1 is considered for the DM trimer with very small distortion, when χI(β1x) = 0 ⊥ R↑ ⊥ ⊥ and M→ Ix [β1x(ϕ1 )]⊥κ̲Iz [β1x(ϕ1 )] at ϕ1 . Figure 6 shows the jump changes of the magnetochiral quantities κI, χI, LIz, and MI ⊥ under the small ±δϕ polar rotations (ϕ⊥± 1 = ϕ1 ± δϕ) of the 1T direction of the field H1 (β1) (H1 , β1 ≈ 1.51) within the XZ ⊥ ⊥ plane with respect to the in-plane (XY) field H← Ix (ϕ1 )∥X at ϕ1 in the scheme 1 in Figure 6. In accordance with Figure 630 in P1, the scheme of frustration of the chirality vector κRI in Figure 6 is the following: In the field rotation scheme 2 ← 1, the small CW ⊥ (right) uniform polar rotation (−δϕ, ϕ⊥− 1 ← ϕ1 ) of the field 1T ← ⊥ H1 (H1 ) direction from HIx (ϕ1 )∥X (1) results in the opposite CCW (left) jump rotation (flop) +γs̲ of the vector κRI within the upper half-plane (1/2)(XZ)+ from the axial orientation ⊥ R ⊥− κ̲R↑ Iz (ϕ1 )∥Z (1) up to the left canting of κI (+γs̲ ) at ϕ1 (2). In the field rotation scheme 1 → 3, the small CCW (left) polar H1 rotation (+δϕ, ϕ⊥1 → ϕ⊥+ 1 ) from the scheme 1 results in the opposite CW (right) polar jump rotation (flop) −γs̲ of the ⊥ R vector κRI from κ̲R↑ Iz (ϕ1 )∥Z (1) up to the right tilt of κI (−γs̲ ) at ⊥+ R ϕ1 (3). The left and right inclined vectors κI (+γs̲ ) and κRI (−γs̲ ) of the ground state (TK = 0 K) are shown by the tilted straight red arrows in the schemes 2 and 3, respectively, in Figure 6. To estimate the magnitude of the vector κRI frustration in Figure 6, H1T 1 , let us consider the CW polar rotation scheme 2 ← 1 of the H1 direction by the small angle −δϕ = −1°, which

4. FRUSTRATION OF THE SPIN CHIRALITIES AND MAGNETIZATION IN THE ROTATING MAGNETIC FIELD 4.1. Frustration of the Chirality Vector κI under Field Rotation. The degenerate frustration of the discrete spin trimers was proposed by Kahn5 on an example of the equilateral Heisenberg achiral trimer CuH3 , where the same spin population is instable with respect to a weak ±δJ perturbation, transforming the equilateral trimer into an isosceles one: small distortions lead to a large variation in the spin population5 at TK = 4.2 K. In Figure 6 of P1,30 it was shown that the sawtooth oscillations of the chirality vector κRI and the sawtooth oscillating rotation of the magnetization vector MI are connected with frustration of the DM trimers in the rotating field. We will consider here frustration of the chirality and magnetization of the DM trimers with respect to the small polar rotation (±δϕ) of the field H1 of a given strength with respect to the X axis, using Figure 2 and the results of P1.30 Figure 6 shows small rotations 2 ← 1 and 1 → 3 of the field H1 for the trimer VR3 (Dz = −0.5 K). In the schemes 2 and 3 in Figure 6, the ground-state chirality κRI and magnetization MI vectors, as well as the scalar chirality χ±I and the axial orbital angular momentum L±Iz, are shown for the field canting angles ⊥+ ⊥ ϕ⊥− 1 and ϕ1 , respectively, in the vicinity of ϕ1 , in accordance with Figures 5b, 6, and 8 in P1. In the scheme 2, the canting angles +γs̲ (=γs2̲ in Figure S6 in P1) and −ηs̲ (=−ηs2̲ ) of the G


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Inorganic Chemistry results in the large opposite CCW jump rotation (flop) R +γs̲ (ϕ⊥− 1 ) ≈ +56° of the vector κI from the scheme 1 into the scheme 2 (eq 22 in P1). On the other hand, the polar CCW H1 rotation scheme 1 → 3 by the same small angle +δϕ = +1° (Figure 6) results in the opposite large polar CW jump rotation R (flop) −γs̲ (ϕ⊥+ 1 ) ≈ −56° of κI from the scheme 1 into the ⊥− scheme 3. Large left +γs̲ (ϕ1 ) and right −γs̲ (ϕ⊥+ 1 ) jump rotations of κRI within the upper half-plane (1/2)(XZ)+ are schematized by the round left and right orange arrows in the schemes 2 and 3, respectively, in Figure 6. This shows that the small uniform polar rotations [−(+)δϕ = 1°] of the field H1 ⊥ direction with respect to the in-plane direction H← 1x(ϕ1 )∥X (1) result in the jump rotations (flops) [+(−)γs̲ ≈ +(−)56°] of the vector κRI with respect to the Z axis (1), which are much larger than the small field rotations, |γs̲ |/δϕ = 56 (Figure 6). The vector κRI demonstrates strong frustration with respect to the small rotations ±δϕ of the field direction relative to the field ⊥ → ⊥ H← 1x(ϕ1 ) [H1x(ϕ2 )] (in the triangle plane, in accordance with Figure 6 of P1). Figure 6 shows large instability of the orientation of the vector κRI (1) with respect to the small polar rotations of H1 in the vicinity of ϕ⊥1 : [(+γs̲ )2 ← (γ0I = 0)1 → (−γs̲ )3]. For the LH chirality trimer VL3 (Dz+) (Figure S830 in P1), under the same small polar rotations −(+)δϕ of the field H1 ⊥ ⊥+ L (ϕ⊥− 1 ← ϕ1 → ϕ1 ) in Figure 6, the LH chirality vector κI performs large jump rotation (flop) +γs̲ (−γs̲ ) within the lower ⊥ half-plane (1/2)(XZ)− from κ̲L↓ Iz (ϕ1 )∥Z (1) in the direction opposite to the field rotation. The vectors κLI are directed opposite to the LH vectors κRI in the schemes 1−3 in Figure 6. The continuous graphs γI̅ and γI̅ ′ of the oscillation of the chirality vector κRI of the slightly distorted trimer V̅ R3 in Figures 6 and 7 in P130 allow consideration of the κRI frustration in V̅ R3 in the rotating field. The rotations 2 ← 1 and 1 → 3 with ±δϕ = ±1° in Figure 6 result in significant fast rotations of the − ⊥+ vector κRI (V̅ R3 ): γI̅ +(ϕ⊥− 1 ) ≈ +24° in the scheme 2 and γI̅ (ϕ1 ) 0 R↑ ⊥ ≈ −24° (3) with respect to γI̅ (ϕ⊥) = 0, κ̲Iz (ϕ1 )∥Z in the scheme 1 in Figure 6, κRIz(ϕ⊥) ≈ 0.56, and κRI (ϕ⊥1 ± ) ≈ 0.61, H1T 1 . This demonstrates significant frustration also for the vector κRI (V̅ R3 ) because γI̅ (ϕ⊥± 1 ) ≫ δϕ. However, fast rotations R R )| ≈ 24°] of κ (V [|γI̅ (ϕ⊥± ̅ 1 I 3 ) are smaller than the large jump rotations [|γs̲ (ϕ⊥1 ± )| ≈ 56°] of the vector κRI of the equilateral trimer VR3 , which shows reduction of the κRI frustration by distortion, γI̅ < γs̲ . The vector κRII of the excited state performs the simultaneous opposite jump that influences the temperature dependence of frustration. As shown in Figures 3−5, an increase of the temperature strongly reduces the magnitudes κTK of the vector κRTK and its oscillations ±γm κ̲ , which means a temperature-induced reduction of the frustration of the chirality vector κRTK. 4.2. Frustration of Magnetization Vector MI in the Rotating Field. Simultaneous frustration of the ground-state magnetization vector MI is shown in Figure 6. Under the small uniform CW polar H1 rotation 2 ← 1 (−δϕ = −1°) in Figure 6, the vector MI performs the polar CW jump rotation (flop) −ηs̲ ⊥ (=−η̲s2) from the in-plane orientation M̲ → Ix (ϕ1 ) (1) to the ⊥− inclined vector MI(−η̲s) at ϕ1 (2) with a large flop magnitude: 0.5T |ηs̲ | = 33.5° (53°) for H1T 1 , β1 ≈ 1.51 (H1 , β1 ≈ 0.75). On the other hand, under the small 1 → 3 CCW polar rotation of H1 (+δϕ = +1°) in Figure 6, the vector MI performs the large polar ⊥ CCW jump rotation +η̲s (=+η̲s3) from M̲ → Ix (ϕ1 )∥X (1) to the ⊥+ inclined vector MI(+η̲s) at ϕ1 (the scheme 3 in Figure 6). The inclined vectors MI(−η̲s) and MI(+η̲s) of the ground state (TK

= 0 K) are schematized by the inclined straight olive arrows in the schemes 2 and 3, respectively, in Figure 6. The jump CCW rotations (flops) ±η̲s of the vector MI are much larger than the small CCW rotations (±δϕ) of the field H1, η̲s ≫ δϕ. This shows large frustration of the vector MI in the rotating field. Frustration (jump rotation) of the vector MI does not depend on the sign of Dz± and is the same for VR3 and VL3 , in contrast to the chirality vectors κRI and κLI of these DM trimers. The considered MI frustration inversely depends on the field strength β1: the decrease of β1 results in an increase of the flop 1T 2T of the vector MI, η̲s(H0.5T 1 ) > ηs̲ (H1 ) > ηs̲ (H1 ). This frustration of the vector MI in the rotating field is directly correlated with the simultaneous frustration of the chirality vector κRI (Figure 6). Under the small CW 2 ← 1 rotation of H1, the large CW flop (−ηs̲ ) of MI is accompanied by the large opposite CCW flop (+γs̲ ) of κRI , with ηs̲ + γs̲ = π/2. Under the small CCW 1 → 3 rotation of H1, large CCW flop (+η̅s) of MI is accompanied by the large opposite CW flop (−ηs̲ ) of κRI , with ηs̲ + γs̲ = π/2 (Figure 6). After these simultaneous jump rotations of the vectors MI and κRI , the vectors κRI (+γs̲ ) and MI(−η̲s) are antiparallel in the scheme 2, while κRI (−γs̲ ) and MI(+̲ ηs̲ ) are parallel in the scheme 3. Continuous graphs η̅I (η̅I′) (Figures 6 and 7 in P130) allow consideration of frustration of the vector MI of V̅ R3 in the 0.5T R rotating field H1T 1 (H1 ). In the slightly distorted trimer V̅ 3 , significant frustration of the vector MI also takes place: small ±δϕ rotations of the field H1 (δϕ = 1°), shown in Figure 6, result in the significant fast rotations +ΔηI̅ and −Δη̅I of the vector MI from M̲ → Ix (ϕ⊥)∥X (1) in the schemes 3 and 2, 1T respectively, in Figure 6: thus, |ηI̅ (ϕ⊥± 1 )| ≈ 12° [20°] for H1 0.5T R [H1 ], with |ΔηI̅ | ≫ δϕ. The MI frustration in V̅ 3 is reduced by the distortions in comparison with that in VR3 , with |ΔηI̅ | < |η̅s|. An increase of the temperature leads to a reduction of frustration of the magnetization vector MTK of the system (Figure 4b). Analogous strong frustration of the correlated quantities κI, MI, and χI takes place under the small polar rotations of H1 ⊥ with respect to the H→ 1x(ϕ2 = 3π/2)∥X (Figure 2). 4.3. Frustration of the Scalar Chirality (Orbital Angular Momentum) In the Rotating Field. Figure 6 depicts frustration of the ground-state scalar chirality χI (orbital angular momentum LIz) correlated with the frustration of the vectors κRI and MI. The scalar chirality χI (LIz) is equal to zero for H1x(ϕ⊥1 )∥X in the scheme 1, χI(ϕ⊥1 ) = LIz(ϕ⊥1 ) = 0 (Figure 6). Small ±δϕ (δϕ = 1°) variations 2 ← 1 and 1 → 3 of the field H1 direction with respect to H1x(ϕ⊥1 )∥X (1), result in the maximal jump change of the magnitude and sign of the ground⊥ state scalar chirality: [χI−(ϕ⊥− 1 ) = −1]2 ← [χI(ϕ1 ) = 0]1 for 2 ⊥ + ⊥+ ← 1, and [χI(ϕ1 ) = 0]1 → [χI (ϕ1 ) = +1]3 for 1 → 3 (Figure 6). The axial orbital angular momentum (LIz) exhibits the ⊥ corresponding sharp change [LIz−(ϕ⊥− 1 ) = −1]2 ← [LIz(ϕ1 ) = 0]1 → [LIz+(ϕ⊥+ ) = +1] , Figure 6. Frustration of the quantities 1 3 χI and LIz in the rotating field manifests itself in the jump inversion of χI and LIz under the small (±δϕ) rotation of the field H1. The sign of χ±I [L±Iz] of the DM trimer is determined2b by the sign of the MIz component, which sharply changes the sign at ϕ⊥1 , χI(ϕ⊥1 ) = 0, and MIz(ϕ⊥1 ) = 0. ⊥+ Small rotation ϕ⊥− 1 ↔ ϕ1 of the field H1 (transitions 2 ↔ 3) ⊥+ leads to the inversions [χI−(ϕ⊥− 1 ) = −1]2 ↔ [χI(ϕ1 ) = +1]3 − ⊥− + ⊥+ and [LIz (ϕ1 ) = −1]2 ↔ [LIz (ϕ1 ) = +1]3. Inversions of the scalar chirality χI (LIz), connected with its frustration in the rotating field, such as [(−1) ↔ (+1) and (−1) ↔ (0); (0) ↔ H


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Inorganic Chemistry (+1), (−1) ↔ (0) ↔ (+1)], are of interest for field control of the scalar chirality (LIz) inversion in molecular devices. Continuous graphs χI̲ (Figures 4, 6, and S3b, S5 in P130) allow us to consider frustration of the scalar chirality of V̅ R3 for 0.5T H1T 1 and H1 . Under the same small (±δϕ = ±1°) rotations 2 → 1 and 1 ← 3 in Figure 6, the scalar chirality χI̲ of the nondegenerate ground state of V̅ R3 is transformed from χI̲ (ϕ⊥) = 0 ⊥+ in the scheme 1 into χI̲ (ϕ⊥− 1 ) ≈ −0.3 and χI̲ (ϕ1 ) ≈ +0.3 in the schemes 2 and 3, respectively. This shows the deformationinduced reduction of frustration of the scalar chirality χ±I̲ (V̅ R3 ) R ⊥± [LIz(ϕ⊥± 1 )] in comparison with that of V3 : |χI̲ (ϕ1 )| ≈ 0.3 < ⊥± |χI(ϕ1 )| = 1. An increase of the temperature strongly decreases frustration of the scalar chirality (Figures 3 and 4a). This consideration shows that the V3 and Cu3 DM trimers are characterized by the strong simultaneous frustration of the vector and scalar chiralities (orbital angular momentum), as well as the magnetization, under polar rotation of the field H1. Joint frustration of the quantities κI, MI, and χI (LIz) is governed by the correlation χI = 2(κI·MI). The considered frustration of the spin chiralities and magnetization of the V3 and Cu3 DM trimers in the rotating field, which is connected with the Dz-induced axial magnetic anisotropy, is maximal in the ground state of the equilateral DM trimers (TK = 0), without the cluster ±δJ distortion, and is reduced by the distortions. Frustration of the quantities κI, χI, and MI of the DM trimer in the rotating field differs from the isotropic frustration of the spin population in the achiral CuH3 trimer induced by the small ±δJ distortion, which does not depend on the magnetic field direction. Frustration of the spin chirality cannot be observed in the achiral Heisenberg trimer (χ = κ = Lz = 0). The considered DM-induced frustration of the magnetization vector MI also cannot be observed in CuH3 , where the vector M0I exhibits the polar uniform rotation θ0M = ϕ together with the rotating field H1. These effects of frustration of the spin chiralities and magnetization with respect to the small rotations ±δϕ of the field direction with respect to the X axis have the axial symmetry for the DM trimers without the in-plane DM coupling (D⊥ = 0): it takes place under the same small polar rotations relative to any axis (⊥Z), lying within the triangle XY plane of the DM trimer (the hard plane of magnetoanisotropy). 4.4. Rotation Behavior of the Chirality under Field Rotation with Respect to the Z Axis. This frustration of the spin chiralities and magnetization with respect to the small rotations ±δϕ of the field H1 direction relative to the triangle plane, shown in Figure 6, is accompanied by the very different rotation behavior of these quantities under the polar field H1 rotation with the respect to the Z axis. As shown in P1, under large uniform rotations (±Δϕ) of the direction of the field R H1(βLF 1 ) (with respect to the Z axis), the vectors κI and MI perform joint small gradual polar oscillations (±ΔγI = ±ΔηI) up to the canting angle ±γm 1 with respect to the Z axis within the upper (1/2)(XZ)+ and lower (1/2)(XZ)− half-plane, respectively, with conservation of the scalar chirality χI− LF (LIz−). Thus, e.g., when the field H1 of the strength H1T 1 (β1 R ≈ 0.1 for V3 with Dz = −7.5 K) is rotated CCW by the large angle +Δϕ = 80° in the XZ plane, the opposite vectors κRI ∥MI perform almost uniform CCW rotation by the small angle +δγI ≈ 5.5°. This shows a large delay in the rotation of κRI and MI. Under the rotations ±δϕ by the small angle δϕ = 1° of H1 with respect to the Z axis, the vectors κRI and MI perform joint rotations (±δγI) relative to the Z axis with very small angle amplitude γI′ < 0.1°, γI′ ≪ δϕ. The origin of this behavior is the

strong axial magnetoanisotropy of the Dz origin in the case of 2Δ ≫ 2h1 and βLF 1 ≪ 1. An increase of β1 increases the amplitude of the uniform polar slow rotations ±γm 1 of the vectors κRI and MI for the same ±Δϕ rotations of the field H1 relative to the Z axis. This effect of the small uniform rotations ±δγI of the vectors κRI and MI with respect to the Z axis under large polar rotations ±Δϕ of the field H1 relative to the Z axis (δγI ≪ Δϕ), with the constant scalar chirality χI, can be considered as the inverse frustration effect because, in the case of frustration, small rotations ±δϕ of H1 with respect to the X axis lead to the large sharp rotations γs̲ and ηs̲ of the vectors κRI and MI with respect to the Z and X axes, respectively, and inversions χI− ⇄ χI+, γs̲ , ηs̲ ≫ δϕ. (As shown in P1, during the 90° rotation of the field, large flop takes place after this small gradual rotation that shows a direct connection between these processes.) As shown in Figure 2, this inverse frustration effect takes place in the large vicinity of the angles ϕ = 0 and ϕ = π (field rotations relative to the Z axis−axial easy-axis of magnetoanisotropy) and has the axial symmetry for D⊥ = 0. This small (hindered) uniform rotation of the vectors κRI and MI under large field rotation can be potentially used in the applications, when, for example, it is necessary to have very slow (precise) manipulation of these vectors without a change of the scalar chirality χI and orbital angular momentum (LIz). 4.5. Application of Frustration of the Chirality in the DM Nanomagnets. This consideration shows that the equilateral (V3 and Cu3) and slightly distorted V̅ R3 chiral DM nanomagnets can be used as molecular devices, which react very differently on the polar rotation of the field H1 in the various ranges of field rotation. (i) In the case of the complete polar rotation of the field, Figure 2 of the present paper and Figure 6 in P1 show the sawtooth oscillations of the ground state (TK = 0 K) chirality vector κRI , the sawtooth oscillating rotation of the vector MI, and the plateaus and jumps for the scalar chirality. In this case, the DM trimer transforms the complete uniform polar rotation of the field H1 in the sawtooth oscillations graphs of these vectors and can be considered as the transducer. (ii) As shown above, the small uniform rotations ±δϕ of the field H1 direction of the intermediate or small strength β1 with respect to the X axis at ϕ⊥1 and ϕ⊥2 (triangle plane) result in large sharp rotations +(−)γs̲ and −(+)η̲s of the vectors κRI and MI with respect to the Z and X axes, respectively (Figure 2 of the present paper and Figure 6 in P1), which is accompanied by fast inversions of χI (LIz). The amplitudes of these sharp rotations are much larger than δϕ, γs̲ ≫ δϕ, and ηs̲ ≫ δϕ (Figure 6). In this case, the axial VR3 chiral equilateral and slightly distorted nanomagnets work as an enhancer (amplifier) of these small uniform field H1 rotations (±δϕ) into the large sharp (fast) rotations of the vectors κRI and MI in the vertical plane (Figure 2). (iii) Under large uniform polar rotations ±Δϕ of the field H1 of the intermediate and small strength β1 with respect to the Z axis, slow polar uniform rotations (±δγI) of the vectors κRI and MI with the small amplitude ±γm I take place with respect to the Z axis, with the constant scalar chirality. In this case, the VR3 chiral nanomagnet works as a reducer of large polar rotations ±Δϕ of the field into small rotations ±δγI of these vectors (κRI and MI) with respect to the Z axis. All processes can be regulated by variation of the rotating field strength β1, the temperature, and distortions. I


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Inorganic Chemistry

Figure 7. Ground-state spin configurations of VR3 in the rotating field H1, H0.5T 1 . The spin configurations and corresponding magnetochiral quantities ⊥+ ⊥ κRI , χI, and MI are shown in the schemes 1 and 3 for the field H1 rotation angles ϕ⊥− 1 and ϕ1 , respectively. The coplanar spin arrangement at ϕ1 are R shown in the scheme 2a,b with the corresponding axial vector chirality κIz, χI = 0, and the in-plane magnetization MIx; see the text.

rotation (ϕ⊥1 → ϕ⊥+ 1 ), this coplanar nonparallel spin arrangement IRb exhibits the 3D flop into the inclined spin-parallel s nonplanar configuration at ϕ⊥+ 1 , which corresponds to the +η3̲ s and −γ3̲ flops of the vectors MI and κI, respectively, in Figure 5 and S6 in P1. Further, in the range Δb−, this spin-parallel nonplanar configuration exhibits the nonuniform gradual CW rotation ξIi = ηI, which coincides with the ηI rotation of the vector MI (for details, see Chapter S2 in the SI). The frustration, nonuniform rotation ηI and oscillation γI of the vectors MI and κRI , respectively, in the rotating field are strongly connected with the rotation behavior of the local vectors m1, m2, and m3 within the rotating spin configuration (m1, m2, m3)RI . As shown in Figure S4a,b in the SI, under the uniform polar H1 rotation, the local vectors m1 and m2 perform the complicated 3D CCW rotation at the S1 and S2 sites, including the 3D transformation into the planar spin configuration with the maximal in-plane canting ζ1 and ζ2, as shown in Figure 7 in the scheme 2a,b. At the same time, the local vector m3 performs very different complicated 2D rotation ξ3, including the CCW and CW rotations with the in-plane direction of m3p at ϕ⊥1 , as shown in the graph ξ3 and in the scheme 2 in Figure 7 (see Chapter S2 in the SI). The nonuniform polar CCW rotation ηI of the 2D magnetization vector MI within the XZ plane (Figure 1) is formed by the different complicated nonuniform 3D rotations of the local vectors m1, m2, and m3: MI = ∑imi, where i = 1−3 (Figure S4a,b in the SI). The chirality and magnetization vectors show strong frustration, as shown in Figure 2, in the previous section, and in Figure 6 in P1 (Figure 6). Figure 7 demonstrates the connection between frustration of the cluster magnetochiral quantities κI, χI, and MI, on the one hand, and the change of the corresponding spin configurations with respect to the small ϕ⊥± 1 = ϕ⊥1 ± δϕ polar rotations of the field in Figure 6, on the another hand. Figure 7 shows the ground-state spin arrangements and corresponding frustrated quantities κI, χI, and MI of V̅ R3 for the field rotation angle ϕ⊥− 1 (3D spin configuration 1),

4.6. Azimuthal Rotation of the In-Plane Field. Under azimuthal uniform φ rotation of the field H⊥1 (H1T 1 ) in the triangle plane, chirality frustration is absent: the scalar chirality of the ground doublet vanishes, χ = Lz = 0, the reduced magnetization vectors MI⊥ and MII⊥ of the ground degenerate state of VR3 rotate uniformly within the XY plane, being opposite to H⊥1 , and the corresponding reduced axial vector chiralities (κR↑ z ∥Z) are conserved. This demonstrates the anisotropy of chirality frustration in the rotating field. The two degenerate states differ by the opposite intermediate spin magnetizations17 II− b M̲ I+ 12(β1⊥, S12 → 1) ≈ +0.834 and M̲ 12 (β1⊥, S12 → 0) ≈ −0.834 for H1T , which describe the behavior of the intermediate spin 1 S12 for the in-plane rotating field β1⊥ (see P1). These intermediate spin quantum numbers are governed by the small ΔJ trimer distortions (see P1).

5. ROTATION AND FRUSTRATION OF LOCAL SPIN CONFIGURATIONS IN THE ROTATING FIELD Local spin configuration (μ1z, μ2z, μ3z)I of the ground state S = 1 /2 of the V3 ring of the V15 molecular magnet was observed53 at low temperature in the NMR experiment; μiz = −gmizμB is the individual spin moment, where i = 1−3. Under polar rotation of the field H1 (H1 < HLC) in the Δa− range, the local parallel magnetization vectors mi = ⟨Si⟩ at the Si sites (i = 1−3) of the ground-state collinear spin arrangement of VR3 , shown in Figure 3a in P1, perform the gradual rotation ξi(ϕ), which coincides with the rotation ηI = γI of the vectors MI and κRI in Figure 1 in this range (see the discussion in Chapter S2 in the SI). The gradual rotation ξi(ϕ) of the spin-parallel nonplanar configuration (m1, m2, m3) occurs up to the maximal canting at ϕ⊥− angle γmax 1 1 shown in Figure 1. Then (under the small ⊥− rotation ϕ1 → ϕ⊥1 ), the jump three-dimensional (3D) rotation of this spin-parallel arrangement takes place in the coplanar nonparallel spin configuration IRb for Hx = 1 T, as shown in Figure 3b in P1, which corresponds to the CCW + ηs2̲ and CW − γs2̲ flops of the vectors MI and κRI , respectively, within the XZ plane (Figure 5 and S6 in P1). Under the subsequent small J


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Inorganic Chemistry ϕ⊥1 (coplanar 2D spin configurations in the scheme 2a,b), and 0.5T ϕ⊥+ 1 [3D configuration in the scheme 3], δϕ = 1°, H1 . The rotation angles ξ1(2), ξ3, ξ1(2), ηI, and γI of the rotating vectors m1, m2, m3, MI(ηI), and κRI (γI), respectively, are also shown (see Figure S4b in the SI). The 3D spin configuration in the scheme 3 at ϕ⊥+ 1 represents the mirror image of the 3D spin configuration in the scheme 1 at ϕ⊥− with respect to the 1 triangle XY plane. Highly distorted configurations in the ⊥+ schemes 1 (ϕ⊥− 1 ) and 3 (ϕ1 ) are characterized by the inclined R chirality vectors κI (1) and κRI (3) (κI ≈ 0.98) with the canting angles γI(1) ≈ +32° and γI(3) ≈ −32°, as shown in the schemes 1 and 3, the reduced opposite scalar chirality (χI(1) ≈ −0.83 and χI(3) ≈ +0.83) and the inclined magnetization vectors MI(1) and MI(3) (MI ≈ 0.44) tilted by ηI(1) ≈ 43° and ηI(3) ≈ 137° (90 ± 47)°, respectively (see Chapter S2 in the SI). For comparison, the ground-state coplanar spin arrangement in the scheme 2a,b in Figure 7 possesses the axial vector chirality κR↑ Iz ≈ 0.79, the in-plane magnetization M→ Ix ≈ −0.31, and zero scalar chirality χI(H1x) = 0. Figure 7 shows the correlations between the distorted local 3D spin configurations and reduced spin chiralities. Under the polar H1 rotation, the nonuniform rotation behavior and frustration of the 2D vectors MI and κI and the scalar chirality χI are determined by the nonuniform 3D rotation behavior and frustration of the spin arrangement (m1, m2, m3)I of the DM trimer.

6. SPIN CHIRALITY OF DISTORTED DM TRIMERS IN THE ROTATING FIELD Possible applications of the chirality of DM trimers in molecular devices have been discussed usually only for the undistorted equilateral trimers, whereas a large number of the Cu3 and V3 trimers are distorted.9−11,12−19 As shown in P1, small rhombic distortion (ΔJ = 0.05 K; ΔJ ≪ Dz) results in the continuous graphs γI and ηI of the slightly smoothed sawtooth oscillations and rotation of the vectors κI and MI of V̅ R3 , respectively, with a fast change at the avoided level crossing (Figures 6, 7, and 8b in P1). Let us consider the dependence of the spin chiralities on the rhombic distortions and the possibility of application of the chirality of Cu3 trimers with large ΔJ distortions in molecular devices. The influence of the rhombic distortion ΔnJ = J13 − J12 of the CuR3 complex with relatively large exchange parameters (Dz− = −5 K, J13 = J23 = J = 100 K, and J12 = J − ΔnJ) on the chiralities κIΔJ and χIΔJ and magnetization MIΔJ in the rotating field (H2T 1 ) is shown in Figure 8a,b for the distortions ΔnJ = 0 K (a), 1 K (b), 2 K (c), 5 K (d), 10 K (e), and 20 K (f). Figure S5 depicts the dependence of the sawtooth oscillations ±γIΔJ of the vector κIΔJ on the ΔnJ distortions (see discussion in Chapter S3 in the SI). Figures 8a and S5 and S6 show that an increase of the distortions ΔnJ strongly changes the scalar chirality χIΔJ (orbital angular momentum LΔJ Iz ) and the magnitude κIΔJ of the groundstate vector chirality κIΔJ. The relatively weak distortions ΔbJ and ΔcJ (in comparison with Dz) result in a small reduction of the magnitude κIΔJ in the vicinity of ϕ⊥1 (Figure 8a). An increase of ΔnJ results in a significant reduction of the length κIΔJ of the vector κIΔJ in comparison with κIa(0) ≈ 1 (see Chapter S3 in the SI). As shown in Figure S5, an increase of the distortions results also in a reduction of the amplitude of the smoothed max sawtooth oscillations ±γIΔJ of the chirality vector κIΔJ. However, even large distortions (ΔeJ and ΔfJ), which significantly exceed the DM parameter Dz, result in the chirality vector κIe (κIf) of the relatively large magnitude κIe ≈ 0.66 (κIf

Figure 8. (a) Dependence of the ground-state scalar chirality χIΔJ and magnitude κIΔJ of the chirality vector κIΔJ of CuR3 in the rotation field H1 (H1 = 2 T) on the distortions ΔnJ = J − J12, Dz− = −5 K, and J = 100 K; see the text. (b) Dependence of the magnitude MIΔnJ of the ground-state magnetization vector MIΔnJ on the ΔnJ distortion. The dependence of the smoothed sawtooth oscillating rotation ηIΔnJ of the vector MIΔnJ on the distortions ΔnJ is shown in an inset.

≈ 0.4), which performs small oscillations ±γIe (±γIf) in the rotating field (Figures 8a and S5). The deformation behavior of the chirality is determined29 by the dimensionless deformation parameter δ = ΔnJ/Dz 3 , thus the deformation-reduced vector 2 chirality of the ground doublet is κR↑ I,z,IIz(0,π) = 1/ 1 + δ for ϕ

2 ⊥ ⊥ = 0 and κR↑ I,z[IIz](ϕ1 = 1/ 1 + (δ + [−]β1) ) for ϕ1 = π/2. The change of the chiralities κIΔJ and χIΔJ does not depend on the sign of ΔnJ. Figure 8a shows that the ground-state chirality χIΔJ in the rotating field is also reduced when ΔnJ increases. For trimers with small ΔbJ distortions, the scalar chirality χIb is close to −1 [+1] in a wide range of rotation. For trimers with large ΔnJ distortions, the scalar chirality χIΔJ is equal in magnitude with κIΔJ only at ϕ = 0 and π, and then χIΔJ is significantly smaller than the magnitude κIΔJ. Figure 8a shows that even DM trimers with relatively large distortions (ΔnJ ≥ |Dz|) are characterized by the relatively large {intermediate} magnitudes κIΔJ of the

K


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Inorganic Chemistry

Figure 9. Energy spectrum and spin chiralities of VR3 with the distortion ΔJ = 0.5 K equal in magnitude with the DM parameter Dz− = −0.5 K in the rotating field H1 (H0.5T 1 ), J = 4.8 K.

chirality vectors and the corresponding scalar chiralities χIΔJ, excluding the vicinity of ϕ⊥1 , χI(ϕ⊥1 ) = 0. Figures 8a and S5, which show control of the magnitude κIΔJ and oscillation angle γIΔJ of the chirality vector κIΔJ, as well as the magnitude and sign of the scalar chirality χIΔJ (LIz), by field rotation and cluster deformation, demonstrate that the distorted Cu3 trimers9,12−19 with large exchange parameters Dz, J and distortions ΔnJ possess large and intermediate spin chiralities κIΔJ and χIΔJ (orbital angular momentum LIz) in the nondegenerate ground state. This shows the possibility of application of the spin chiralities of these distorted Cu3 DM trimers in molecular devices. Figure 8b depicts the dependence of the magnitude MIΔJ of the ground state magnetization vector MIΔJ of CuR3 (Figure 8b) in the rotating field H2T 1 for the same ΔnJ distortions. An increase of the distortion ΔnJ reduces the DM action and results in the tendency of the MIΔJ magnitude toward the constant magnitude M0I = 0.5 of the vector M0I of the distorted Heisenberg trimer CuH3 , which experiences the uniform rotation θ0M = ϕ. The inset in Figure 8b shows the deformation (ΔnJ) dependence of the CCW smoothed sawtooth oscillating rotation ηIΔJ of the vector MIΔJ in the rotating field in comparison with the θ0M graph of uniform rotation. As shown in Figure 2, during the uniform CCW polar rotation of the field H1, the vector MIΔJ participates in the smoothed sawtooth oscillating rotation ηIΔJ, which includes the uniform CCW rotation θ0M and the continuous smoothed sawtooth oscillations ηIΔJ − θ0M. An increase of the distortions ΔnJ does not influence the uniform θ0M rotation but reduces the oscillations ηIΔJ − θ0M with respect to the θ0M graph. This shows the tendency of the nonuniform rotation ηIΔJ toward the Heisenberg θ0M type with increasing distortions, which takes place together with the tendency of MIΔJ toward the constant magnitude M0I , when ΔnJ ≫ Dz (Figure 8b). Application of the trimer chirality (mostly scalar chirality) in molecular devices has been proposed for the Cu311a and V310a nanomagnets (Dz ∼ 0.5 K) in the equilateral Heisenberg model. However, these trimers possess significant distortions of the same order as Dz.10a,11a The behavior of the spin chiralities, particularly the scalar chirality, of these distorted DM trimers in the rotating field under an increase of the temperature is of

interest for understanding the conditions of possible application of their chirality. Figure 9 shows the energy spectrum in the R rotating field H1 (H0.5T 1 ) of V3 (J = 4.8 K) with the distortion ΔJ = 0.5 K of the same magnitude as the DM parameter Dz− = −0.5 K, which results in the significant gaps Δn′ between the ground E1 and excited E2 states (E3 and E4) at ϕ⊥1 (Figure 9). The scalar chiralities χN and the examples of the chirality are shown for all frustrated states. Thus, the initial vectors κR[L] N deformation-reduced chiralities of the states I, II at ϕ = 0 in Figure 9 are κR↑ Iz,IIz(0) ≈ +0.87, χI[II](0) = −[+]0.87 for MIz[IIz](0) = −[+]0.5, while at ϕ⊥1 = π/2 one obtains χI,II(ϕ⊥1 ) = ⊥ → ⊥ R↑ ⊥ 0, κR↑ Iz (ϕ1 ) ≈ +0.60, MIx (ϕ1 ) ≈ −0.40 and κIIz (ϕ1 ) ≈ +0.985, → ⊥ MIIx(ϕ1 ) ≈ −0.10 (see discussion in Chapter S3). This shows large spin chiralities of the DM trimers with distortion equal to Dz, ΔnJ = |Dz|(δ = 1/ 3 ). Figure 9 shows that even in the DM trimers VR3 with the relatively large distortion ΔJ = Dz the vectors κIΔJ and MIΔJ perform the oscillations and rotation, respectively, in the separated ground state, as shown in Figure 9. The temperature dependence of the scalar and vector chiralities of this distorted trimer is shown in Figure S6 (see discussion in Chapter S3 in the SI). The rotation behavior of the chiralities and magnetization in the ground state of the significantly distorted DM trimer V̅ R3 in the rotating field (Figures 9 and S6) is governed by the magnetochiral correlation χIΔJ = 2(κRIΔJ·MIΔJ). The chirality−magnetization connection for V̅ R3 with increasing TK also has the form χΔJ,TK = 2(κΔJ,TK·MΔJ,TK). The chirality of the equilateral V3R and distorted V̅ R3 trimers (Dz ∼ 0.5 K) can be observed (and used) at low temperatures (Figures 3, 5, and S6), whereas the chirality of the equilateral and distorted CuR3 trimers with large parameters Dz, J. and ΔJ distortions (Figure 8) can be observed (and used) at higher temperatures.

7. SPIN CHIRALITY AND MAGNETIZATION IN THE TILTED MAGNETIC FIELD Hζ 7.1. Rotation of the Vectors κRI and MI under an Increase of the Strength Hζ of the Tilted Field. The Zeeman levels εI−εIV of the CuL3 trimer (Dz+ = 0.53 K and J = 4.2 K) in the tilted field Hζ are shown in Figure 10; the solid and dashed graphs correspond to the given tilts ζa = π/4 and ζb L


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Inorganic Chemistry

βζ ≥ 0, 0 ≤ ζn < π/2, α = x and z, κI(βζ) = 1, MI(βζ) = 0.5, MI = ∑mi, mi(βζ) = 1/6, i = 1−3, and m1∥m2∥m3 in the scheme 1 in Figure 11, tan γI = κIx/κIzL, tan ηI = MIx/MIz. In eqs 7−11,

Figure 10. Zeeman spin levels and spin chiralities of the trimer CuL3 (Dz+ = 0.5 K and J = 4.2 K) in the field Hζ tilted at the given angles ζa = π/4 (solid) and ζb = π/3 (dashed) in the scheme 1. The groundstate level crossing S = 3/2 → S = 1/2 under an increase of the strength Hζ(βζ) of the tilted field Hζ is shown at the level crossing field HLC ζ .

= π/3, respectively. The direction (nζ) of the tilted magnetic field Hζ = Hζnζ of increasing strength Hζ is tilted at the given angle ζa(b) with respect to the Z axis within the XZ plane (the scheme 1 in Figure 10). The dimensionless energies (εi = Ei/ dz) of the Zeeman levels, characterized by the vector and scalar chiralities, depend on the strength βζ of the dimensionless tilted magnetic field βζ (=gμBHζ/Dz√3) and on the given tilt ζa(b). The crossing of the states (S = 3/2 → S = 1/2) at the level crossing field HLC ζ is also shown in Figure 10. In the Heisenberg achiral trimer CuH3 (DDM = 0), the tilted field Hζ orients the ground-state magnetization vector M0I (dark-yellow arrow in the scheme 1) only against the field Hζ direction, M0I ∥Hζ. The canting angle θ0M of M0I coincides with the given tilt ζ of the field Hζ, θ0M = ζ, and does not depend on the strength Hζ (Figure 10). The axial chirality vectors of the trimer CuL3 (Dz+) at zero field (Hζ = 0) are directed along the Z axis, κLIz,IIz(0) = −1 and R κIIIz,IVz (0) = +1. The scalar chirality, components, and magnitudes of the ground-state chirality κLI (βζ) and magnetization MI(βζ) vectors of CuL3 in the sweeping tilted field Hζ are shown in Figures S7 and S8, ζb = π/3. The magnetic behavior depends on the tilted field strength βζ and the given tilt ζn. The βζ dependence of the ground-state energy εLI (βζ,ζ) (Figure 10), the components (Figures S7 and S8), and the canting angles γI and ηI of the ground-state vectors κLI and MI of CuL3 for a given tilt ζn have the form κILz = −(1 + βζ cos ζn)/Π+ζ ,

Figure 11. Rotation angles γIa, ηIa and γIb, ηIb of the ground-state chirality vector κLI and magnetization vector MI of the CuL3 trimer (Figure 10) under an increase of the strength Hζ(βζ) of the tilted field Hζ, with ζa = π/4 and ζb = π/3. The vectors κLI and MI for the low and intermediate fields βζ, the local magnetization vectors mi (short green arrows) of the ηI tilted spin configuration, and the scalar chirality χI+ of CuL3 , as well as the magnetization vector M0I of the Heisenberg CuH3 trimer tilted by θ0M = ζ, are shown in the scheme 1.

the strength βζ of tilted field Hζ is changed for the given tilt ζn of the field that differs from the equations in P1, where the rotation angle ϕ of the field H1 of a given strength β1 was changed. The ground-state LH chirality vector κLI = κLI (βζ,ζn) of CuL3 corresponds to the negative Z-component κLIz(βζ) < 0 in the entire range of increasing fields ±Hζ up to the level crossing (Figure S7). With an increase in the strength βζ of the tilted Hζ L field, βζ < βLC ζ , the ground-state vectors κI and MI perform the correlated CCW polar rotation within the lower half-plane (1/ LC 2)(XZ)− up to the level crossing at ±HLC ζ (±βζ ) (the scheme 1 in Figure 11), and the stacked together vectors κLI and MI are parallel because κLI = 2χMI in eq 10. The graphs γIa = ηIa and γIb = ηIb in Figure 11 describe the joint polar rotation γI and ηI (ηIΔJ eq 8) of the vectors κLI and MI under an increase of βζ for ζa = π/4 and ζb = π/3, respectively. The vectors κLI and MI are shown for the ηI tilted spin configuration (m1, m2, m3)LI in the scheme 1 in Figure 11. At low tilted fields, βLF ζ ≪ 1, the canting angles γI and ηI are LF L very small, γI(βLF ζ ) ≈ βζ sin ζ, γI(βζ=0) = 0, and κI (0)∥Z. The LF low-field direction of the vector MI(βζ ) is very close to the Z LF LF 1 axis, ηI(βLF ζ ) = γI(βζ ), MIz(βζ ) ≈ − /2, and MI(0) = 0. At high HF L tilted fields, βζ ≫ 1, the vectors κI and MI tend to be oriented against the direction (nζ) of the tilted field Hζ: the rotation angles γI = ηI tend toward the given tilt ζn of the field Hζ, HF γI(βHF ζ ) = ηI(βζ ) → ζn, as shown in Figure 11 and eq 7, with HF βζ < βLC and Hζ > 0. The vectors κLI and MI conserve their magnitudes when they perform nonuniform polar rotation (γI = ηI) under an increase of βζ (βζ ) = ζa(b) after the level crossing because 3 the ground state is the achiral level SI(βζ>βLC ζ ) = /2 and κI = 0. The graphs γIa, ηIa and γIb, ηIb in Figure 11 show large changes of the rotation angles γI and ηI of the rotating vectors κLI and MI at low fields βζ and small changes of γI and ηI at high fields; thus, e.g., ΔγIa(b) ≈ 26° (35°) for ΔHζ = 1 T in the field range 0 ≤ Hζ ≤ 1 T, while ΔγIa(b) ≈ 1.1° (1.5°) in the range 4 T ≤ Hζ ≤ 5 T. The vectors κLI and MI rotate faster in the first range and slower in the second range. The rotation γI = ηI of the vectors κLI and MI is significantly nonuniform (Figure 11). This joint rotation of the vectors κLI and MI with increasing βζ corresponds to the rotation ξi = ηI of the local vectors mi in the nonplanar spin-collinear configuration 1 in Figure 11, which is considered in the SI. Figure S7 shows that the vectors κLI and κII perform opposite rotations, respectively, under an increase of βζ. Figures S7−S9 compare the rotations γI = ηI and γII = ηII of the ground-state vectors κLI and MI and the excited-state vectors κII and MII with ζb = π/3. The rotation angle ηIa(b) and opposite angle ηIIb of the vectors MI and MII in the increasing tilted field Hζ are shown in Figures S8 and S9. The ground-state scalar chirality χI+(βζ) = 1 [orbital angular momentum LIz+(βζ)] does not change the sign and magnitude under an increase of the strength Hζ+ (Hx+ > 0) because the Z components miz− and MIz are always negative for Hζ+ in Figure 11. The magnetochiral correlation equation (10) governs the correlated behavior of the magnetochiral quantities κI, χI, and MI in the tilted field Hζ in Figure 11. 7.2. Frustration of the Scalar Chirality and Magnetization in the Sweeping Tilted Field Hζ. Figures S7 and S8 show frustration of the ground-state scalar chirality χI (orbital angular momentum LIz) and the magnetization vector MI of the system in the sweeping tilted field Hζ (see discussion in Chapter S4.2 in the SI). Small variation of the low sweeping tilted field βζ− ← βζ+ in the vicinity of zero field βζ = 0 results in a sharp inversion (χI− ← χI+) of the ground-state scalar chirality (LIz) under a transition from the scheme 1 (βζ+) to the scheme 2 (βζ−) in Figure S8, with conservation of the direction of the chirality vector κLI . This transition is accompanied by the jump of the vector MI from the scheme 1 (βζ+) in Figure S8, where the MI and κLI directions coincide, to the scheme 2 (βζ−), where they are opposite, in accordance with the chirality−magnetization correlation equation (10). This frustration of the ground-state scalar chirality χI (orbital angular momentum LIz) and the vector MI (with conservation of the vector κIL direction) in the sweeping tilted ±βζ field in the vicinity of βζ = 0 (Figure S8) differs from the joint frustration of the magnetochiral quantities κRI , MI, and χI (LIz) in the rotating field in the vicinity of ϕ⊥1 = π/2 and ϕ⊥2 = 3π/2 in Figure 2. 7.3. Action of the DM-Induced Axial Anisotropy on the Chirality in the Tilted Field Hζ. The Dz-induced axial magnetic anisotropy acts against the orientation of the vectors κLI and MI along the direction nζ of the tilted magnetic field (Hζ = Hζnζ) by the low (βLF ζ ≪1) and intermediate βζ fields (Figures 11 and S7 and S8). This is shown from eq 8 presented in the form tan γI = tan ηI = [hζ sin ζn/(dz + hζ cos ζn)] = [Hζ sin ζn/(H0z + Hζ cos ζn)], where hζ = HζgμB/2, and H0z = Dz 3 / gμB is the effective axial field of magnetoanisotropy (P1), which corresponds to the DM 2Δ splitting. The dimensionless magnetoanisotropy term β0z = 1in the denominator (1 + βζ cos ζn) in eq 8, as well as the large Dz - contribution dz [H0z ] to the denominator (dz + hζ cos ζn)[(H0z + Hζ cos ζn)], which describes the out-of-plane components, determines the deviation of the vectors κLI and MI from the Hζ direction

and the rotation behavior of these vectors in the increasing tilted field. An increase of the strength βζ of the tilted field Hζ results in the nonuniform CCW rotation (γI = ηI) of the parallel vectors κLI and MI from an initial almost axial orientation at βLF ζ HF (Figure 11). Only the high tilted field βHF ζ (1 ≪ βζ < βLC) orients the vectors κLI and MI almost antiparallel to the field, very close to the direction nζ of the field, as shown in Figure 11. An increase of Dz (decrease of βζ) stabilizes the initial orientations of the vectors κLI and MI very close to the Z direction, so that it requires more high tilted field Hζ to rotate these vectors. For CuL3 with the large parameter Dz+, even the high tilted field Hζ results in canting of the vectors κLI and MI, which is significantly smaller than the tilt angle ζ of the field Hζ; thus, e.g., for Dz+ = 10 K (H0z ≈ 13.24 T), the canting angle of the vectors κLI and MI at the high tilted field Hζ = 10 T is γIa(b) = ηIa(b) ≈ 19° (25°) for ζa(b) = 45° (60°). Nonlinear rotations ηI and γI of the vectors MI and κLI , induced by an increase of the tilted field strength βζ, differ the DM trimers with Dz ≠ 0 from the Heisenberg achiral trimer CuH3 , in which even a very low 0 tilted field HLF ζ always orients the ground-state vector MI 0 against the tilted field Hζ, with θM = ζ (Figure 10); the following increase of Hζ does not change the ζ direction and magnitude of M0I . 7.4. Spin Chiralities in the Tilted Field Hζ under an Increase of the Temperature. The different behavior of the spin chiralities and magnetization in the ground and excited states (Figures S7 and S8) under an increase of the tilted field strength βζ determines the net spin chiralities κLTK = κL(βζ,ζ,TK), χTK = χ(βζ,ζ,TK), and magnetization MTK = M(βζ,ζ,TK) of the LH trimer CuL3 (Dz+ = 0.53 K) under an increase of the temperature (Figure 12). The rotation angles γTK and ηTK of the vectors κLTK and MTK, respectively, of this trimer CuL3 at the given temperatures under an increase of Hζ(βζ) are shown in Figure 12 for the given tilt angle ζb = 60°, tan γTK = κx/κLz , and tan ηTK = Mx/Mz. In Figure 12a, the scalar chiralities χTK and magnitudes κTK of the rotating chirality vector κLTK as a function of the field strength Hζ are shown for the various temperatures TK = 0, 0.1, 0.2, 0.3, 0.5, and 1 K in a tilted field Hζ, increasing in the range 0 ≤ Hζ ≤ 5.3 T, up to the level crossing at HLC ζ with ζb = 60°. The corresponding rotation angles γTKand ηTK of the vectors κLTK and MTK, respectively, at the temperature TK are shown in Figure 12b; the scheme 1 shows the net vectors κL ( =κLTK) and M (=MTK) of the system in comparison with the ground-state vectors κLI and MI. The schemes in Figures S7 and S8 explain the temperature dependence of the scalar chirality χTK, the magnitudes and canting angles of the net chirality κLTK(βζ), and the magnetization vector MTK(βζ) of the system under an increase of βζ in Figure 12 (see Chapter S4 in the SI). In Figure 12a (ζb = π/3), an increase of the temperature TK leads to a strong reduction of the magnitude κTK of the chirality vector κLTK of the system in the increasing tilted field Hζ (Figure S7). The low-temperature chiralities κ0.1 K and χ0.1 K in Figure 12a coincide in the large range and then show fast reduction, approaching the level crossing field HLC ζ . For each given temperature TK, an increase of the strength βζ results in a significant decrease of κTK in comparison with the low-field maximal values κTK(βLF ζ )max. N


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Inorganic Chemistry

manifestation of their frustration in the tilted magnetic field discussed in section 7.2. Figure 12b depicts the corresponding rotation angles γTK and ηTK of the different nonuniform rotations of the nonparallel vectors κLTK and MTK, respectively, of this trimer for the given temperatures TK with increasing βζ in comparison with the graph γIb = ηIb of the joint rotations of the parallel ground-state vectors κLI and MI for ζb = 60°. The graphs γ0.1 K and η0.1 K differ from the ground-state graph γI = ηI for low fields βLF ζ , while they tend toward γI = ηI for the fields Hζ > 1 T. An increase of the temperature TK significantly increases the low-field rotation ηTK of the vector MTK in comparison with ηI of the vector MI. The graphs ηTK tend toward the linear graph ζb = 60° under an increase of the temperature TK, ηTK → ζb. This means that the magnetization vector MTK of the system tends to be orient closer to the given ζb orientation of the tilted field Hζ(ζb) even at very low fields Hζ (Figure 12b). The magnetization vector MTK demonstrates more Heisenberg behavior upon increasing temperature TK. In Figure 12b, the graphs γTK, which describe the nonuniform rotation of the vector κTL K(βζ) with increasing field βζ, demonstrate strong deviations from the rotation angle γI of the ground-state vector κLI under an increase of TK, especially at low and intermediate fields. This rotating vector κLTK(Hζ) demonstrates (i) a reduction of the magnitude κ1 K (Figure 12a) and a decrease of the canting angle γTK from the Z axis under an increase of the temperature TK and (ii) an increase of the differences ηTK − γTK, ηTK − ηI, and γI − γTK of the rotations γTK and ηTK of the nonparallel vectors κLI and MT K, where γTK < ηTK, which were the same in the ground state, TK = 0 and γI = ηI (Figure 12b). The scheme 1 in Figure 12b shows an increase of the canting angle η (=ηTK) of the vector M (MTK) and a decrease of the canting angle γ (=γTK) of the vector κL (κLTK) of the trimer CuL3 at the temperature TK under an increase of the strength Hζ, in comparison with the equal (γI = ηI) canting angles of the vectors κLI and MI at TK = 0. A decrease of the magnitudes of the chiralities (κTK and χTK) under an increase of the temperature TK shows more Heisenberg type of the magnetic behavior of the spin chiralities. The influence of the in-plane DM coupling (Dx = Dz) on the chiralities, which is essential in the range of the avoided level crossing at βLC ζ in the field Hζ is shown in Figure S9. Figures 12 and 13 show that the nonuniform rotation angles γI and ηTK of the nonparallel vectors of chirality κLTK(βζ) and magnetization MTK(βζ), respectively, of variable magnitude (Figure 12a) under an increase of the tilted field strength βζ, which is correlated with the temperature-dependent variation (Figure 12a) of the net scalar chirality χTK(βζ) of the DM trimer at the temperature TK, is governed by the same connection χTK(βζ) = 2(κTK(βζ)·MTK(βζ)) as that in the case of the rotating field. Figures 12 and S7 and S8 demonstrate manipulation of the spin chiralities by variation of the low temperature TK and the strength (and sign) Hζ(βζ) of the tilted field Hζ. This consideration shows the following conditions for possible application of the spin chiralities of the CuL3 (Dz+ ∼ 0.5 K) nanomagnet in the tilted field Hζ: The spin chiralities of this

Figure 12. (a) Scalar chirality χTK and magnitude κTK of the net chirality vector κLTK of the LH CuL3 trimer (J = 4.2 K and Dz+ = 0.53 K; Figure 10) at the temperature TK under an increase of the tilted field strength Hζ; see the text. (b) Rotation angles γTK and ηTK of the vectors κLTK and MTK of CuL3 at TK under an increase of the strength Hζ, with ζb = 60°, in comparison with the rotation angles γI = ηI of the groundstate vectors κLI and MI. The scheme 1 shows the vectors κL, M, κLI , and MI and the corresponding canting angles γ < γI = ηI < η.

The scalar chirality χTK(βζ) (orbital angular momentum Lz) increases nonlinearly from zero at βζ = 0 up to the maximal reduced magnitude close to κTK, as shown by the large plateau of χTK(Lz). Then, after the χTK plateau, the reduced scalar chirality χTK decreases together with κTK, approaching the level crossing field HLC ζ , after which the scalar chirality vanishes (Figure 12a). The spin chiralities κ1 K and χ1 K of this trimer (Dz = 0.53 K) are strongly reduced already at TK = 1 K (Figure 12a). The magnitude M0.1 K of the magnetization vector M0.1 K in the increasing tilted field demonstrates the smoothed jump at low fields βζ from zero up to the plateau and, then, after the plateau, the next jump (M ≈ 0.5 → M ≈ 1.5), approaching the level crossing at βLC ζ (Figure 12a), whereas the vector M0.1 K performs the η0.1 K rotation, as shown in Figure 12b. The jumps and plateaus of the scalar chirality and magnetization are a O


Article

Inorganic Chemistry

magnitude κTK close to |κLI | ≈ 1 and a canting angle γTK (

Nanomagnets. 2. Frustration, Temperature, and Distort - American

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