Magnetization Plateaus by the Field-Induced Partitioning of Spin Lattices

Abstract

To search for a conceptual picture describing the magnetization plateau phenomenon, we surveyed the crystal structures and the spin lattices of those magnets exhibiting plateaus in their magnetization vs. magnetic field curves by probing the three questions: (a) why only certain magnets exhibit magnetization plateaus, (b) why there occur several different types of magnetization plateaus, and (c) what controls the widths of magnetization plateaus. We show that the answers to these questions lie in how the magnets under field absorb Zeeman energy, hence changing their magnetic structures. The magnetic structure of a magnet insulator is commonly described in terms of its spin lattice, which requires the determination of the spin exchanges’ nonnegligible strengths between the magnetic ions. Our work strongly suggests that a magnet under the magnetic field partitions its spin lattice into antiferromagnetic (AFM) or ferrimagnetic fragments by breaking its weak magnetic bonds. Our supposition of the field-induced partitioning of spin lattices into magnetic fragments is supported by the anisotropic magnetization plateaus of Ising magnets and by the highly anisotropic width of the 1/3-magnetization plateau in azurite. The answers to the three questions (a)–(c) emerge naturally by analyzing how these fragments are formed under the magnetic field.

1. Introduction

The properties of a magnet are primarily characterized by measuring thermodynamic quantities (e.g., magnetization M and/or magnetic specific heat C_mag) as a function of the temperature and external magnetic field. The values of M(H, T) and C_mag(H, T) for a given magnet depend on its magnetic energy spectrum and on how the individual states of this spectrum are thermally populated. The individual magnetic states differ in their magnetic properties, and their population at a given temperature is governed by the Boltzmann factor, so the thermodynamic quantity is a weighted average of the properties of various states with weights given by their Boltzmann factors at that particular temperature. As a function of temperature, the Boltzmann populations of the individual states changes. States with lower energy become exponentially more populated as the temperature is decreased. Thus, measuring the temperature dependence of the M or C_mag of a magnet is an indirect way of probing its magnetic energy spectrum. To confirm whether or not a magnet undergoes a long-range magnetic ordering as the temperature is lowered is often judged from the temperature dependence of its specific heat. The occurrence of such an ordering is signaled, e.g., by the presence of an anomaly in the C_mag vs. T curve, which reflects the loss of the magnetic entropy associated with the long-range magnetic ordering.

Often, to a very good approximation, the energy spectrum of a magnet can be described by the Heisenberg spin Hamiltonian ${\widehat{H}}_{\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}}$ (Equation (1)), which is written as the sum of the pairwise spin exchange interactions between spin operators ${\widehat{S}}_{\mathrm{i}}$ and ${\widehat{S}}_{\mathrm{j}}$ located at the magnetic ion sites i and j, respectively.

$${\widehat{H}}_{\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}}=\sum _{\mathrm{i}>\mathrm{j}}{J}_{\mathrm{i}\mathrm{j}}{\widehat{S}}_{\mathrm{i}}·{\widehat{S}}_{\mathrm{j}}$$

(1)

Often, the spin operators ${\widehat{S}}_{\mathrm{i}}$ and ${\widehat{S}}_{\mathrm{j}}$ can be regarded as the classical spin vectors ${\overrightarrow{S}}_{\mathrm{i}}$ and ${\overrightarrow{S}}_{\mathrm{j}}$, respectively, and hence the spin Hamiltonian has the classical expression,

$$H_{\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}}=\sum _{\mathrm{i}>\mathrm{j}}{J}_{\mathrm{i}\mathrm{j}}{\overrightarrow{S}}_{\mathrm{i}}·{\overrightarrow{S}}_{\mathrm{j}}=\sum _{\mathrm{i}>\mathrm{j}}{J}_{\mathrm{i}\mathrm{j}}{S}_{\mathrm{i}}{S}_{\mathrm{j}}\cos \mathrm{\theta }$$

(2)

where θ is the angle between the two spin vectors ${\overrightarrow{S}}_{\mathrm{i}} \mathrm{a}\mathrm{n}\mathrm{d}{\overrightarrow{S}}_{\mathrm{j}}$.

Here, we consider the simplest case of the symmetric spin exchange and omit contributions of antisymmetric exchange to the Hamiltonian or anisotropies in the exchange. Being a dot product, the extrema of ${\overrightarrow{S}}_{\mathrm{i}}·{\overrightarrow{S}}_{\mathrm{j}}$ occur when the two spins are parallel and antiparallel to each other, respectively. Thus, with the spin Hamiltonian defined as in Equations (1) and (2), the two spins prefer an AFM coupling if the spin exchange is positive ($J_{\mathrm{i}\mathrm{j}}>0$), but a ferromagnetic (FM) coupling if it is negative ($J_{\mathrm{i}\mathrm{j}}<0$). The “spin lattice” of a magnet refers to the repeat pattern of its spin exchange paths of nonnegligible strengths. If we consider such spin exchange paths as magnetic bonds, then the spin lattice of a magnet is the lattice of its magnetic bonds of various strengths. The spin lattice of a magnet is crucially important because it allows one to generate the energy spectrum relevant for the magnet using a model Hamiltonian $H_{\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}}$ with a minimal number of spin exchange parameters.

Magnets including Heisenberg magnetic ions, with nonzero quantized spin moments in all directions, are described by the Heisenberg spin Hamiltonian, for which ${\overrightarrow{S}}_{\mathrm{i}}·{\overrightarrow{S}}_{\mathrm{j}}$=$S_{\mathrm{i}\mathrm{x}}{S}_{\mathrm{j}\mathrm{x}}+S_{\mathrm{i}\mathrm{y}}{S}_{\mathrm{j}\mathrm{y}}+S_{\mathrm{i}\mathrm{z}}{S}_{\mathrm{j}\mathrm{z}}$. The magnets of uniaxial (i.e., Ising) magnetic ions, which possess nonzero spin moments only in one direction (by convention, the z-direction) so that ${\overrightarrow{S}}_{\mathrm{i}}·{\overrightarrow{S}}_{\mathrm{j}}=S_{\mathrm{i}\mathrm{z}}{S}_{\mathrm{j}\mathrm{z}}$, are described by the Ising spin Hamiltonian,

$$H_{\mathrm{I}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}=\sum _{\mathrm{i}>\mathrm{j}}{J}_{\mathrm{i}\mathrm{j}}{S}_{\mathrm{i}\mathrm{z}}{S}_{\mathrm{j}\mathrm{z}}$$

(3)

Transition-metal magnetic ions M in oxide magnets ions form MO_n (typically, n = 3–6) polyhedra. The Ising magnetism is found for a magnetic ion M when the d-states of its MO_n polyhedron has an unevenly occupied degenerate d-state (in the non-spin-polarized, one-electron picture of electronic structure description) [1,2]. Such a magnetic ion is susceptible to a Jahn–Teller distortion, which tends to lift, though weakly, the degeneracy responsible for the Ising magnetism [3]. Thus, pure Ising magnets are rather rare.

The dependence of magnetization M on the external magnetic field μ₀H is usually measured at the lowest possible temperature to minimize the contributions of magnetic excited states lying close to the magnetic ground state through the Boltzmann averaging. As a function of the magnetic field μ₀H, the magnetization of a paramagnet is well described by a Brillouin function, which increases steadily from zero to the magnetic saturation M_sat. On increasing the magnetic field μ₀H, the magnetization of an antiferromagnet exhibits spin flop and spin flip transitions (see below) while that of a ferromagnet rapidly reaches the saturation, depending on anisotropy and dipolar energies.

For some magnets among the wide variety of magnetic materials, their magnetization versus magnetic field (M vs. H) curves exhibit s plateaus at rational fractions f = m/n, where m and n are integers with m < n (most commonly, m = 1) of their saturation magnetization M_sat. This phenomenon occurs not only in magnets undergoing a long-range magnetic order at low temperatures but also in low-dimensional or spin-frustrated magnets that do not undergo a magnetic order down to the lowest temperatures. In discussing the M vs. H curves observed for such magnets, it is convenient to distinguish magnets with and without uniaxial (i.e., Ising) anisotropy [1,2]. The idealized M vs. H curves observed for non-Ising magnets are illustrated in Figure 1a–c, and those for Ising magnets in Figure 1d–f. Consider first the M vs. H behaviors of non-Ising magnets. On increasing the magnetic field from zero, a gradual increase in M from zero to (m/n)M_sat precedes before reaching the m/n-magnetization plateau at M = (m/n)M_sat (Figure 1a), the (m/n)-magnetization plateau occurs as soon as the field increases from zero (Figure 1b), or the zero-magnetization plateau at M = 0 precedes until the field reaches a value from which a gradual increase in M to (m/n)M_sat begins (Figure 1c). For an Ising magnet, the spin moment is nonzero only along one specific direction in space. An Ising magnet exhibits a highly anisotropic M vs. H behavior; its M vs. H curve exhibits a “step-like” feature when the applied field is parallel to the direction of the spin moment (Figure 1d,e), i.e., the easy axis direction, but the magnetization does not change with field showing no magnetization plateau when the applied field is perpendicular to the spin moment direction (Figure 1f). Experimentally, a very slight linear increase with field is observed, but this is often due to a minute misalignment of the crystal with respect to the external magnetic field.

Magnetic plateaus have been found in a large variety of magnets. Their spin lattices can be one-dimensional, two-dimensional (2D) or three-dimensional (3D), their ground state can be AFM or ferrimagnetic in the absence of the external magnetic field, their spin lattices may or may not be spin-frustrated, and their structures can be extended or discrete. For magnets of high symmetry, a number of theoretical studies examined their magnetization plateaus from the viewpoint of their magnetic energy spectra generated by model spin Hamiltonians [4,5]. So far, however, there has been no systematic study aimed at providing a conceptual picture for the magnetization plateau phenomenon. The primary objective of our survey is to come up with a conceptual framework useful for chemists, materials scientists and experimental physicists in organizing and thinking about magnetization plateaus. Therefore, we pursue the qualitative answers to the three questions (a)–(c) by analyzing not only the structural chemistry associated with the magnetic ions but also the relative magnitudes and the signs of the spin exchange interactions between them. Our study strongly suggests that the spin lattice of a magnet exhibiting a magnetic plateau is partitioned into ferrimagnetic or AFM fragments by breaking the weakest magnetic bonds one at a time by absorbing Zeeman energy provided by an external magnetic field. The M vs. H curve of a magnet is divided into two different regions: the regions where a magnet does not absorb Zeeman energy so nonzero (m/n > 0) magnetic plateaus occur and the regions where the magnet absorbs Zeeman energy so no magnetization plateau, except for the zero (m/n = 0) magnetization plateau, occurs.

Our survey is organized as follows: In Section 2, we analyze why the spin lattice of a certain magnet is partitioned into smaller magnetic fragments and what types of magnetization plateaus are possible. Section 3 describes the magnetization plateaus of magnets whose spin lattices are partitioned into AFM fragments (with an even number of spin sites) under field, and in Section 4 those of magnets whose spin lattices are partitioned into ferrimagnetic fragments (with an odd number of spin sites) under field. The magnetization plateaus of magnets possessing kagomé and trigonal layers are discussed in Section 5, and those of magnets with complex magnetic fragments in Section 6. Finally, our conclusions are summarized in Section 7.

We note that this work is not a comprehensive review on magnetization plateaus, but a survey on studies of magnetization plateaus that enabled us to put forward the concept that a magnet under field absorbs Zeeman energy by breaking its weak magnetic bonds. The associated partitioning of its spin lattice into magnetic fragments gives rise to magnetization plateaus. For magnets with low-symmetry crystal structures possessing a large number of magnetic ions per unit cell, describing their magnetic structures quantitatively using a model spin Hamiltonian is practically impossible. This difficulty has led us to search for a qualitative description of such magnets on the basis of their spin exchanges (i.e., magnetic bonds), because they can be readily determined by employing density functional theory (DFT) calculations. Our studies on numerous such magnets over the past two decades revealed that the spin exchanges obtained from DFT calculations are quite accurate in their relative magnitudes and are therefore reliable in finding which magnetic bonds are weak and hence will be broken preferentially under field. This realization led us to the concept of the field-induced partitioning of a spin lattice into magnetic fragments, initially from our own studies on magnets exhibiting magnetic plateaus. We then checked whether this concept is applicable to other magnets for which magnetization plateaus were reported. When the spin exchanges of these magnets were not available, we determined them by performing DFT calculations as summarized in the supporting information. This survey is the outcome of these efforts. Our choice of references is not as comprehensive as expected for a review article but is rather confined to those central to our supposition of the field-induced partitioning of a spin lattice into magnetic fragments.

2. Field-Induced Partitioning of Spin Lattices

2.1. Zeeman Energy and Magnetic Bonds

Two spins of an AFM exchange path tend to align antiparallel to each other, so it requires energy to force them to be ferromagnetically aligned. At very low temperatures where magnetization measurements are usually carried out, the energy needed for such a conversion in a spin exchange path of a magnet is supplied by Zeeman energy, E_Z. For a magnetic ion with spin moment ${\overrightarrow{\mu }}_s=-g{\mu }_B\overrightarrow{S}$ under a magnetic field ${\mu }_0\overrightarrow{H}$, the Zeeman energy is given by

$$E_Z=g{\mu }_0{\mu }_B\overrightarrow{H}·\overrightarrow{S},$$

(4)

which is positive and negative if $\overrightarrow{H}$ and $\overrightarrow{S}$ are parallel and antiparallel, respectively. As the magnetic field is gradually increased from zero, the conversion from antiparallel to parallel spin alignment occurs initially in weak AFM exchange paths, i.e., those with small exchange J are converted first. For the convenience of our discussion, an AFM exchange path will be termed “a magnetic bond” if the two spins of the path are antiferromagnetically coupled. Likewise, an AFM exchange path may be termed “a broken magnetic bond” or “a magnetic antibond” if the two spins of the path are forced to be ferromagnetically coupled. Thus, an AFM magnetic bond can be broken by the Zeeman energy E_Z (Figure 2a). Similarly, an FM exchange path may be termed “a magnetic bond” if the two spins of the path are ferromagnetically aligned, but “a broken magnetic bond or a “magnetic antibond” if the two spins of the path are antiferromagnetically aligned. Thus, an FM magnetic bond can be broken by Zeeman energy (Figure 2b). For the convenience of our discussion, we will represent the up-spin and down-spin at a magnetic ion site by unshaded and shaded circles, respectively (Figure 2). [Here, the up-spin and down-spin are parallel and antiparallel to the direction of an external magnetic field (taken to be the z-direction by convention), respectively. In the absence of an external field, what matters is that the up-spin and down-spin are antiparallel to each other, regardless of their absolute directions in space.] Breaking an AFM magnetic bond increases the spin moment (Figure 2a), while breaking an FM bond can either decrease or increase the spin moment (Figure 2b) because an FM coupling in an FM magnetic bond can be represented by the (↑↑) or (↓↓) spin arrangement. It is the FM bond breaking from (↓↓) to (↑↓), not from (↑↑) to (↑↓), that is relevant for our discussion of magnetization because the total magnetic moment should not decrease with field (see below for further discussion). Summarizing, magnetic bonds should be referred to as either AFM magnetic bonds or FM magnetic bonds. However, for most magnets showing magnetization plateaus, one deals with breaking AFM magnetic bonds, and it is very rare to find magnets whose magnetization requires the breaking of FM magnetic bonds. Thus, in the following, we use the term “magnetic bonds” to describe AFM bonds, unless stated otherwise.

To avoid a possible confusion in using the terminology, the broken or unbroken magnetic bond, it is necessary to distinguish between the eigenstates and the broken-symmetry states. For all practical evaluations of spin exchanges for any magnet, broken-symmetry states are used instead of the eigenstates simply because the latter are very difficult to determine [6]. For example, consider a spin dimer made up of two S = 1/2 magnetic ions representing, for example, the molecular Cu₂Cl₆²⁻ ion of edge-sharing CuCl₄ square planes (see Section 3.2.2). This dimer can be described by the singlet and triplet states |S〉 and |T〉, which are the eigenstates of the dimer (Figure 3a). Then, the energy difference between the two states is the spin exchange J, i.e., E_T − E_S = J (Figure 3b). If the dimer is described by the broken-symmetry states ↑↑ and ↑↓, then the energy difference between the two states is given by E_↑↑ − E_↑↓ = J/2 (Figure 3c). In the following, by breaking an AFM J bond in an extended magnet, we mean the conversion from the AFM coupling ↑↓ to the FM coupling ↑↑. In the case of an isolated dimer, the breaking of the AFM J bond means the excitation from the singlet to the triplet state.

2.2. Causes for Magnetization Plateaus

In the following, we put forward the supposition that during the magnetization process the spin lattice of a magnet becomes partitioned into either AFM or ferrimagnetic fragments by breaking its weak magnetic bonds. As an example, consider how a 0-magnetization plateau arises by considering a chain in which AFM dimers described by spin exchange J₁ are antiferromagnetically coupled in a tail-to-tail bridging pattern to make J₂ bonds between adjacent dimers such that J₁ > J₂. An AFM chain made up of alternating J₁ and J₂ bonds is presented in Figure 4a. Since J₂ is weaker than J₁, the J₂ bond will be broken “successively” (Figure 4b,c) as the magnetic field is increased from 0 until all J₂ bonds are broken (Figure 4d). In Figure 4b–d, the red ellipses are used to indicate that the (↓↑) dimers, resulting from the (↑↓) dimers of Figure 4a, break the interdimer bonds J₂. The energy needed for breaking a J₂ bond is supplied by Zeeman energy, but the magnetization remains at zero while the field increases until all J₂ bonds are broken because the spins stay paired in the J₁ bonds. This leads to a 0-magnetization plateau (e.g., Figure 1c).

As the magnetic field increases further, the J₁ bonds become broken one at a time, as depicted in Figure 4e–g, where the green ellipses are used to indicate the broken dimers, (↑↑), resulting from the (↓↑) dimers of Figure 4d. In principle, a broken dimer can be equally well represented by the configuration (↓↓). However, throughout our discussion, a broken dimer will be represented by (↑↑), because the spin moments of a magnet under the magnetic field should not decrease with field and because we use the convention that up-spin and down-spin have positive and negative moments, respectively. Since each J₁ bond breaking creates unpaired up-spins, the magnetization M increases with the field: M = M_sat/4, if one out of four J₁ bonds is broken (Figure 4e), M = M_sat/3 if one out of three J₁ bonds is broken (Figure 4f), and M = M_sat/2 if one out of two J₁ bonds is broken (Figure 4g).

In general, when the spin lattice of a magnet becomes partitioned into identical AFM fragments by breaking its weak magnetic bonds interconnecting them, the magnet acquires a zero-magnetization given by the AFM fragment regardless of how many inter-fragment magnetic bonds there are. If there exist a large number of different arrangements between the AFM fragments, which differ only in the number of their inter-fragment magnetic bonds, then the magnetization of a magnet remains unchanged while the inter-fragment magnetic bonds are being broken by increasing the magnetic field. The 0-magnetization plateau of the AFM chain made up of AFM dimers discussed above is an example. To increase the magnetization beyond this level, a weak magnetic bond of each AFM fragment needs to be broken.

Suppose that the spin lattice of a magnet becomes partitioned into identical ferrimagnetic fragments when the field increases by breaking its weak inter-fragment magnetic bonds. Then, the magnetization increases gradually with field until all inter-fragment bonds are broken so that all ferrimagnetic fragments become ferromagnetically coupled, leading to a certain level of magnetization given by the ferrimagnetic fragments. To increase the magnetization beyond this level, it is necessary to break a weak magnetic bond of each ferrimagnetic fragment. If this magnetic bond is strong, the bond breaking will not happen unless the field reaches a high enough value, hence leading to a magnetization plateau.

2.3. Magnetic Bonding Pattern Affecting the Nature of Magnetization Plateaus

Consider now an AFM chain in which ferrimagnetic linear trimers with the (↑↓↑) configuration are antiferromagnetically coupled in a tail-to-tail pattern (Figure 5a) so that the (↑↓↑) and (↓↑↓) trimers alternate. We assume that the intra-trimer J₁ bond is stronger than the inter-trimer bond J₂. Then, as the field increases from 0, the J₂ bonds become broken one at a time (Figure 5b) until all J₂ bonds are broken (Figure 5c) by converting each (↓↑↓) trimer to a (↑↓↑) trimer, as indicated by the red ellipses. Since each trimer constitutes a ferrimagnetic unit, the magnetization increases with the number of broken J₂ bonds until it reaches M_sat/3, where all J₂ bonds are broken. When the magnetic field is further raised, the bonds to break are the two J₁ bonds in each (↑↓↑) trimer as indicated by the green ellipse in Figure 5d. The magnetic bonds J₁ are strong, and two J₁ bonds of a trimer should be broken simultaneously. Therefore, until the field reaches a high enough value, they are not broken hence leading to no increase in the magnetization. The magnetization plot of Figure 1a is a characteristic feature of an AFM chain in which ferrimagnetic fragments are antiferromagnetically coupled in a tail-to-tail manner.

When the ferrimagnetic linear trimers are combined in a head-to-tail bridging pattern, the resulting chain is a ferrimagnetic chain (Figure 6a), with magnetization M = M_sat/3. Under a magnetic field, a J₂ bond of this ferrimagnetic chain cannot be broken because, if broken, the resulting ferrimagnetic trimer will have the (↓↑↓) configuration (indicated by the red ellipse in Figure 6b) and hence will reduce the overall moment of the chain. Therefore, the only way to absorb the Zeeman energy is to break the two J₁ bonds of a ferrimagnetic trimer successively (indicated by the green ellipses in Figure 6c,d) hence increasing the magnetization toward M_sat. The J₁ bond is strong and a simultaneous breaking of two J₁ bonds requires high energy, so this will not occur until the applied field is high enough. Thus, the 1/3-magnetization plateau will be wide. The magnetization curve of the ferrimagnetic chain is well described by Figure 1b.

2.4. Field-Induced Ferrimagnetic Fragments in Spin-Frustrated Lattices

As described above, field-induced partitioning of a spin lattice into magnetic fragments lies at the heart of the magnetization plateau phenomenon. In understanding this field-induced partitioning, it is crucial to identify the weak magnetic bonds of a given spin lattice that can be readily broken by moderate magnetic fields. The absorption of Zeeman energy by a magnet is a consequence of the Le Chatelier’s principle. There are cases when it is not immediately obvious how a spin lattice under field will be partitioned into magnetic fragments when the spin lattice defined by a few spin exchanges of comparable magnitude is spin-frustrated, e.g., trigonal, kagomé and diamond chain spin lattices. For such cases as well, Le Chatelier’s principle enables us to put forward the supposition that a spin lattice is partitioned into small ferrimagnetic fragments of nonzero spin $\overrightarrow{S}$ such that these fragments fill the spin lattice without overlapping between them. This partitioning reduces the extent of spin frustration by absorbing Zeeman energy and hence breaking the inter-fragment bonds and making the ferrimagnetic fragment absorb Zeeman energy further when the field is raised. For example, consider a magnet consisting of diamond chains made up of two AFM spin exchange J₁ and J₂ with J₁ > J₂ (Figure 7a). The weaker magnetic bonds J₂ form a continuous chain, but the 1/3-magnetization phenomenon observed for such a magnet can be readily understood by supposing that the diamond chain undergoes a field-induced partitioning into triangular ferrimagnetic fragments as depicted by shaded triangles in Figure 7b,c.

Each triangular fragment can have six possible spin arrangements (Figure 8). In the absence of an applied magnetic field, the (↑↑↑) and (↓↓↓) trimers are less stable than the other four ferrimagnetic trimers, and the (↑↓↓) and (↓↑↓) trimers with net negative moment are as stable as the (↓↑↑) and (↑↓↑) trimers with net positive moment. Applying a magnetic field stabilizes the latter but destabilizes the former. Likewise, an applied field stabilizes the (↑↑↑) trimer while destabilizing the (↓↓↓) trimer. Thus, for the discussion of the 1/3-magnetization plateau of the diamond chains, only the ferrimagnetic (↑↓↑) or (↓↑↑) trimers are relevant.

Let us consider theoretical and experimental justifications for our supposition that a spin-frustrated spin lattice will undergo a field-induced partitioning into small ferrimagnetic fragments of nonzero spin $\overrightarrow{S}$. If a magnet can produce such ferrimagnetic fragments, the energy of each ferrimagnetic fragment is raised by Zeeman energy, $E_Z=g{\mu }_0{\mu }_{B}HS$, which can be used for the magnet to break the magnetic bonds necessary for the partitioning. However, if a magnet cannot interact with the magnetic field, such fragmentation cannot occur so that the magnet cannot develop any magnetization plateau. An experimental test for these predictions is provided by magnetization studies on Ising magnets. The spins of an Ising magnet are nonzero in one direction in space (say, the ||z direction) but zero in all directions perpendicular to this direction (i.e., ⊥z). For field ${\overrightarrow{H}}_{\left|\right|}$ along the ||z direction, the Zeeman energy is nonzero (${\overrightarrow{H}}_{\left|\right|}·\overrightarrow{S}$ ≠0). For field ${\overrightarrow{H}}_{\perp }$ along the ⊥z direction, however, the Zeeman energy is zero (${\overrightarrow{H}}_{\perp }·\overrightarrow{S}$ = 0). (Here, we have assumed that the anisotropy energy that forced the spins to align either parallel or antiparallel to z is much larger than the Zeeman energy). Therefore, an Ising magnet can have a magnetization plateau when the field is along the ||z direction but cannot if the field is along the ⊥z direction (Figure 1d–f). The step-like feature of the magnetization curves found for Ising magnets under field ${\overrightarrow{H}}_{\left|\right|}$ indicates that a large number of ferrimagnetic fragments develop simultaneously, because spin flipping is necessary for magnetic bond breaking.

2.5. Spin-Lattice Interactions

In our discussion of magnetic plateaus so far, the field-induced partitioning of a spin lattice into ferrimagnetic or antiferromagnetic fragments is discussed without considering spin-lattice interactions. The magnetic fragments broken off differ in their surroundings from the spin lattice (e.g., Figure 5a–c) and hence would require some relaxation in their atomic/electronic structures through magnetoelastic coupling. The concomitant change in the spin-lattice interaction can therefore affect the stability and nature of a magnetization plateau. A magnetization plateau generates a certain pattern of up-spin and down-spin arrangement and hence the associated spin-lattice interactions. When the spin-lattice interaction, associated with a certain magnetization plateau, leads to an energy-lowering relaxation, this magnetization plateau will arise. Otherwise, it will not be observed. It goes without saying that the spin-lattice interaction will be important for magnets of compact and high-symmetry atomic structure, because the pattern of up-spin and down-spin arrangement should be compatible with the symmetry of the underlying spin lattice.

Magnets of spinel structure such as CdCr₂O₄ and LiGaCr₄O₈ have a pyrochlore spin lattice (see Section 3.1.2). The ideal spinel structure is cubic. Studies on CdCr₂O₄ [7] and LiGaCr₄O₈ [8,9] reveal that their observed magnetization plateaus, namely, the 1/2-magnetization plateaus, are strongly stabilized by the magnetoelastic (i.e., spin-lattice) coupling. Another high-symmetry magnet showing the importance of the spin-lattice interaction is the quasi-2D tetragonal magnet SrCu₂(BO₃)₂. Early magnetization measurements on single-crystal samples of SrCu₂(BO₃)₂ identified 1/8-, 1/4- and 1/3-plateaus of M_sat (see Section 3.1.1) [10]. More recent experiments with fields up to ~140 T [11,12] revealed that the transition into the regions of the 1/2-magnetization plateau is accompanied by strong magnetoelastic effects [13]. In stabilizing the magnetization plateaus of other magnets with less rigid and low-symmetry atomic structures, the magnetoelastic coupling would also be important, although elaborate studies carried out for the spinel magnets and SrCu₂(BO₃)₂ are mostly not available yet. Thus, in what follows, we will discuss the field-induced fragmentation of a spin lattice based solely on the interaction of the external magnetic field with the spin arrangement in the spin lattice.

2.6. Different Magnetization Behaviors of Heisenberg and Ising Magnets

Heisenberg and Ising magnets change the direction of their magnetic moments as the external magnetic field increases and exhibit a slightly different dependence on the magnetic field. Though often described by an idealized Heisenberg spin Hamiltonian, such a magnet exhibits weak magnetic anisotropy if the orbital moment quenching of its magnetic ions is incomplete. In a similar manner, an ideal Ising magnet described by an Ising Hamiltonian is difficult to find because the associated Jahn–Teller distortion can lift, though weakly, the degeneracy of the d-state responsible for the Ising magnetism [3]. Therefore, by Heisenberg and Ising magnets, we mean those whose magnetic properties are well approximated by Heisenberg and Ising spin Hamiltonians, respectively.

The plateau formation at fractional values of the saturation magnetization M_sat compete and coexist with spin-flop transitions in Heisenberg antiferromagnets and metamagnetic transitions in Ising antiferromagnets. In the absence of a magnetization plateau, the magnetization processes of Heisenberg antiferromagnets can be described as illustrated in Figure 9, and those of Ising antiferromagnets as illustrated in Figure 10. Most Heisenberg magnets possess weak magnetic anisotropy, so describing them with a Heisenberg spin Hamiltonian is an approximation. Similarly, the MO_n polyhedra of the magnetic ions M in most oxide Ising magnets are weakly distorted due to the Jahn–Teller instability. Therefore it is an approximation to describe Ising magnets using an Ising spin Hamiltonian. The majority of magnetic measurements are conducted mainly on polycrystalline samples. For measurements on such samples, it is often difficult to distinguish whether a magnetic transition involved is a spin-flop or a metamagnetic type. However, it is generally observed that the M(H) curve is concave for spin-flop transitions, but convex for metamagnetic transitions.

2.6.1. Spin Flop and Spin Flip Processes of Antiferromagnets

From the viewpoint of phenomenological description, Heisenberg and Ising magnets differ in the strength of the single-ion magnetic anisotropy D with respect to that of the spin exchange J. The positions of the spin-flop and metamagnetic transitions in the magnetization curves are determined by J and D. General aspects of field-induced transitions were described by Néel [14], detailed discussions of spin-flop transitions by Morosov and Sigov [15], and those of metamagnetic transitions by Strujewski and Giordano [16]. Consider an antiferromagnet with single-ion anisotropy aligning the magnetic moments along a preferred crystal direction commonly called an “easy axis”, which is generally defined as the z-direction. For such a magnet, its behaviors in a field directed either along or perpendicular to the easy magnetization axis are important to analyze. First, let us consider a Heisenberg-type antiferromagnet with a small magnetic anisotropy D compared with the exchange J, which has two magnetic sublattices, M₁ and M₂ (namely, the up-spin and down-spin sublattices). Consider, for example, Fe₂O(SeO₃)₂ [17], in which the Fe atoms Fe1, Fe2 and Fe3, lead to three spin exchange paths J₁, J₂ and J₃ (for the nearest-neighbor Fe1-Fe2, Fe2-Fe3 and Fe3-Fe3 paths, respectively). These paths form 2D nets parallel to the ab-plane (Figure 11a) which are stacked along the c-direction. The Fe1, Fe2 and Fe3 atoms exist as Fe³⁺ (d⁵, S = 5/2) ions, so their magnetic anisotropy is weak (i.e., small D). The neutron diffraction studies [17] reveal that, in all these spin exchange paths, the spin moments parallel to the b-axis are antiferromagnetically coupled. The magnetic susceptibility of this magnet (Figure 11b) shows that, for an external magnetic field along the easy axis, μ₀H||b, the transition to the AFM state at the Néel temperature T_N manifests itself as a sharp decrease in the magnetic susceptibility χ_||. For an external field perpendicular to the easy axis, μ₀H⊥b, however, the magnetic susceptibility χ_⊥ at T < T_N remains almost unchanged. At low temperatures, χ_|| < χ_⊥. Thus, upon reaching a certain critical field, the difference in the energy of the magnetic moments M₁ and M₂ oriented either parallel or perpendicular to an external magnetic field reaches a critical value

$$\Delta E=-{\displaystyle \frac{1}{2}}({\chi }_{\perp }-{\chi }_{\left|\right|}){\mu }_0{H}_{spin-flop}^2,$$

(5)

at which there is a 90° rotation of the magnetic moments M₁ and M₂ to the direction perpendicular to the magnetic field. Taking into account the fact that these two moments are related by the exchange interaction J, the field of the spin-flop transition is determined by the expression,

$${\mu }_0{H}_{spin-flop}={\left(2\mathrm{D}\mathrm{J}\right)}^{1/2}.$$

(6)

At this field and at temperatures that are low compared with the Néel temperature T_N, there is a sharp jump in the magnetization M_||, after which the magnetization monotonically increases up to the saturation magnetization

$$M_{saturation}={|M}_1|+|M_2|,$$

(7)

which is reached in the field of the spin-flip transition,

$${\mu }_0{H}_{spin-flip}^{Heisenberg}=J.$$

(8)

In an external magnetic field μ₀H⊥b, the magnetization M_⊥ increases monotonically, reaching the saturation magnetization in the same field μ₀H_spin-flip (we neglect here the anisotropy of the g factor). The field dependence of the magnetization of the easy-axis antiferromagnet Fe₂O(SeO₃)₂ is shown in Figure 11c.

We now turn to an Ising-type antiferromagnet with a magnetic anisotropy D comparable in strength to or exceeding the spin exchange J. When the magnetic field is directed along the magnetic moments of the two sublattices M₁ and M₂ (i.e., along the b axis), the magnetic susceptibility χ_|| of an Ising antiferromagnet is similar to that observed for a Heisenberg antiferromagnet, as shown in Figure 12a for the francisite-type compound [18]. When an external magnetic field μ₀H||b reaches the critical value

$${\mu }_0{H}_{\mathrm{metamagnetic}}=J,$$

(9)

one of the sublattices (either M₁ or M₂) will reverse its moment direction by 180°, producing a sharp jump as shown in Figure 12b [19]. In this case, μ₀H_metamagnetic is equivalent to μ₀H_spin-flip and corresponds to M_saturation. In a magnetic field μ₀H⊥b, both M₁ and M₂ moments continuously rotate to the direction of the external magnetic field reaching the saturation magnetization M_saturation at

$${\mu }_0{H}_{spin-flip}^{Ising}=J+D.$$

(10)

2.6.2. Magnetization Plateaus

The magnetization behaviors of Heisenberg and Ising magnets differ in their M vs. H curves preceding a magnetization plateau (Figure 1); the magnetization exhibits a linear increase with field for a Heisenberg magnet but does not depend on field for an Ising magnet. We briefly comment on why this difference comes about. In a Heisenberg exchange coupled system, the exchange energy depends only on the relative orientation of the participating spin moments. To a first approximation, the effective spin Hamiltonian for an Ising magnet contains the z-components of the spins only. The only options of the crystal field anisotropy are to favor either a parallel or an antiparallel alignment of the spin moments along the easy axis. Therefore, an external magnetic field not only competes with the spin exchange but also with the anisotropy. Consequently, the magnetic response of an Ising magnet to an external magnetic field depends sensitively on the alignment of the magnetic field with respect to the easy axis as well as their relative magnitudes. Spin-flop transitions with sudden jumps of the magnetization from very low to large values are a phenomenon connected to the presence of a crystal field anisotropy. Thus, the magnetization curve of an Ising magnet deviates somewhat from the step-like features (Figure 1d,e) when the magnetic field is parallel to the easy axis, and from the flat line (Figure 1f) when the magnetic field is perpendicular to the easy axis.

2.7. Quantitative Evaluations of Spin Exchange Interactions

In understanding the magnetic properties of a magnet, it is essential to know the strengths of its magnetic bonds, i.e., the values of its spin exchanges J_ij. The parameters J_ij specify the spin Hamiltonian (Equation (1)) and determine the energy spectrum for a given magnet. These days, it is almost routine to evaluate the spin exchanges of a magnet composed of transition-metal magnetic ions by using the energy-mapping analysis [6,20,21] based on density functional theory (DFT) calculations. The quantitative values of the spin exchanges are determined by mapping the energy spectrum of a magnet generated by the spin Hamiltonian onto that obtained for a set of broken-symmetry states of the magnet by spin polarized DFT calculations.

The spin exchanges of some magnets discussed in our survey have not been determined before. To gain better insight into these magnets, we determined them by carrying out the energy-mapping analyses. For our DFT calculations, we employed the frozen core projector augmented plane wave (PAW) method [22] encoded in the Vienna ab Initio Simulation Packages (VASP) [23] and the PBE exchange-correlation functional [24]. To take into consideration the electron correlation associated with transition-metal magnetic ions, we performed DFT+U calculations [25] with an effective on-site repulsion U_eff = U − J = 3 and 4 eV on the magnetic ions to ensure that all broken-symmetry states employed for a magnet are magnetic insulating. For a certain magnet, the effect of spin–orbit coupling (SOC) was tested by performing DFT+U+SOC calculations [26]. Unless stated otherwise, the values of the calculated spin exchanges (in K), which are included in a figure describing each magnet, were those determined using DFT+U or DFT+U+SOC calculations with U_eff. Other details of our calculations are reported in the Supporting Information (SI).

3. Magnets of AFM Fragments

3.1. Spin Clusters with Even Number of Spin Sites

3.1.1. Orthogonal Spin Dimers in SrCu₂(BO₃)₂

In the quasi-2D tetragonal compound SrCu₂(BO₃)₂, the CuBO₃ layers alternate with Sr layers along the c axis. In each CuBO₃ layer, planar Cu₂O₆ dimers of two edge-sharing CuO₄ units are interconnected by BO₃ triangles (Figure 13a). The two dominant spin exchanges of this layer are the intradimer exchange J₁ of the Cu-O-Cu type and the interdimer exchange J₂ of the Cu-O…O-Cu type [6]. In each CuBO₃ layer, the (Cu²⁺)₂ dimer ions have an orthogonal arrangement such that the J₁ and J₂ exchange paths are interconnected by a “head-to-tail” bridging pattern (Figure 13b). The values of J₁ and J₂, estimated to be 84.7 and 53.4 K (J₂/J₁ = 0.63), respectively, by LDA+U calculations [27], are very close to the experimental values [28]. This orthogonal arrangement of the dimers represents a realization of the so-called Shastry–Sutherland spin lattice [29]. The magnetic susceptibility of SrCu₂(BO₃)₂ has a sharp peak at about 20 K. Once the contribution of magnetic defects is removed, the susceptibility drops to zero indicating a nonzero spin gap Δ (Figure 14a) [30]. At low temperatures, early magnetization measurements on single-crystal samples of SrCu₂(BO₃)₂ identified 1/8-, 1/4- and 1/3-plateaus of M_sat (Figure 14b) [10]. More plateaus (in particular, 2/5 and 1/2) were found in recent experiments with fields up to ~140 T [11,12]. In the plateau regions, the triplet dimers [namely, the (Cu²⁺)₂ dimers with broken J₁ bonds] crystallize into magnetic superstructures, so the transitions into the plateau regions are accompanied by substantial magnetoelastic effects. The latter are detected by changes in the magnetostrictive length and volume and a drastic decrease in the sound velocity [13]. To theoretically model the 1/2-magnetization plateau, the interlayer spin exchange needs to be taken into account [13].

The essential aspect of the Shastry–Sutherland spin lattice is illustrated in Figure 13c. In the ground state, each J₁ bond is surrounded by an equal number of unbroken and broken J₂ bonds. Thus, the contribution of the J₂ bonds to the total energy vanishes. The magnetization of SrCu₂(BO₃)₂ is zero between 0 and ~20 T (Figure 14b), because it requires the breaking of J₁ bonds to increase magnetization and because all different arrangements of the J₂ and broken J₂ bonds are identical in energy as long as the J₁ bonds remain unbroken. The 1/n-magnetization plateau of SrCu₂(BO₃)₂ requires that one out of every n dimers has a broken J₁ bond, because the resulting spin configuration (↑↓)_n−1(↑↑) has two net spins out of every 2n spins hence leading to the 1/n-plateau. The magnetic superstructure describing the 1/n-plateau is determined by how a (↑↑) dimer is arranged with respect to every (n − 1) (↑↓) dimer. Figure 14b reveals that the widths of the magnetization plateaus are not uniform, i.e., they decrease in the order, 1/3- > 1/2- > 2/5-, 1/4- > 1/8-plateau. This variation would be caused by their spin-lattice interactions and hence the associated energy lowering, because they will be different due to the differences in their magnetic superstructures. Several theoretical studies examined the magnetic textures and superstructures of the CuBO₃ layer predicting many more plateaus [10], which were mostly detected in magnetization, magnetostriction, magnetocaloric effect and nuclear magnetic resonance measurements.

3.1.2. Spin Tetrahedra in Spinel CdCr₂O₄

CdCr₂O₄ is a spinel-type compound based on CrO₆ octahedra containing Cr³⁺ (S = 3/2) ions. It is convenient to describe the structure of this compound in terms of the Cr₄O₁₆ cluster (Figure 15a), which is made up of four CrO₆ octahedra by sharing their edges to form a Cr₄O₄ distorted cube containing a Cr₄ tetrahedron. The spinel structure of CdCr₂O₄ is obtained by corner-sharing the Cr₄O₁₆ clusters, which is accompanied by the corner-sharing Cr₄ tetrahedra such that each Cr₄ tetrahedron shares a corner with four Cr₄ tetrahedra in a tetrahedral arrangement (Figure 15b). Thus, the resulting arrangement of the Cr³⁺ ions is a pyrochlore spin lattice, which is highly spin-frustrated.

CdCr₂O₄ undergoes a long-range AFM ordering and a tetragonal lattice distortion at T_N = 7.8 K [31]. Using neutron diffraction measurements, the ground state spin configuration was found to be a helical spin structure [32]. The magnetization curve determined at low temperatures reveals a gradual increase with field as the field increases from zero, which is followed by a sharp transition into a 1/2-plateau at M_sat/2 = 1.5 μ_B/Cr³⁺ (Figure 16a). To account for the nature of the observed M vs. H curve, we treat the pyrochlore arrangement of Cr³⁺ ions as composed of isolated (Cr³⁺)₄ tetrahedra by neglecting the interactions between them, as shown in Figure 15c. In the zero-magnetization state, each tetrahedron has the (2↑2↓) configuration (Figure 16b), which has four J₁ and two broken J₁ bonds. When a tetrahedron has the (3↑1↓) configuration (Figure 16b), which has three J₁ and three broken J₁ bonds, the magnetization increases. The 1/2-magnetization, M = M_sat/2, is reached when all isolated tetrahedra have the (3↑1↓) configuration. The energy change required for each (Cr³⁺)₄ tetrahedron to undergo the (2↑2↓) to (3↑1↓) transition is to break one J₁ bond per tetrahedron. The sharp jump in the magnetization of CdCr₂O₄ takes place at about 29 T. At 1.8 K, M(H) increased linearly with H until ~28 T, where M ≈ 0.75 μ_B, namely, when half the (Cr³⁺)₄ tetrahedra have the (3↑1↓) configuration. On increasing the field beyond this point, the J₁ bond breaking occurs simultaneously everywhere such that all (Cr³⁺)₄ tetrahedra have the (3↑1↓) configuration. As the number of broken J₁ bonds increases, the J₁ breaking at one tetrahedron becomes correlated with those at other places due to the inter-cluster tetrahedra, which were neglected in our discussion. Above 28 T, CdCr₂O₄ shows a flat magnetization. For the magnetization to increase beyond M_sat/2, a tetrahedron must undergo the configuration change from (3↑1↓) to (4↑0↓). The (4↑0↓) configuration has no J₁ bond (Figure 16b), while (3↑1↓) has three J₁ bonds, so the (3↑1↓) to (4↑0↓) transition requires to break three J₁ bonds per tetrahedron. That is, this transition requires more energy than does the (2↑2↓) to (3↑1↓) transition (i.e., one J₁ bond per tetrahedron). This explains why the magnetization of CdCr₂O₄ is flat above 28 T.

A spinel magnet with the ideal pyrochlore spin lattice would not undergo a 3D magnetic long-range order due to the severe spin frustration. However, most spinel magnets undergo a 3D magnetic long-range order because the degeneracy of their ground state is lifted by a structural distortion. In a sense, this distortion can be considered a 3D spin-Peierls transition. A theoretical analysis of CdCr₂O₄ [7] showed that its 1/2-magnetization plateau is stabilized by magnetoelastic coupling and this plateau is robust. The pyrochlore spin lattice of the spinel magnet LiGaCr₄O₈ differs from that found for CdCr₂O₄ in that it consists of small and large tetrahedral (Cr³⁺)₄ clusters which alternate by corner-sharing [8]. The magnetization and magnetostriction studies of LiGaCr₄O₈ under magnetic fields of up to 600 T [9] show that it exhibits a two-step coupled magnetic and structural phase transition between 150 T and 200 T, followed by a robust 1/2-magnetization plateau up to ~420 T, and that the intermediate-field phase is stabilized by the strong spin-lattice coupling. This phase can be considered a tetrahedron-based superstructure with a 3D periodic array of (3↑1↓) and canted (2↑2↓) configurations.

3.1.3. Spin Hexamers in Pyroxene CoGeO₃ and Anisotropic Magnetization Plateau

CoGeO₃ consists of two nonequivalent Co atoms, Co1 and Co2, forming Co1O₆ and Co2O₆ octahedra. By sharing their edges, these octahedra form zigzag ribbon chains parallel to the bc-plane, as shown in Figure 17a. The 3D structure of CoGeO₃ is obtained from these zigzag ribbon chains when their oxygen atoms are shared with GeO₄ tetrahedra [33]. The magnetic properties of CoGeO₃ present a novel feature [34]. The 1/3-magnetization plateau of CoGeO₃ is uniaxially anisotropic, that is, CoGeO₃ exhibits a pronounced 1/3-plateau when measured with field applied along the c-direction but does not show any magnetization plateau when measured with field perpendicular to the c-direction (Figure 18a). This observation provides an experimental support for our supposition that field-induced partitioning of a spin lattice into ferrimagnetic fragments is essential for magnetization plateaus. To probe the cause for the anisotropic character of the 1/3-magnetization plateau in CoGeO₃, mentioned above, we evaluate the four spin exchanges J₁–J₄ defined in Figure 17b using DFT+U calculations. The intrachain exchanges J₁–J₃ are of the Co-O-Co type exchange, while the interchain exchange J₄ is of the Co-O…O-Co type. Results of these calculations are summarized in Figure 17c (see Section S1 of the Supplementary Materials).

The spin exchanges J₁–J₄ are all AFM, and the interchain exchange J₄ is considerably weaker than the intrachain exchanges J₁–J₃. Since J₁ is considerably weaker than J₂ and J₃, the magnetic ground state for the layer of the double chains has the AFM spin arrangement as depicted in Figure 18b, where each double chain is made up of J₂ and J₃ magnetic bonds as well as broken J₁ bonds. The smallest fragment that can generate a ferrimagnetic fragment of 1/3-magnetization is the hexamer, composed of three J₂ bonds with (3↑3↓) spin configuration (indicated by shading in Figure 18b). The ferrimagnetic fragment of (4↑2↓) configuration is generated when one of the three J₂ bonds is broken, ultimately leading to the ferrimagnetic state (Figure 18c) when every hexamer has the (4↑2↓) spin configuration. The conversion from a (3↑3↓) to a (4↑2↓) is facilitated because the breaking a J₂ bond is accompanied by the formation of two J₁ bonds.

CoGeO₃ exhibits uniaxial (i.e., Ising) magnetism with the spin moments oriented along the c-direction [33]. According to the selection rules governing the preferred spin orientations of magnetic ions [35], either Co1²⁺ or Co2²⁺ or both ions of CoGeO₃ prefer to have their spins oriented along the c-direction. Consider an ideal axially compressed CoO₆ octahedron containing Co²⁺ (d⁷, S = 3/2) ion with the short Co-O bonds oriented along the z-axis, as depicted in Figure 19a. The t_2g-state of such an octahedron is split into the degenerate (xz, yz) state lying above the xy state, assuming that the axially compressed octahedron has an ideal shape with four-fold rotational symmetry. With two d-electrons to occupy the down-spin d-states, the split t_2g states become occupied as depicted in Figure 19a. Thus, between the highest-occupied and the lowest-unoccupied d-states, the minimum difference in their magnetic quantum numbers, |ΔL_z|, is zero so that the preferred spin orientation is parallel to the z-axis [35]. Of the Co1O₆ and Co2O₆ octahedra of CoGeO₃, only the Co2O₆ octahedra have a structure close to an axial-compression [namely, Co2-O_ax = 1.994 (×2), and Co2-O_eq = 2.118 (×2), 2.278 (×2) Å]. Since the Co2O₆ octahedra have no four-fold rotational symmetry, their xz and yz states are not degenerate, but they still lie above the xy state due to the strong axial compression. The latter guarantees |ΔL_z| = 0, hence predicting that the spins of the Co2O₆ octahedra are oriented along the short Co2-O bonds, i.e., along the c-direction. The Co1²⁺ ions adopt the spin orientation of the Co2²⁺ ions to maximize their spin exchanges (J₂ and J₄) with the Co2²⁺ ions. This explains why CoGeO₃ exhibits uniaxial magnetism with spin moment along the c-direction.

Based on the uniaxial magnetism of CoGeO₃, one can understand why it shows a 1/3-magnetization plateau only when the field is along the c-direction. As already discussed in Section 2, the Zeeman energy E_Z between the spin moment ${\overrightarrow{\mu }}_s=-g{\mu }_{\mathrm{B}}\overrightarrow{S}$ and magnetic field ${\mu }_0\overrightarrow{H}$ is given by $E_Z=g{\mu }_0{\mu }_B\overrightarrow{H}·\overrightarrow{S}$ (Equation (4). Therefore, when the spin moment and magnetic field are parallel to each other (Figure 19b), $E_Z>0$ so that the formation of a ferrimagnetic fragment is energetically favored. However, $E_Z=0$ if the magnetic field is perpendicular to the spin moment. In such a case, the energy needed to break the magnetic bonds and hence form ferrimagnetic fragments is not available. This explains why the 1/3-magnetization plateau of CoGeO₃ has a uniaxial character. In addition, this finding is in support of our suggestion that, for a magnet to exhibit magnetic plateaus, its spin lattice should undergo a field-induced partitioning into ferrimagnetic clusters. It should be noted that the magnetization curves of CoGeO₃ (Figure 18a) have a “step-like” feature, because the uniaxial magnetism favors a spin-flip mechanism for magnetization.

3.2. Bose–Einstein Condensates

In a certain magnet composed of discrete units possessing two magnetic ions, such “dimers” have an S = 0 ground state, and the interactions between adjacent dimers are weak so that the first excited state of each dimer, which has S > 0, lies close to the S = 0 ground state. In such a case, the magnetic states of the magnet are well approximated by those of its dimer. The |S, S_z〉 = |S, −S〉 substate of the excited state is lowered in energy under magnetic field μ₀H. When μ₀H exceeds a certain value, μ₀H_c, the |S, −S〉 substate becomes lower in energy than the ground state |0, 0〉, so that the magnetic ground state of each dimer becomes an S > 0 state. Magnets showing such a behavior, known as Bose–Einstein condensates, have been reviewed by Zapf et al. [36]. It should be noted that S = 0 dimers can be discrete molecular units such as Cu₂Cl₆²⁻ anions containing two magnetic ions or dimers composed of two monomers such as (MnO₄³⁻)₂ (see below). In both cases, the interdimer spin exchange is weaker than the intradimer exchanges. In this section, we discuss the magnetization phenomena observed in two Bose–Einstein condensates.

3.2.1. 0- and 1/2-Plateaus of Ba₃Mn₂O₈

The trigonal compound Ba₃Mn₂O₈ [37] is composed of MnO₄ tetrahedra containing Mn⁵⁺ (S = 1) ions. Every two tetrahedra combine to form a dimer unit (MnO₄)₂ such that one Mn-O bond of each MnO₄ is parallel to the c-axis (hereafter the Mn-O_|| bond). The two Mn-O_|| bonds of each dimer are pointed in opposite directions (Figure 20a), and these dimers form trigonal layers. Adjacent layers are shifted from each other such that each dimer of one layer is pointed to the center of three dimers of the two adjacent layers (Figure 20a). Consequently, every (Mn⁵⁺)₂ dimer ion of one layer is surrounded by six (Mn⁵⁺)₂ dimer ions (Figure 20b). The spin exchanges of Ba₃Mn₂O₈ are dominated by the intradimer exchange J₀ and the interdimer exchange J₁ (Figure 20c). (J₀ = 15.2 K and J₁ = 1.4 K according to our DFT+U calculations, see Section S2 of the Supplementary Materials).

The allowed spin states of each (Mn⁵⁺)₂ dimer ion are singlet, triplet and quintuplet since Mn⁵⁺ is an S = 1 ion, apart from usually small zero-field splitting of these multiplets. The temperature dependence of the magnetic susceptibility χ measured for Ba₃Mn₂O₈ is shown in Figure 21a [38], which evidences that Ba₃Mn₂O₈ is in a singlet ground state with a spin gap Δ = 11.2 K, in which all (Mn⁵⁺)₂ dimer ions are in the singlet state. At low temperatures, the magnetization plateaus are observed at M = 0 zero and M_sat/2 = 2 μ_B per formula unit (Figure 21b) [38]. We now examine how these plateaus are related to the breaking of the J₁ and J₀ bonds. The most stable and least stable arrangements of J₁ bonds around a J₀ bond are shown in Figure 22a, and those around a broken J₀ bond in Figure 22b. There are many other arrangements of the J₁ and broken J₁ bonds whose stabilities lie in between these two extremes. In general, the arrangement becomes more stable if it has more J₁ bonds but becomes less stable if it has more broken J₁ bonds. As the field increases from 0 to H_c1, each J₁ bond begins to break without breaking the J₀ bonds. Thus, M = 0 between 0 and H_c1. As the field increases from H_c1, the J₀ bond breaking proceeds, hence increasing M. Two dimers with one J₀ and one broken J₀ bond have the (3↑1↓) configuration. When half the J₀ bonds are broken, the M = M_sat/2 point at H_c2 is reached. The 1/2-plateau between H_c2 and H_c3 means that there are more J₁ bonds than broken J₁ bonds at H_c2, while the opposite is the case at H_c3. That is, magnetic energy is absorbed without increasing magnetization from M_sat/2. Since J₁ is a weak magnetic bond, the width of the 1/2-magnetization plateau is narrow. When the field is stronger than H_c3, more J₀ bonds begin to break, increasing the magnetization.

3.2.2. Gapped and Gapless Ground States of ACuCl₃ (A = K, Tl, NH₄)

A.

Singlet to triplet excitations under magnetic field

The molecular magnets ACuCl₃ (A = K, Tl, NH₄) [39,40,41] consist of planar Cu₂Cl₆²⁻ anions, which are made up of two CuCl₄ square planes containing Cu²⁺ (S = 1/2) ions by edge-sharing (Figure 23a). Thus, each Cu₂Cl₆²⁻ anion contains a spin dimer (Cu²⁺)₂. The ground spin state for such a dimer can be either singlet (Δ_ST > 0, “singlet dimer”, Figure 23b) or triplet (Δ_ST < 0, “triplet dimer” Figure 23c). When such a spin dimer is exposed to a magnetic field μ₀H, the triplet state |S, S_z〉 (S = 1, S_z = −1, 0, 1) is split while the singlet state |S, S_z〉 (S = 0, S_z = 0) remains unaffected. For a singlet spin dimer (Δ_ST > 0) under magnetic field, the triplet state becomes more stable than the singlet state if the field is greater than a critical value μ₀H_c (Figure 23d), so that every spin dimer occupies the S_z = −1 state, and the system undergoes a Bose–Einstein condensation. Likewise, for a triplet spin dimer (Δ_ST < 0) under the magnetic field, the singlet state becomes more stable than the triplet state if the field is higher than a critical value (Figure 23e). For a singlet dimer below μ₀H_c, there are three possible spin-flip transitions from the singlet to the triplet under the magnetic field, namely, |0, 0〉 → |1, S_z〉 (S_z = −1, 0, 1), and the energy difference between the two states can be accessed, e.g., by inelastic neutron spectroscopy techniques. The energy difference immediately provides the magnitude of the spin exchange J in the spin dimer. For such transitions to be observed by inelastic neutron scattering measurements, the singlet state |0, 0〉 should be thermally populated and should be more populated than the triplet state(s) into which the transition occurs. This is the case for a singlet dimer because, for field lower than μ₀H_c, the |0, 0〉 state is the lowest-lying in energy than any of the three triplet branches (Figure 23d). For a triplet dimer (Δ_ST < 0), the |0, 0〉 state can be thermally more populated than one branch of the triplet, i.e., the |1, +1〉 state, only when the field is substantially greater than μ₀H_c (Figure 23e). Under this condition, the |0, 0〉 → |1, +1〉 transition can take place in a triplet dimer.

NH₄CuCl₃ is known to consist of three different spin dimers, termed A, B and C, in the 1:2:1 ratio. Results of inelastic neutron scattering experiments carried out for NH₄CuCl₃ at 0.13 and 1.80 K are summarized in Figure 23f [42], which shows the three branches of the triplet state for the dimers B and C at both temperatures (1.80 and 0.13 K). This finding proves that B and C are singlet dimers with ΔE_ST values of ~1.6 and ~3.0 meV, respectively. For dimer A, however, only one branch is found, which becomes visible only above ~8.5 T and only at 1.80 K, but apparently is not observed for the measurement at 0.13 K. A singlet–triplet splitting of 0.5 meV (~5.8 K) with the spin triplet lower than the singlet (Δ_ST < 0) derived by extrapolating the observed |1, +1〉 branch to zero field is consistent with this finding. The small energy difference between the triplet and singlet implies that any excitations with energies below ~0.5 meV could have been masked under the elastic peak or accidentally coincide with excitations for dimers B and C (see the inset in Figure 4 of [30]). These results suggest that the (Cu²⁺)₂ dimer A is weakly coupled by ferromagnetic spin exchange, i.e., it is a triplet dimer. Further temperature dependent inelastic neutron scattering investigations are necessary to verify this conclusion. We found that the magnetization curve observed for NH₄CuCl₃ at 0.5 K using the H||a field is reasonably well reproduced by assuming that dimers A, B and C are all singlet dimers with the intradimer spin exchanges of 2.7 (1), 11.3 (1) and 19.3 (2) K, respectively (see Figure S1 in Section S3 of the Supplementary Materials). This suggests that dimer A is a singlet dimer but is inconsistent with the inelastic neutron scattering study described above. With dimer A as a triplet dimer, it is straightforward to understand the gapless excitation in the magnetization measurements of NH₄CuCl₃ (see below) because its triplet dimers A have a nonzero spin moment even in the absence of field. Though isostructural with NH₄CuCl₃ as far as the atom positions of the heavier atoms are concerned, KCuCl₃ and TlCuCl₃ consist of only one kind of singlet spin dimer. The excitation energy gaps measured for KCuCl₃ and TlCuCl₃ are 2.6 and 0.7 meV, respectively [43,44].

B.

Different magnetization behaviors of ACuCl₃ (A = K, Tl, NH₄)

Magnetization processes of KCuCl₃, TlCuCl₃ and NH₄CuCl₃ have been investigated up to 39 T at low temperatures (Figure 24). Both KCuCl₃ and TlCuCl₃ exhibit a 0-magnetization plateau, and this plateau has a much wider width for KCuCl₃ (Figure 24a,b). The transition from a singlet ground state to a magnetic excited state occurs when the field is greater than ~6 and ~20 T for TlCuCl₃ and KCuCl₃, respectively. These critical fields μ₀H_c are consistent with the excitation energy gaps of 0.7 and 2.6 meV observed for TlCuCl₃ and KCuCl₃, respectively [43,44]. Except for the 0-magnetization plateau, the magnetization of TlCuCl₃ and KCuCl₃ increases continuously with field showing no more plateau. The magnetization of NH₄CuCl₃ shows a very different behavior. As the field increases from zero, the magnetization reveals gapless excitations toward a 1/4-plateau, which was followed by a 3/4-plateau before reaching full saturation M_sat (Figure 24c) [45].

The magnetization behaviors of KCuCl₃ and TlCuCl₃ can be readily understood by considering how the magnetic bonds of their spin dimers are broken under the magnetic field. When the field is zero, both KCuCl₃ and TlCuCl₃ have zero moment because each spin dimer has the singlet configuration, $\left(\uparrow \downarrow -\downarrow \uparrow \right)/\sqrt{2}$. The magnetization of KCuCl₃ and TlCuCl₃ can increase from zero only if dimers start to break their magnetic bonds, one at a time, to assume the triplet configuration (↑↑). The critical field μ₀H_c needed to break each dimer magnetic bond is much higher for KCuCl₃ than for TlCuCl₃ (~20 vs. ~6 T) because the singlet-triplet energy difference Δ_ST is much larger for KCuCl₃ than for TlCuCl₃ (2.0 vs. 0.7 meV). This explains why the 0-magnetization plateau is much wider for KCuCl₃. With increasing the field, the singlet to the triplet magnetic bond breaking will continue until all magnetic bonds are broken, namely, until the saturation magnetization is reached.

The magnetization behaviors of NH₄CuCl₃, though apparently more complex, can be similarly explained by noting that spin dimers A, B and C constitute 25%, 50% and 25% of all the dimers, and that dimers A are triplet dimers while dimers B and C are singlet dimers, and the singlet–triplet energy gap is greater for C than for B (3.0 vs.1.6 meV) [42]. When μ₀H = 0, triplet dimers A should exist half in the (↑↑) configuration and half in the (↓↓) configuration. As μ₀H increases from 0, dimers A with (↓↓) configuration will switch their configuration to (↑↑), successively, until all dimers A attain the (↑↑) configuration at μ₀H_c1, where M = M_sat/4 because all A dimers have the (↑↑) configuration while the dimers B and C are in the (↑↓) configuration and because 25% of the dimers are dimers of type A. The 1/4-plateau continues until μ₀H_c2. When the field is greater than μ₀H_c2, the magnetic bonds of dimers B start to break, one at a time, until all dimers B break their bonds at μ₀H_c3, where M = 3M_sat/4 because all dimers A and B have the (↑↑) configuration while the dimers C have the (↑↓) configuration and because dimers A and B with (↑↑) configuration represent 75% of the total dimers. The 3/4-plateau continues until μ₀H_c4. When the field is greater than μ₀H_c4, the magnetic bonds of dimers C start to break, successively, until all dimers C break their bonds at μ₀H_c5, and the saturation magnetization is finally reached.

C.

Crystal structures of ACuCl₃ (A = K, Tl, NH₄)

As discussed above, the spin dimers of KCuCl₃ are very different from those of TlCuCl₃ in the singlet-to-triplet excitation energies. However, the Cu₂Cl₆²⁻ ions of KCuCl₃ are very similar in crystal structure to those of TlCuCl₃ [35,36]. Neutron scattering measurements reveal the existence of three different spin dimers in NH₄CuCl₃ [42], but the neutron diffraction studies to determine the crystal structure [46] carried out for ND₄CuCl₃ at various temperatures show that there is only one kind of Cu₂Cl₆²⁻ anion in ND₄CuCl₃. These apparently puzzling observations imply that the spin dimers used in interpreting the experimental magnetic data are the effective spin dimers which are affected by the interactions between dimers and by those with the cations A⁺ (A = K, Tl, NH₄). Therefore, it is necessary to examine the crystal structures of ACuCl₃ (A = K, Tl, NH₄) in more detail with focus on why the magnetic behavior of NH₄CuCl₃ differs from those of KCuCl₃ and TlCuCl₃.

In ACuCl₃, the Cu₂Cl₆²⁻ anions form stacks along the a-direction (Figure 25a). The spin exchanges describing the interactions within each stack are the intradimer exchange J₁ and the two interdimer exchanges J_a and J′_a. An important interdimer exchange between adjacent stacks of Cu₂Cl₆²⁻ anions is J₂ (Figure 25b). The spin exchanges J₁ and J₂ are contained in a layer of Cu₂Cl₆²⁻ anions and A⁺ cations, which is parallel to the ad-plane, where the repeat vector d is defined as d = a + c/2 (Figure 25b). In this layer each Cu₂Cl₆²⁻ anion is surrounded by six A⁺ cations, and the adjacent J₁-J₂-J₁-J₂ alternating chains interact by the interdimer exchanges J₃ and J₄.

The crystal structure of NH₄CuCl₃ is slightly more complex than those of KCuCl₃ and TlCuCl₃ due to the orientation of each NH₄⁺ cation. The crystal structures of ND₄CuCl₃ including the D atom positions were determined via neutron diffraction at various temperature [46]. In these structure determinations, the presence of three different dimers A, B and C were not taken into consideration (see below for further discussion).

D.

Interdimer exchanges of KCuCl₃ and TlCuCl₃

The values (in K) of the spin exchanges defined in Figure 25a–c, evaluated by using the energy-mapping analysis based on DFT+U calculations (see Sections S4 and S5 of the Supplementary Materials), are presented in Figure 25d. The intradimer exchange J₁ is stronger than the interdimer exchanges in both KCuCl₃ and TlCuCl₃, and the interdimer spin exchanges are substantially stronger for TlCuCl₃ than for KCuCl₃. These findings are consistent with the results of the neutron scattering study of Matsumoto et al. [43]. (The intradimer exchange J₁ is slightly smaller for KCuCl₃ in their study, while the opposite is the case in our calculations). Thus, the excitation energy is substantially smaller for TlCuCl₃ than for KCuCl₃ essentially because the interdimer spin exchanges are substantially stronger for TlCuCl₃ as found previously [43].

To find why the interdimer exchanges were stronger for TlCuCl₃ than for KCuCl₃, we consider the spin exchanges J₂ and J′_a as representative examples. The x²-y² magnetic orbital of each Cu²⁺ ion lies in the CuCl₄ square plane. Thus, the two Cu²⁺ ions of a spin exchange path are represented by two CuCl₄ square planes, a (CuCl₄)₂ dimer, for short. The (CuCl₄)₂ dimers of the exchange paths J₂ and J_a′ make short contacts with A⁺ cations, as depicted in Figure 26a and Figure 26b, respectively. In each (CuCl₄)₂ dimer, the x²-y² magnetic orbitals of two Cu²⁺ ions form in-phase and out-of-phase combinations (see Figure S2, Section S3 of the Supplementary Materials), which we represent by the labels (+) and (−), respectively. The frontier orbitals of K⁺ and Tl⁺ that can interact with the (+) and (−) d-states are K 4s, Tl 6s and Tl 6p orbitals (Figure 26c). By symmetry, the K 4s orbital interacts with the (+) state, so the (+) level is lowered in energy (Figure 26c). The Tl 6s orbital interacts with the (+) state, which raises the (+) level, but the Tl 6p orbital also interacts with the (−) state, which lowers the (−) level (Figure 26c). Such interactions occur at every Cl…A⁺…Cl bridge each (CuCl₄)₂ dimer makes with the surrounding A⁺ cations. Consequently, the energy gap between the (+) and (−) d-states for the interdimer spin exchanges is larger for TlCuCl₃ than for KCuCl₃. The intra-stack exchange J_a′ is calculated to be slightly stronger than the strongest inter-stack exchange J₂ in both KCuCl₃ and TlCuCl₃ (Figure 25d). This reflects that the J_a′ path has four A⁺ cations making the Cl…A⁺…Cl bridges (Figure 26b), while the J₂ path has only two such bridges (Figure 26a).

E.

Intradimer exchange of NH₄CuCl₃

Let us now examine how the spin exchanges of NH₄CuCl₃ depend on the orientations of the NH₄⁺ cations with respect to the Cu₂Cl₆²⁻ anions they surround (Figure 25c). Each N-H bond of a NH₄⁺ cation has a σ*_N-H orbital, which is highly anisotropic in shape because it is oriented along the N-H bond. Based on the crystal structure determined via X-ray diffraction [41], and assuming that the rotational mobility of the NH₄⁺ cations ceases at low temperatures, we constructed three model orientations, termed YY, NY and NN, of the two NH₄⁺ cations that bridge either side of the Cl…Cl contact in every J₂ exchange path (Figure 27a). Under the constraint that two NH bonds of each NH₄⁺ group are coplanar with the Cl…Cl contact and the other NH₂ group bisects the Cl…Cl contact, only three different NH₄⁺ arrangements were possible: both NH₄⁺ cations make N-H…Cl hydrogen bonds with the Cl…Cl contact in the YY arrangement, only one NH₄⁺ cation does in the NY arrangement, and no NH₄⁺ cation does so in the NN arrangement (Figure 27b). The orientations of six NH₄⁺ cations surround each Cu₂Cl₆²⁻ ion in the YY, NY and NN arrangements (see Figure S3, Section S3 of the Supplementary Materials).

The relative energies of these three structures and the values of their intradimer spin exchange J₁ are summarized in Figure 27c, from which we note the following: (a) The YY, NY and NN arrangements of NH₄CuCl₃ have considerably different relative stabilities, with the stability increasing in the order, YY < NY < NN. (b) The intradimer exchange J₁ of NH₄CuCl₃ depends strongly on the NH₄⁺ orientations, with its value increasing in the order, NN < NY < YY. (c) The interdimer exchanges J₂ and J_a′ of NH₄CuCl₃ are much weaker than those of KCuCl₃ and TlCuCl₃. This reflects the fact that the σ*_N-H orbitals of NH₄⁺ are strongly contracted compared with the K 4s and the Tl 6s/6p.

Since there are two equivalent ways of having the NY arrangement, the statistical probabilities for the YY, NY and NN arrangements are 1:2:1. The observations (a) and (b) are consistent with the experimental observation suggesting that NH₄CuCl₃ consists of three different Cu₂Cl₆²⁻ anions in the 1:2:1 ratio [42]. The Cu₂Cl₆²⁻ anions in the YY, NY and NN structures are surrounded by NH₄⁺ cations with different orientations (Figure S3, Section S3 of the Supplementary Materials) and hence will undergo different local relaxations further changing the values of their spin exchanges. To test this hypothesis, we optimized the YY, NY and NN structures by relaxing only the Cu and Cl positions and then calculated the spin exchanges for the resulting structures (Figure 27c) to find a reduction of J₁ by 0.05, 13 and 43% for the YY, NY and NN structures, respectively. These reductions reflect that the mid Cl atom of the Cu₂Cl₆²⁻ anion with (without) the short N-H…Cl contact moves away from (toward) the N atom thereby increasing (decreasing) the ∠Cu-Cl-Cu angle of the Cu₂Cl₆²⁻ anion [namely, 96.28° (×2) for the YY, 95.43° and 96.27° for the NY, and 95.35° (×2) for the NN structure]. The effect of the structure relaxation on other spin exchanges is weak (see Sections S6–S8 of the Supplementary Materials).

F.

Consequence of the interaction between NH₄⁺ and Cu₂Cl₆²⁻ in NH₄CuCl₃

It is of interest to find why the intradimer exchange J₁ of NH₄CuCl₃ depends so sensitively on the NH₄⁺ orientations. The two d-states of a Cu₂Cl₆²⁻ anion, termed the [+] and [−] states in Figure 28a, are the in-phase and out-of-phase combinations of the two x²-y² magnetic orbitals. With respect to the long axis of the Cu₂Cl₆²⁻ ion, the two p-orbitals at each bridging Cl atom (hereafter, the mid-Cl atom) in Figure 28a are hybridized to become a perpendicular p-orbital (p_⊥) in the [+] state, but a parallel p-orbital (p_||) in the [−] state. The p_⊥ orbital is spatially more extended out toward the surrounding cations NH₄⁺ than is the p_|| orbital and is hence more effective in the cation–anion interactions in the short NH₄⁺…Cl contacts. The observation (b) reflects that the energy lowering by the (p_⊥-σ*_N-H) interaction occurs in one and two places in the NY and YY structures, respectively (Figure 28b). Thus, the energy gap between the [+] and [−] states of a Cu₂Cl₆²⁻ anion interacting with the surrounding NH₄⁺ cations increases in the order NN < NY < YY, as depicted in Figure 28c.

We now examine an important implication of the observation made in Figure 28c. In general, the spin exchange J of a spin dimer made up of two S = 1/2 ions is written as J = J_F + J_AF [6,47]. If the spin sites at i and j are represented by magnetic orbitals ϕ_i and ϕ_j, respectively, the FM component J_F (<0) increases in magnitude with the overlap density ρ_ij = ϕ_iϕ_j, and the AFM component J_AF (>0) with the magnitude of the overlap integral S_ij = 〈ϕ_i|ϕ_j〉. The interaction between ϕ_i and ϕ_j leads to the energy split (Δe)_ij between them, which is related to S_ij as (Δe)_ij ∝ (S_ij)². Therefore, the overall spin exchange J can be FM if (Δe)_ij is small. Figure 28c shows that the energy gap between the [+] and [−] d-states of NH₄CuCl₃ decreases in the order, YY > NY > NN. If the energy split (Δe)_ij becomes smaller, then the associated spin exchange can become FM, hence the associated dimer becomes a triplet dimer. It is most likely that the three spin dimers A, B and C of NH₄CuCl₃, as experimentally observed, might be assigned to the Cu₂Cl₆²⁻ anions surrounded with the NN, NY and YY orientations of the NH₄⁺ cations, respectively. This is a consequence that a given NH₄CuCl₃ sample does not have a uniform orientation of the NH₄⁺ cations. It rather consists of regions possessing mainly YY, NY and NN orientations of the NH₄⁺ cations.

4. Magnets of Ferrimagnetic Fragments

4.1. Linear Trimers and Chains

4.1.1. Isolated Linear Trimers in Mn₃(PO₄)₂

Manganese diphosphates Mn₃(PO₄)₂ are found in several different phases, namely α, β’, and γ [48], which undergo a long-range AFM order at T_N = 21.9, 12.3, and 13.3 K, respectively. The 3D crystal structures of these phases consist of corner- and edge-sharing MnO₅ and MnO₆ polyhedra, which are further bridged by PO₄ tetrahedra. In γ-Mn₃(PO₄)₂, each Mn2O₆ octahedron corner-shares with two Mn1O₅ trigonal bipyramids to form a Mn1-Mn2-Mn1 linear trimer (Figure 29a), and these trimers are edge-shared either in a head-to-tail (Figure 29b) or tail-to-tail (Figure 29c) fashion. The magnetization curves of these phases show spin-flop-like features at a low magnetic field, but a 1/3-magnetization plateau is found only for the α- and γ-Mn₃(PO₄)₂ modifications. As shown in Figure 29d, the 1/3-plateau of the γ-phase is very wide.

A simplified view of the layer that linear Mn1-Mn2-Mn1 trimers form by a head-to-tail bridging is presented in Figure 30a. The spin lattice of this layer is defined by the intra-trimer exchange J₃ and the inter-trimer exchange J₂. Such layers make a 3D structure by a tail-to-tail bridging between the trimers lying in adjacent layers, which leads to the interlayer exchange J₁ (Figure 30b). The spin exchanges J₁, J₂ and J₃ of γ-Mn₃(PO₄)₂ (Figure 29a) are all AFM and are estimated to be 1.7, 4.7 and 10.5 K, respectively [48]. Each layer defined by the exchanges J₃ and J₂ are ferrimagnetic because each linear trimer is ferrimagnetic due to the strong AFM exchange J₃, and because the head-to-tail coupling between two ferrimagnetic trimers does not cancel their moments (Figure 30c). Such ferrimagnetic layers are coupled antiferromagnetically via the exchange J₁ to form an AFM magnetic ground state responsible for the AFM ordering.

The gradual increase in the magnetization M of γ-Mn₃(PO₄)₂ with increasing μ₀H in the region of 0–7.5 T mirrors the breaking of the interlayer J₁ bonds, leading to the ferrimagnetic layers. This field-induced ferrimagnetic state at 7.5 T explains the 1/3-plateau. For each ferrimagnetic layer to go beyond the 1/3-plateau, it is necessary to break two J₃ bonds of a trimer, which is accompanied by the breaking of a J₂ bond (Figure 30d). The wide plateau between 7.5 and 23.5 T reflects the difficulty of simultaneously breaking one J₂ and two J₃ bonds when a linear AFM trimer becomes FM.

4.1.2. Bent Trimers in Cu₃(P₂O₆OH)₂

The building blocks of Cu₃(P₂O₆OH)₂ are Cu2O₆ octahedra and Cu1O₅ trigonal bipyramids, as found in γ-Mn₃(PO₄)₂. However, each Cu2O₆ octahedron edge-shares with two Cu1O₅ trigonal bipyramids in Cu₃(P₂O₆OH)₂ (Figure 31a) [49] to form linear Cu1-Cu2-Cu1 trimers, in contrast to the corner-sharing found in γ-Mn₃(PO₄)₂ (Figure 29a). Cu₃(P₂O₆OH)₂ exhibits a 1/3-magnetization plateau above 12 T (Figure 32a) [50], which was initially interpreted by supposing that its spin lattice is a J₁-J₂-J₂ chain made up of ferrimagnetic linear Cu1-Cu2-Cu1 trimers (Figure 31b). However, this model is not consistent with the spin exchanges of Cu₃(P₂O₆OH)₂ evaluated using DFT+U calculations [51]; the latter found that the exchange J₂ is practically zero, and that Cu₃(P₂O₆OH)₂ has a 2D spin lattice made up of three spin exchanges J₁, J₃ and J₆ (479, 69 and 90 K, respectively) as shown in Figure 31c.

The three AFM exchanges J₁, J₃ and J₆ lead to an AFM spin arrangement in the 2D lattice (Figure 32b), where half the Cu2 sites have up-spins, and the remaining half down-spins (i.e., those in green circles). Since J₃ and J₆ are considerably weaker than J₁, the increase in M with μ₀H is achieved by breaking these magnetic bonds, i.e., by flipping the down-spin to up-spin at the Cu2 sites, one at a time. This spin flipping simultaneously breaks one J₆ and two J₃ bonds. When all down-spins at the Cu2 sites are flipped, a ferrimagnetic configuration with M = M_sat/3 is reached (Figure 32c). Note that this spin arrangement is equivalent in energy to another ferrimagnetic spin arrangement shown in Figure 32d. Either ferrimagnetic arrangement can be decomposed into bent ferrimagnetic trimers, as illustrated in Figure 32d. The plateau above 12 T is wide because the J₁ bond is strong and because a spin flip from (↑↓↑) to (↑↑↑) in each ferrimagnetic trimer, which must occur to increase the magnetization beyond M = M_sat/3, simultaneously breaks one J₁ and one J₃ bond.

4.1.3. Heisenberg Chains in Volborthite Cu₃V₂O₇(OH)₂·2(H₂O)

Volborthite, Cu₃V₂O₇(OH)₂·2H₂O, has a layered crystal structure, in which the layers of composition Cu₃O₆(OH)₂ parallel to the ab-plane are pillared by pyrovanadate V₂O₇ groups and crystal water molecules occupy the voids between the layers. The Cu²⁺ ions in each Cu₃O₆(OH)₂ layer have a kagomé-like arrangement (Figure 33a). Below room temperature, volborthite undergoes two structural phase transitions, one at ~292 K from a C2/c phase to a I2/a phase, and the other at ~155 K from the I2/a phase to a P2₁/c phase [52]. The latter structural phase transition generates two kagomé layers slightly different in structure. Below 1.5 K, volborthite exhibits magnetic order, indicated by two anomalies in the magnetic specific heat [53].

Volborthite had been regarded as a kagomé spin lattice system [54]. However, according to a recent study [53], it is not a kagomé spin lattice but an S = 1/2 AFM uniform Heisenberg (AUH) chain that describes the magnetic properties of volborthite at low temperatures. This observation reflects the fact that the spin lattice of a magnet does not necessarily follow the geometrical pattern of its magnetic ion arrangement but is determined by that of strongly interacting spin exchange paths between the magnetic ions [55]. The CuO₆ octahedra of volborthite accommodating the Cu²⁺ ions are axially elongated, so their x²-y² magnetic orbitals lie in their CuO₄ square planes perpendicular to the elongated Cu-O bonds. The arrangement of these CuO₄ planes in volborthite, depicted in Figure 33b, is highly anisotropic forming the Cu2-Cu1-Cu2 linear trimers bridged by Cu2-O-Cu1 linkages. Within each Cu₃O₆(OH)₂ layer, the Cu2-Cu1-Cu2 trimers are arranged as in Figure 33c. The spin exchanges of volborthite determined using DFT+U calculations show [53] that the strongest AFM spin exchange, J₂ (550 and 582 K for the two different layers), makes each Cu2-Cu1-Cu2 linear trimer ferrimagnetic, and these ferrimagnetic trimers are coupled antiferromagnetically by the next strongest spin exchange J₄ (78 K for both layers) to form two-leg spin ladders. All other spin exchanges are negligibly small, and adjacent spin ladders are entangled in their legs (Figure 33c). In essence, the kagomé-like arrangement of Cu²⁺ ions in Cu₃V₂O₇(OH)₂·2H₂O gives rise to weakly interacting two-leg spin ladders with Cu2-Cu1-Cu2 trimers as rungs, which have an AFM spin arrangement as depicted in Figure 33d.

At low temperatures where thermal excitations within each trimer rung are absent, each rung acts as an effective S = 1/2 species due to a strong AFM coupling between adjacent Cu²⁺ sites, so that each two-leg spin ladder should behave as an effective S = 1/2 AUH chain (Figure 33e) [53]. Indeed, the magnetic susceptibility of volborthite at low temperatures (below 75 K) is very well described by an S = 1/2 AUH chain model to find the nearest-neighbor spin exchange J_C = 27.8(5) K, as shown in Figure 34a [53]. On lowering the temperature, the susceptibility shows a broad maximum and converges to a nonzero value as the temperature approaches zero, a characteristic feature expected for an S = 1/2 AUH chain. Volborthite exhibits an extremely wide 1/3-magnetization plateau above 28 T continuing over 74 T at 1.4 K (Figure 34b) [56]. Before reaching the value of M = M_sat/3, the magnetization increases with field because each linear (↓↑↓) trimer is converted to a linear (↑↓↑) trimer, breaking four J₄ bonds, eventually reaching the ferrimagnetic state (Figure 33f) at ~28 T. A further increase in the magnetic field does not increase magnetization leading to the 1/3-plateau because it requires breaking two J₂ bonds to convert a (↑↓↑) rung to a (↑↑↑) rung and because the J₂ bond is very strong. There is a theoretical study on the magnetization plateau of a two-leg spin ladder [57]. In our analysis of the magnetization plateau of volborthite obtained at 1.4 K [55], we employed the S = 1/2 AUH chain model (Figure 34c) because each (↑↓↑) rung acted as an effective S = 1/2 entity at 1.4 K. As shown in Figure 34c, the experimental magnetization data were very well described by the S = 1/2 AUH chain model (Figure 34c) using the nearest-neighbor spin exchange J_C of 27.5 K [53], just as were the magnetic susceptibility data below 75 K.

4.1.4. Head-to-Tail Coupling of Bent Trimers and Anisotropic 1/3-Plateau in Cs₂Cu₃(SeO₃)₄·2(H₂O)

Cs₂Cu₃(SeO₃)₄·2H₂O consists of two nonequivalent Cu atoms, Cu1 and Cu2 in the 1:2 ratio, each forming Cu1O₄ and Cu2O₄ square planes, respectively. The 3D framework of Cs₂Cu₃(SeO₃)₄·2H₂O is formed by the corner-sharing of Cu1O₄ and Cu2O₄ square planes [58]. As depicted in Figure 35a,b, each Cu1O₄ square plane is corner-shared with four Cu2O₄ square planes such that the four Cu2 atoms around a Cu1 atom make a Cu1(Cu2)₄ tetrahedron (Figure 35c). Condensing such Cu1(Cu2)₄ tetrahedra by sharing their Cu2 corners leads to the 3D network of Cu²⁺ ions of Cs₂Cu₃(SeO₃)₄·2H₂O, which can be described as resulting from the fusing of chair-shape hexagonal rings (Figure 35d).

Cs₂Cu₃(SeO₃)₄(H₂O)₂ is a ferrimagnet ordering at T_C = 20 K with residual magnetization at about M_sat/3 (Figure 36a). In general, ferrimagnetism occurs when ferrimagnetic fragments are combined antiferromagnetically in a head-to-tail bridging pattern so that the magnetic moment of each ferrimagnetic fragment is not quenched. Such ferrimagnetic units in Cs₂Cu₃(SeO₃)₄(H₂O)₂ should consist of one Cu1 and two Cu2 atoms, given that the Cu1 and Cu2 atoms occur in the 1:2 ratio. DFT+U calculations [58] show that the nearest-neighbor exchange J₁ (Figure 35c) is strong (256 K) but other exchanges are negligibly weak. This makes all nearest-neighbor Cu1…Cu2 linkages antiferromagnetically coupled, so the ferrimagnetic fragments needed to explain the ferrimagnetism of Cs₂Cu₃(SeO₃)₄(H₂O)₂ are the bent Cu2-Cu1-Cu2 units with (↑↓↑) spin configuration.

When measured for a single crystal sample of Cs₂Cu₃(SeO₃)₄(H₂O)₂ parallel (||) and perpendicular (⊥) to the c-direction (Figure 36a,b), the values of the magnetization at μ₀H = 7 T are quite different, namely, M_⊥ = 1.18 μ_B, whereas M_|| = 0.93 μ_B [58]. There are three factors contributing to this highly anisotropic magnetization plateau: the nearly orthogonal arrangements of the Cu2O₄ square planes around each Cu1O₄ square plane (Figure 35b), the strong nearest-neighbor antiferromagnetic exchange J₁ and the anisotropic g-factor of Cu²⁺ ions in a square planar coordination site. The magnetic anisotropy of a magnetic ion arises from SOC. In the spin-only description, in which orbital information is suppressed, the effect of SOC on magnetic anisotropy is discussed by introducing g-factor g, different from 2 [2]. That is, the magnetic moment μ of a spin site with spin S is given by μ = gS, where g is the anisotropic g-factor of the magnetic ion. The g-factors of Cu²⁺ at a square planar coordination site along the c-axis (||c for short) and perpendicular to the c-axis (⊥c for short) are written as

g_|| = 2 + Δg_||

g_⊥ = 2 + Δg_⊥

where Δg_|| > Δg_⊥ (approximately, 0.25 vs. 0.05) (Figure 37a). Δg_|| and Δg_⊥ are proportional to the amounts of unquenched orbital angular momenta [2], so the associated magnetic moments are also anisotropic, namely,

μ_|| = g_||S = (2 + Δg_||)S

μ_⊥ = g_⊥S = (2 + Δg_⊥)S

Thus, the magnetic moment of the Cu²⁺ ion is greater along the ||z direction than along the ⊥z direction.

To simplify our analysis of the observed magnetization anisotropy, we assume that each Cu2O₄ unit is truly a square planar in shape, and the planes of the Cu1O₄ units are truly orthogonal to the plane of the idealized Cu2O₄ unit (Figure 37b,c). Then, all three CuO₄ square planes of a bent Cu2-Cu1-Cu2 ferrimagnetic fragment are identical except for their spatial arrangement. Since J₁ is AFM, the three Cu²⁺ spins of a bent Cu2-Cu1-Cu2 fragment have a (↑↓↑) spin arrangement. For the magnetic field H||c, the three spins are aligned along the crystallographic ||c direction (Figure 35b) so that the magnetic moments on the two up-spin sites are both μ_⊥, while that on the down-spin site is -μ_|| (Figure 37b). For the magnetic field H⊥c, however, the three spins are aligned along the ⊥c direction so that the magnetic moments on the two up-spin sites are both μ_||, while that on the down-spin site is -μ_⊥ (Figure 37c). Therefore, the total moments of the ferrimagnetic fragment are given by

μ_tot (||c) = 2μ_⊥ − μ_|| = (2g_⊥ − g_||)S ≈ 1 + (Δg_⊥ − Δg_||/2) ≈ 0.94

μ_tot (⊥c) = 2μ_|| − μ_⊥ = (2g_|| − g_⊥)S ≈ 1 + (Δg_|| − Δg_⊥/2) ≈ 1.195

This difference explains why the 1/3-magnetization plateau has a moment larger than 1 μ_B for H⊥c, but a moment smaller than 1 μ_B for H||c, and why the magnetization plateau deviates more from 1 μ_B for H⊥c than for H||c.

4.1.5. Haldane Chain of Cu₆ Clusters and a 1/3-Magnetization Plateau in Fedotovite K₂Cu₃O(SO₄)₃

Fedotovite, K₂Cu₃O(SO₄)₃, consists of Cu₆ clusters (Figure 38a), which are made up of three different Cu atoms, Cu1, Cu2 and Cu3, in strongly distorted square planar coordination. Two Cu3O₄ planes are edge-shared to form a twisted Cu3₂O₆ dimer, and one bridging O atom of this dimer is corner-shared with two Cu1O₄ square planes while the other bridging O atom is corner-shared with two Cu2O₄ square planes. Thus, the atoms of a Cu₆ cluster have the shape of an edge-sharing tetrahedra (Figure 38b). Such Cu₆ clusters form chains along the b-direction (Figure 38c) [59].

K₂Cu₃O(SO₄)₃ undergoes a 3D AFM ordering at T_N = 3.1 K. Above this temperature, K₂Cu₃O(SO₄)₃ behaves as an S = 1 Haldane chain system (Figure 39a), with each Cu₆ cluster acting as an S = 1 species [60]. This implies that the spin exchange coupling between six Cu²⁺ ions of the Cu₆ cluster is very strong so that thermal excitations within each Cu₆ cluster are weak. In addition, K₂Cu₃O(SO₄)₃ exhibits a 1/3-plateau above T_N (Figure 39b) [60], implying that each Cu₆ cluster forms a ferrimagnetic fragment with (4↑2↓) spin configuration. To confirm this interpretation, we examine the eight spin exchanges J₁–J₈ defined in Figure 38. The values of these exchanges determined using DFT+U calculations are summarized in Figure 38 (see Section S9 of the Supplementary Materials). The exchange J₁ between the Cu1²⁺ ions is strongly AFM, and so is the exchange J₂ between the Cu2²⁺ ions. In contrast, the exchange J₃ between the Cu3²⁺ ions is strongly FM. There are four strong AFM exchanges between the Cu1²⁺ and Cu3²⁺ ions (namely, 2J₄ + 2J₅) and between the Cu2²⁺ and Cu3²⁺ ions (namely, 2J₆ + 2J₇). Since these AFM interactions dominate over J₁ and J₂, the energetically favorable spin arrangement for a Cu₆ cluster is either a (2↑2↓2↑) or a (2↓2↑2↓) configuration (Figure 40a), which are both ferrimagnetic. Due to the AFM inter-cluster exchange J₈, the ferrimagnetic Cu₆ clusters prefer to couple antiferromagnetically (Figure 40b). The gradual increase in the magnetization with the magnetic field from 0 to about 20 T is explained by the field-induced breaking of the inter-cluster magnetic bonds J₈, one at a time, eventually reaching the ferrimagnetic state (Figure 40c), in which all J₈ bonds are broken with M = M_sat/3.

It should be noted that the magnetic susceptibility of K₂Cu₃O(SO₄)₃ is rather weak (Figure 39a). This is a direct consequence of the fact that the spin exchanges J₁–J₇ leading to the ferrimagnetic fragment Cu₆ are rather strong. The latter is necessary for the effective S = 1 behavior of the Cu₆ clusters. The magnetic susceptibility of this Haldane chain system is weak despite the presence of six Cu²⁺ cations in each cluster due to the (↑↓↑) arrangement of three FM dimers.

4.1.6. Trigonal Arrangement of Ferromagnetic Chains in Ca₃Co₂O₆

Ca₃Co₂O₆ consists of Co₂O₆ chains in which Co2O₆ trigonal prisms alternate with Co1O₆ octahedra by sharing their triangular faces (Figure 41a) [61]. These chains running along the c-direction have a trigonal arrangement (Figure 41b), with Ca²⁺ cations occupying the positions in between these chains. Each Co₂O₆ chain is FM [62] so that the spin lattice of Ca₃Co₂O₆ can be described as a trigonal lattice by treating each FM chain a pseudo-magnetic ion with a giant spin moment. Both Co1 and Co2 atoms of Ca₃Co₂O₆ are in the oxidation state of +3 [3,63], indicating that each Co2O₆ trigonal prism has six electrons to occupy its d-states, and so does each Co1O₆ octahedron. This made it difficult to explain why Ca₃Co₂O₆ exhibited a uniaxial magnetism [1,3,63] because the configuration (3z² − r²)¹(xy, x² − y²)⁰ predicted for a Co2O₆ trigonal prism does not lead to uniaxial magnetism (Figure 42a, left). A systematic study [3] of Ca₃Co₂O₆ based on DFT+U and DFT+U+SOC calculations, including geometry relaxations allowing for Jahn–Teller distortions, showed that the uniaxial magnetism of Ca₃Co₂O₆ is a consequence of three effects: (a) the FM spin arrangement between the Co³⁺ ions of adjacent Co2O₆ and Co1O₆ polyhedra, (b) a direct metal–metal interaction between adjacent Co³⁺ ions mediated by their 3z² − r² orbitals (Figure 41c), and (c) the SOC of the Co³⁺ ion at the trigonal prism site (Figure 42a, right).

A single crystal sample of Ca₃Co₂O₆ exhibits a 1/3-magnetization plateau when the field is parallel to the chain direction, with the magnetization curve showing a step-like feature. When the field is perpendicular to the chain direction, there occurs no magnetization plateau [62]. (Experimentally, it is very difficult to align a single crystal sample of a uniaxial magnet precisely perpendicular to the field. A very slight misalignment can easily give rise to nonzero magnetization). As discussed for CoGeO₃ in the previous section, these observations are a direct consequence of the fact that Ca₃Co₂O₆ is a uniaxial magnet with a spin moment along the chain direction. Ca₃Co₂O₆ can be described in terms of a regular trigonal spin lattice of simple magnetic ions once each FM Co₂O₆ chain is treated as a single magnetic ion (see Section 5). It is noteworthy that Ca₃Co₂O₆ reaches a full saturation magnetization at a rather low field (namely, at about 3.5 T). This reflects that the interchain magnetic bonds are weak.

4.2. Distorted Triangular Fragments

4.2.1. Diamond Chains of NaFe₃(HPO₃)₂(H₂PO₃)₆

NaFe₃(HPO₃)₂(H₂PO₃)₆ has two nonequivalent Fe atoms, Fe1 and Fe2, forming Fe1O₆ and Fe2O₆ octahedra [64]. The HPO₃ unit occurs in two different forms, i.e., H-PO₃ and PO₂(OH), but the H₂PO₃ unit only in the form H-PO₂(OH). Consequently, both Fe1 and Fe2 atoms are present as Fe³⁺ (S = 5/2) ions. These Fe³⁺ ions are bridged by H-PO₃, PO₂(OH) or H-PO₂(OH), as illustrated by Figure 43a. DFT+U calculations [65] showed that four AFM spin exchanges (i.e., J₂, J₃, J₄ and J₆ depicted in Figure 43a) are relevant and comparable in magnitude (~2 K). The three spin exchanges J₂, J₃ and J₆ couple the Fe³⁺ cations to diamond chains, which are interlinked by J₄ to form 2D layers (Figure 43b). Such layers are stacked to form the 3D spin lattice of NaFe₃(HPO₃)₂(H₂PO₃)₆. In addition, there are weak interlayer AFM spin exchanges (see below for further discussion).

As shown in Figure 44a (inset), NaFe₃(HPO₃)₂(H₂PO₃)₆ undergoes a ferrimagnetic ordering below T_C = 9.5 K and exhibits a 1/3-magnetization plateau in the magnetization [65]. The plateau extends to ∼8 T. Above this field, the magnetization increases linearly with field until the saturation is reached at ∼27 T. As discussed in Section 2.4, we suppose that the spin lattice of NaFe₃(HPO₃)₂(H₂PO₃)₆ undergoes field-induced partitioning into ferrimagnetic triangular clusters (Figure 44b). Then, the spin arrangement, (↑↓↑), (↑↑↓) or (↓↑↑), of each cluster leads to one positive moment per cluster. Among these three, the (↑↓↑) arrangement at each triangular fragment is energetically most favorable because of the inter-diamond spin exchange J₄, thereby leading to the ferrimagnetic state with M = M_sat/3 (Figure 44c). To increase the magnetization beyond M_sat/3, each ferrimagnetic triangle must break two magnetic bonds (Figure 44d) within a cluster, which is accompanied by the breaking of two inter-diamond J₄ bonds. This needs a high enough magnetic field, hence explaining the 1/3-plateau extending to ~8 T. When the field increases beyond 8 T toward the saturation magnetization, each ferrimagnetic triangle begins to break two magnetic bonds (Figure 44d) within a cluster.

In general, the ground state of a magnet composed of ferrimagnetic layers is AFM because the weak high-spin orbital interactions between adjacent ferrimagnetic layers favor an AFM coupling rather than an FM coupling [66]. Indeed, DFT+U calculations found [65] that the interlayer spin exchanges J₁ and J₅, which are weakly AFM (~0.6 and ~0.4 K, respectively), and form spin-frustrated (J₁, J₅, J₄) triangles between adjacent ferrimagnetic layers as depicted in Figure 45a. The interlayer FM coupling (Figure 45b) leads to the (J₁, J₅, J₄) triangles, which have J₅ magnetic bonds and J₁ broken magnetic bonds. In contrast, the interlayer AFM coupling (Figure 45c) leads to the (J₁, J₅, J₄) triangles, which have J₅ broken magnetic bonds and J₁ magnetic bonds. Since J₁ is slightly stronger than J₅, the interlayer AFM coupling is energetically favored over the interlayer FM coupling. This is consistent with the general observation that the magnetic ground state of a magnet composed of ferrimagnetic layers is AFM rather than ferrimagnetic. Furthermore, we note from Figure 44a that, below ~2 T, the magnetization rises sharply with field toward M_sat/3. This observation can be related to the breaking of the weak interlayer magnetic bonds (i.e., J₅ and J₁) in the AFM ground state. It will be interesting to examine whether the magnetic ground state of NaFe₃(HPO₃)₂(H₂PO₃)₆ is AFM or ferrimagnetic.

4.2.2. Three-Dimensional Spin Lattice and Anisotropic Plateau Width in Azurite Cu₃(CO₃)₂(OH)₂

The important structural building blocks of azurite Cu₃(CO₃)₂(OH)₂ [67] are the Cu1O₄ and Cu2O₄ square planes containing their x²-y² magnetic orbitals. Each Cu1²⁺ ion is surrounded by four Cu2²⁺ ions to form a Cu₅ ribbon (Figure 46a) where the four Cu2O₄ square planes of a Cu₅ ribbon are nearly perpendicular to the central Cu1O₄ square plane. This structural feature implies that the orientations of the x²-y² magnetic orbitals are the keys to understanding the magnetic properties and especially the magnetization plateau observed for azurite. By sharing their edges, such Cu₅ ribbons form “diamond chains” along the b-direction (Figure 46b). Each Cu₅ ribbon is described by three spin exchanges J₁–J₃, as used early on by Rule et al. to discuss the temperature dependence of the magnetic susceptibility [68]. Kang et al. carried out DFT+U calculations [69] to find that adjacent diamond chains interact through the spin exchanges J₄, which takes place through the bridging of CO₃²⁻ ions (Figure 46c) to form a layer of interacting diamond chains, and that the dimer exchange J₂ (=363 K) dominates with J₁/J₂ ≈ J₃/J₂ = 0.24 and J₄/J₂ = 0.13. Jeschke et al. proposed a modified diamond chain model by including a spin exchange between the Cu1 cations [70]. Topologically, the 2D spin lattice of azurite is identical with that for NaFe₃(HPO₃)₂(H₂PO₃)₆ discussed in the previous section. So, one might expect that each layer of azurite is ferrimagnetic as shown in Figure 44c, and such ferrimagnetic layers are antiferromagnetically coupled to form an ordered 3D AFM state, and that azurite exhibits a 1/3-plateau as NaFe₃(HPO₃)₂(H₂PO₃)₆ does. Indeed, azurite orders antiferromagnetically at T_N ≈ 1.9 K [71] and exhibits a 1/3-magnetization plateau below this temperature, as shown in Figure 47.

The 1/3-plateau of azurite presents two features remarkably different from that of NaFe₃(HPO₃)₂(H₂PO₃)₆ (Figure 44a): (1) The field H_c1 where the M = M_sat/3 point starts on increasing the field from zero is much greater for azurite (over 10 T) than for NaFe₃(HPO₃)₂(H₂PO₃)₆ (~1 T). In NaFe₃(HPO₃)₂(H₂PO₃)₆, the gradual increase in M with μ₀H in the region of 0–H_c1 is ascribed to the breaking of the interlayer magnetic bonds. Since H_c1 is much higher for azurite, the interlayer AFM spin exchange of azurite must be substantial. (2) The width of the 1/3-plateau is much wider when the field is perpendicular to the b-axis (⊥b) than parallel to the b-axis (||b); the H_c1 is greater for H||b than for H⊥b (16 vs. 11 T), whereas the H_c2 is smaller for H||b than for H⊥b (26 vs. 30 T). Thus, the plateau width is significantly larger for H⊥b. However, in contrast to these differences in the plateau widths, the saturation fields H_c3 in both orientations are identical (32.5 T) [71]. In the following, we examine why these observations occur.

• Interlayer spin exchange in azurite

Two-dimensional layers of interlinked diamond chains are stacked as depicted in Figure 48a. There occur two Cu-O…O-Cu type spin exchange paths, J₅ and J₆, between adjacent layers (Figure 48b). The J₅ paths take place between Cu1²⁺ and Cu2²⁺ ions (Figure 48c), and the J₆ paths between two Cu2²⁺ ions (Figure 48d). The values of J₅ and J₆, determined using energy-mapping analysis (see Section S10 of the Supplementary Materials) are not negligible compared with the inter-diamond exchange J₄ within a layer; J₅ is AFM while J₆ is FM, and J₅/J₄ = 0.7 and J₆/J₄ = −0.5. The presence of the AFM interlayer exchange J₅, which is only slightly weaker than J₄, confirms our suggestion that the increase in magnetization with a field in the 0–H_c1 region is related to the breaking of the interlayer magnetic bonds, and azurite reaches the state consisting of (↑↓↑) ferrimagnetic triangular fragments at H_c1.

B.

Magnetic anisotropy affecting Dzyaloshinskii–Moriya (DM) interactions

The different widths of the 1/3-plateaus were explained by Kikuchi et al. [71] in terms of DM interactions by assuming a DM vector perpendicular to both the J₂ bond and the b-axis. So far, however, it is unknown why such a DM vector should exist in azurite, or why the 1/3-plateau starts at a higher field for H||b than for H⊥b. To resolve these questions, we examine how the Zeeman energies of the Cu²⁺ ions in azurite are affected by the field direction based on the following three observations:

(1)

The g-factor for the Cu²⁺ ion of a CuO₄ square plane is anisotropic; the g-factor along the four-fold rotational axis, g_|| = 2 + Δg_|| ≈ 2.25, is substantially greater than that perpendicular to this axis, g_⊥ = 2 + Δg_⊥ ≈ 2.05 [72].

(2)

In general, the g-factor of a magnetic ion measured with magnetic field H in a certain direction can be written as g = 2 + Δg, where Δg is related to the unquenched orbital moment δL on the magnetic ion along that direction as [2]

$$∆g=\lambda {\displaystyle \frac{\delta L}{{\mu }_{B}H}}\propto \delta L,$$

(11)

where λ is the SOC constant of the magnetic ion, i.e., Δg is a measure of δL.

(3)

In a DM interaction ${\overrightarrow{D}}_{ab}·({\overrightarrow{S}}_a\times {\overrightarrow{S}}_b)$ between two spins located at the sites a and b and coupled by spin exchange J_ab, the DM vector ${\overrightarrow{D}}_{ab}$ is related to the unquenched orbital moments $\delta {\overrightarrow{L}}_a$ and $\delta {\overrightarrow{L}}_b$ of the magnetic ions at the sites a and b, respectively, as [2,73]

$${\overrightarrow{D}}_{ab}=\lambda J_{ab}\left(\delta {\overrightarrow{L}}_a-\delta {\overrightarrow{L}}_b\right)\propto (∆g_a-∆g_b)$$

(12)

The essential key to understanding the observation of the different widths of the plateaus in azurite is that each Cu1O₄ square plane is nearly perpendicular to the four Cu2O₄ square planes within each diamond chain, and also nearly perpendicular to the two Cu2O₄ square planes between two adjacent diamond chains (Figure 49a). To simplify our analysis, we assume that the Cu1O₄ and Cu2O₄ units have an ideal planar square shape, and that the arrangement of these square planes are ideally orthogonal such that the two edges of the Cu1O₄ plane are aligned along the y- and z-axes, but those of the Cu2O₄ planes along the x- and y-axes (Figure 49b). Then, the four-fold rotational axis of the Cu1O₄ plane is parallel to the x-axis (||x), but that of each Cu2O₄ plane is parallel to the z-axis (||z). With this choice of the Cartesian coordinate system, the y-direction is approximately aligned along the b-direction, i.e., the diamond chain direction. Then, for the Cu1O₄, the g-factor of the Cu²⁺ cations is g_|| for H||x, but g_⊥ for H⊥x (Figure 49c). For the Cu2O₄ planes, however, the g-factor of the Cu²⁺ is g_|| for H||z, but g_⊥ for H⊥z (Figure 49c).

Using the results summarized in Figure 49c, the Zeeman energy for the three Cu²⁺ ions of each ferrimagnetic triangle (namely, one Cu1²⁺ and two Cu2²⁺ ions, Figure 46d) is calculated as follows:

For H||x: $E_Z=({\mathrm{g}}_{\left|\right|}+2{\mathrm{g}}_{\perp }){\mu }_0{\mu }_{B}H\mathrm{S}$

For H||y: $E_Z=3{\mathrm{g}}_{\perp }{\mu }_0{\mu }_{B}H\mathrm{S}$

For H||z: $E_Z=({2\mathrm{g}}_{\left|\right|}+{\mathrm{g}}_{\perp }){\mu }_0{\mu }_{B}H\mathrm{S}$

For the Cu²⁺ ion, ${\mathrm{g}}_{\perp }$< ${\mathrm{g}}_{\left|\right|}$ (i.e., ~2.05 vs. ~2.25). Thus, at a given magnetic field strength μ₀H, the Zeeman energy is lower for H||y than either for H||x or for H||z. This implies that in reaching the energy required for breaking a certain magnetic bond, a higher magnetic field is necessary when the field is aligned along the y-direction. This explains why the H_c1 is higher for H||b than for H⊥b (16 vs. 11 T) since the y-direction is approximately aligned along the b-direction of azurite. Interestingly, this identifies the magnetic bonds to break in this process are the interlayer magnetic bonds, i.e., the interlayer exchange coupling is a crucial factor leading to the different widths of the 1/3-plateaus in azurite.

We now examine why H_c1 is lower for H||b than for H⊥b (i.e., 26 vs. 30 T) by noting that the H_c2 marks the point where each (2↑1↓) ferrimagnetic triangle of Figure 46d begins to change into a (3↑0↓) ferromagnetic triangle. This change breaks six magnetic bonds (namely, 2J₁ + 2J₃ +2J₄) around a Cu1²⁺ ion. As pointed out earlier, J₁/J₂ ≈ J₃/J₂ ≈ 0.24 and J₄/J₂ ≈ 0.13, so (2J₁ + 2J₃ +2J₄) ≈ 0.74J₂. The DM interactions of the six magnetic bonds are identical except for the magnetic bond strengths. Therefore, we treat the six DM interactions involving one Cu1²⁺ ion as one DM interaction of the Cu1²⁺ ion with a hypothetical Cu2²⁺ ion with effective bond J_eff = 0.74J₂. Then, by considering the Cu1²⁺ and the hypothetical Cu2²⁺ ions at sites a and b, respectively, we obtain the following results,

For μ₀H||x: $D_{ab}\propto \lambda J_{\mathrm{e}\mathrm{f}\mathrm{f}}(∆g_{\left|\right|}-∆g_{\perp }) \approx 0.2\lambda J_{\mathrm{e}\mathrm{f}\mathrm{f}}<0$

For μ₀H||y: $D_{ab}\propto \lambda J_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(∆g_{\perp }-∆g_{\perp }\right) \approx 0$

For μ₀H||z: $D_{ab}\propto \lambda J_{\mathrm{e}\mathrm{f}\mathrm{f}}(∆g_{\perp }-∆g_{\left|\right|}) \approx -0.2\lambda J_{\mathrm{e}\mathrm{f}\mathrm{f}}>0,$

where we used the fact that λ < 0 for Cu²⁺ with more than a half-filled d-shell. The above results show that the DM interaction vanishes for H||y. The DM vector for H||x is opposite in sign to that for H||z. For H||z, the DM interaction raises the Zeeman energy, so the magnetic bond breaking occurs at a lower field (compared with the H||y case). For H||x, however, the DM interaction lowers Zeeman energy, forcing the magnetic bond breaking to a higher field. What is observed for azurite can be understood if the ⊥b direction is close to the ||x direction. The ||z direction is also approximately the ⊥b direction, but the DM interaction for H||z raises the Zeeman energy while that for H||x lowers it. This leads to the prediction that the plateau widths increase in the order,

H||z < H||y < H||x.

It would be interesting to verify this prediction experimentally.

5. Trigonal vs. Kagomé Magnets

The magnetization plateaus of magnets with triangular [74] and kagomé [75,76,77,78,79,80] spin lattices have received more attention in theoretical studies than in experimental studies. These plateaus are less prominent compared with those found for other magnets of lower symmetry.

5.1. Cause for the Presence or Absence of a Clear-Cut 1/3-Magnetization Plateau

Magnets of triangular and kagomé spin lattices show contrasting behaviors in their magnetization, especially, in the development of magnetization plateaus. The 1/3-magnetization plateaus observed for trigonal spin-lattice magnets are generally narrow in their widths [81,82]. In the case of kagomé spin lattice magnets, it is often difficult to detect magnetization plateaus in terms of their M vs. H curves. Therefore, sometimes dM/dH vs. H plots have been employed to discuss the plateaus [83,84,85]. However, 1/3-plateaus are clearly observed in their M vs. H plots for trigonal spin lattice magnets. In the following, we examine the probable cause of this difference by regarding the kagomé and trigonal spin lattices as made up of non-overlapping ferrimagnetic fragments, namely, ferrimagnetic triangles indicated by shading in Figure 50a and Figure 50b, respectively. As discussed in Section 2.4, each ferrimagnetic triangle can assume three different spin arrangements (Figure 50c). Then, all possible ordered and disordered spin configurations representing the 1/3-magnetization plateau are generated by how each ferrimagnetic triangle adopts one of the three spin arrangements. For example, Figure 51 shows three ordered spin arrangements creating the 1/3-plateau state for a kagomé spin lattice, and Figure 52 shows those for a trigonal spin lattice. To probe the question of whether kagomé and trigonal spin lattices have a 1/3-magnetization plateau, we note that the magnetization of the whole spin lattice remains at M_sat/3 regardless of whether there are more or fewer inter-fragment magnetic bonds. Thus, in the following, we examine the most and least stable arrangements that a given (↑↓↑) ferrimagnetic fragment can have with the surrounding ferrimagnetic fragments.

Let us first examine possible spin arrangements around one ferrimagnetic fragment in a kagomé spin lattice. As depicted in Figure 53a, each shaded triangle, representing a ferrimagnetic fragment, is corner-shared with three unshaded triangles. The two sites on each edge of the unshaded triangle belong to two different ferrimagnetic fragments (see Figure 51) so that each ferrimagnetic fragment interacts with six different ferrimagnetic neighboring fragments. A chosen ferrimagnetic fragment makes the most stable inter-fragment spin arrangement by making six inter-fragment magnetic bonds (Figure 53b), and the least stable spin arrangement by making six inter-fragment broken bonds (Figure 53c). Obviously, it is not possible for every ferrimagnetic fragment to make six magnetic bonds with the six adjacent ferrimagnetic fragments. For, in making six bonds (broken bonds) with a chosen fragment, the six surrounding ferrimagnetic fragments should possess specific spin arrangements. These arrangements cannot be altered to make six bonds (broken bonds) for another ferrimagnetic fragment next to the chosen fragment. This means that there is a variation in the number of inter-fragment magnetic bonds each ferrimagnetic fragment can make, from six bonds to six broken bonds. The kagomé spin lattice as a whole is more (less) stable if it has more inter-fragment bonds (broken bonds) on average, implying that a good indicator for the width of the 1/3-plateau is the energy difference between the most and the least stable inter-fragment magnetic bonding. This energy difference corresponds to effectively 12 magnetic bonds, i.e., from six bonds (Figure 53b) to six broken bonds (Figure 53c) per ferrimagnetic fragment.

In a trigonal spin lattice, each ferrimagnetic fragment of a trigonal lattice is surrounded by 12 unshaded triangles and interacts with six adjacent ferrimagnetic fragments (Figure 54a). Three of these six make interactions through a corner, and the remaining three through an edge (indicated by red rectangles in Figure 54a). That is, in a trigonal spin lattice as well, a given ferrimagnetic fragment is surrounded by six ferrimagnetic fragments. In the interactions through a corner, the corner spin site can be either up-spin or down-spin. In the interactions through an edge, the two spins on the edge can be both up-spins or a combination of one up-spin and one down-spin, because this edge is a part of a (↑↓↑) triangle. Consequently, with six adjacent ferrimagnetic fragments, a ferrimagnetic fragment makes nine bonds and three broken bonds (i.e., effectively six bonds) in the most stable spin arrangement (Figure 54b), but two bonds and 10 broken bonds (i.e., effectively, eight broken bonds) in the least stable arrangement (Figure 54c). Thus, the energy difference between the most stable and the least stable arrangements is effectively 14 bonds.

The above analysis indicates that, between the most stable and the least stable arrangements, the trigonal spin lattice has only a slightly greater energy difference than the kagomé spin lattice (i.e., 14 vs. 12 magnetic bonds). From this, one might be led to speculate if trigonal and kagomé spin lattices have similar 1/3-plateau properties. However, we note that the end point of the 1/3-plateau occurs when a ferrimagnetic fragment starts to have a configuration change from (↑↓↑) to (↑↑↑). In a trigonal spin lattice, the (↑↓↑) to (↑↑↑) spin flip requires the breaking of six bonds in the most stable inter-fragment arrangement (Figure 54b), and that of two bonds in the least stable inter-fragment arrangement (Figure 54c). Namely, the spin flip requires energy in the most and least stable inter-fragment arrangements. This is not the case for a kagomé spin lattice. There, the (↑↓↑) to (↑↑↑) spin flip requires the breaking of four magnetic bonds at the site of the most stable inter-fragment arrangement (Figure 53b), but no energy at the site of the least stable inter-fragment arrangement (Figure 53c) because breaking two bonds within a ferrimagnetic fragment generates two bonds between the fragments. This implies that, during the field sweep from the most stable to the least stable distribution of the inter-fragment magnetic bonding, Zeeman energy causes (↑↓↑) to (↑↑↑) spin flips at certain down-spin sites with less favorable bonding connections with its neighboring fragments (e.g., Figure 53c,d) because, in such a case, the spin flip requires less energy than does the breaking of the inter-fragment bonds. This reasoning predicts that a kagomé spin lattice has a narrower 1/3-magnetization plateau than does a trigonal spin lattice, and this might be the reason why the M vs. H curves of kagomé spin lattices show a steady increase in magnetization with the field through the M_sat/3 point.

5.2. Variation in the 1/3-Plateau Widths in RbFe(MoO₄)₂, Ba₃CoSb₂O₉ and Ba₂LaNiTe₂O₁₂

The 2D antiferromagnets RbFe(MoO₄)₂ [86], Ba₃CoSb₂O₉ [87] and Ba₂LaNiTe₂O₁₂ [82] consist of trigonal layers made up of MO₆ (M = Fe, Co, Ni) octahedra (Figure 55a). In RbFe(MoO₄)₂, the upper and lower surfaces of such a layer are condensed by corner-sharing with MoO₄ tetrahedra (Figure 55b,c) in RbFe(MoO₄)₂, with Sb₂O₉ double octahedra in Ba₃CoSb₂O₉, and with TeO₆ octahedra in Ba₂LaNiTe₂O₁₂ (Figure 55d). RbFe(MoO₄)₂ consists of trigonal layers of Fe³⁺ (d⁵, S = 5/2) ions, Ba₃CoSb₂O₉—those of Co²⁺ (d⁷, S = 3/2) ions—and Ba₂LaNiTe₂O₁₂—those of Ni²⁺ (d⁸, S = 1) ions. RbFe(MoO₄)₂ undergoes a phase transition at T_N = 3.8 K into a 120° spin structure with all the spins confined in the basal plane. Application of an in-plane magnetic field induces a collinear spin state between 4.7 and 7.1 T, producing a 1/3-magnetization plateau (Figure 56a) [88]. Ba₃CoSb₂O₉ exhibits an AFM transition at T_N = 3.8 K and the powder neutron diffraction measurements show that it adopts a 120° spin structure in the ab-plane [87]. Under the magnetic field applied in the ab-plane, Ba₃CoSb₂O₉ exhibits a 1/3-magnetization plateau between 10 and 15 T (Figure 56b) [89]. Ba₂LaNiTe₂O₁₂ undergoes successive magnetic phase transitions at T_N1 = 9.8 K and T_N2 = 8.9 K [90]. The ground state is accompanied by a weak ferromagnetic moment, suggesting that it adopts a slightly canted 120° spin structure. The magnetization curve exhibits a 1/3-magnetization plateau between 35 and 45 T (Figure 56c) [82].

The observed widths Δ(μ₀H) of the 1/3-plateaus found for RbFe(MoO₄)₂, Ba₃CoSb₂O₉ and Ba₂LaNiTe₂O₁₂ are 2.4, 5.0 and ~10 T, respectively. In the previous section, we argued that the energy difference between the most stable and the least stable arrangements involving a (↑↓↑) ferrimagnetic triangle amounts to 14 nearest-neighbor magnetic bonds J. Thus, the Δ(μ₀H) values of these magnets should be related to their nearest-neighbor spin exchanges J as Δ(μ₀H) ∝ J. Precisely speaking, the spin exchange between two magnetic ions of spin S coupled by exchange constant J generates the energy JS² (Equation (1)). Since we compare the relative strengths of the magnetic bonds involving the ions of different spins, it is necessary to use the relationship Δ(μ₀H) ∝ JS². Then, according to the observed experimental Δ(μ₀H) values, the JS² values should increase in the order, RbFe(MoO₄)₂ < Ba₃CoSb₂O₉ < Ba₂LaNiTe₂O₁₂.

As depicted in Figure 57a, the nearest-neighbor exchange J in the three magnets is of the M-O…O-M type exchange. In general, the strength of such an exchange becomes stronger as the O…O contact distance decreases [2,91]. As summarized in Figure 57b, the O…O contact distances of the J exchange paths decrease in the order RbFe(MoO₄)₂ > Ba₃CoSb₂O₉ > Ba₂LaNiTe₂O₁₂. The standard deviation of the O…O distance in Ba₂LaNiTe₂O₁₂ is rather large. However, even if the largest O…O distance of 2.74 Å allowed by the standard deviation is considered, the trend RbFe(MoO₄)₂ > Ba₃CoSb₂O₉ > Ba₂LaNiTe₂O₁₂ still remains valid. We carried out an energy mapping analysis based on DFT+U calculations (see Sections S11–S13 of the Supplementary Materials) to find that J = 1.5 K for RbFe(MoO₄)₂, J = 6.2 K for Ba₃CoSb₂O₉ and J = 56 K for Ba₂LaNiTe₂O₁₂. (As expected, the J value of Ba₂LaNiTe₂O₁₂ is very large due to the unusually short O…O distance of 2.67 Å reported in the structure determination. A more accurate crystal structure would reduce the J value). As summarized in Figure 57b, the JS² values of the three antiferromagnets increase in the order RbFe(MoO₄)₂ < Ba₃CoSb₂O₉ < Ba₂LaNiTe₂O₁₂. This provides experimental and theoretical support for our arguments presented in the previous section.

6. Complex Clusters

6.1. Trimer–Dimer Zigzag Chains for the 3/5-Plateau in Na₂Cu₅(Si₂O₇)₂

Sodium copper pyrosilicate, Na₂Cu₅(Si₂O₇)₂, consists of ferrimagnetic zigzag chains in which trimer units alternate with dimer units (Figure 58a) [92]. Each trimer becomes ferrimagnetic due to the nearest-neighbor AFM exchange (J₁); the exchange (J₂) between adjacent trimer and dimer units is AFM, and the dimer exchange (J₃) is FM [92]. Thus, the ground state of the zigzag chain is ferrimagnetic (Figure 58b). Since Na₂Cu₅(Si₂O₇)₂ undergoes a 3D AFM ordering below T_N = 8K, the interchain interaction is weakly AFM. The fitting analysis of the magnetic susceptibility data using the ferrimagnetic chain model led to J₁ = 236 K, J₂ = 8 K and J₃ = −40 K [92]. The repeat unit of this ferrimagnetic state has the (3↑2↓) configuration with M = M_sat/5. Under a magnetic field, Na₂Cu₅(Si₂O₇)₂ exhibits a 3/5-magnetization plateau as shown in Figure 58c [93]. It occurs because the weak J₂ bond is broken under field leading to the higher-energy ferrimagnetic state (Figure 58d) with the (4↑1↓) configuration and hence M = 3M_sat/5. To increase the magnetization beyond 3M_sat/5, the two J₁ bonds in each trimer should be broken. Since J₁ is a strong bond, this does not occur unless the magnetic field is strong enough. Thus, the 3/5-magnetization plateau arises.

6.2. Linear Heptamer of One Trimer and Two Dimers for the 3/7-Plateau in Y₂Cu₇(TeO₃)₆Cl₆(OH)₂

Viewed solely geometrically, the Cu²⁺ ions of Y₂Cu₇(TeO₃)₆Cl₆(OH)₂ make chains of diamond-like tetramers which are interconnected by linear trimers [94]. In each trimer, a CuO₂Cl₂ plane corner-shares its oxygen atoms with two CuO₃Cl planes such that the adjacent CuO₂Cl₂ and CuO₃Cl planes are nearly perpendicular (Figure 59a). In each diamond-like tetramer, the planes of the two Cu₂O₃Cl dimers are separated and are nearly parallel to each other (Figure 59b). When viewed from the point of the magnetic orbitals contained in square planar units containing Cu²⁺ ions, a somewhat different picture emerges. In each diamond-like tetramer, the spin exchange between two Cu₂O₃Cl dimer units cannot be strong because their magnetic orbital planes are nearly parallel to each other. However, each Cu₂O₃Cl dimer can interact with an adjacent trimer through the Cu-O…O-Cu type spin exchange because the O…O distance is short (2.659 Å) and the Cu-O bonds are nearly directed toward each other (∠Cu-O…O = 158.8, 165.7°). Thus, each trimer is connected to two adjacent Cu₂O₃Cl dimers forming a linear heptamer (Figure 59c), and such heptamers are expected to be important for Y₂Cu₇(TeO₃)₆Cl₆(OH)₂.

The magnetization curve presents a field-induced metamagnetic transition at 0.2 T, which is followed by a magnetization plateau within a wide magnetic field range from 7 T to at least 55 T (Figure 59d). To account for this observation, an energy-mapping analysis based on DFT+U calculations was carried out to find the spin exchanges (in K), summarized in Figure 60a (see Section S14 of the Supplementary Materials). The latter shows that the dominating AFM spin exchanges are the inter-trimer–dimer exchange and the intradimer exchange J₃. The intra-trimer exchange J₂ is FM, and so are the exchanges between the Cu₂O₃Cl units within each diamond-like tetramer, but their magnitudes are weaker. Therefore, these considerations lead to the (5↑2↓) spin arrangement for each linear heptamer (Figure 60b). One heptamer is coupled to two other heptamers through the AFM exchanges J₅, leading to an AFM chain of heptamers (Figure 60c,d). (Since Y₂Cu₇(TeO₃)₆Cl₆(OH)₂ undergoes an AFM ordering below T_N = 4.1 K, the chains of heptamers have a very weak AFM interchain coupling). Therefore, the breaking of all J₅ bonds is necessary to reach the M = 3M_sat/7 point. To increase the magnetization beyond 3M_sat/7, it is necessary to break the intradimer J₃ bonds. These bonds are strong so that their breaking does not take place unless the magnetic field is strong enough. This explains the occurrence of the 3/7-plateau.

6.3. Zigzag Pentamer as an Effective S = 1/2 Unit in Cu₅(VO₄)₂(OH)₄₊

Turanite, Cu₅(VO₄)₂(OH)₄, has layers made up of three nonequivalent CuO₆ octahedra, which are interconnected by VO₄ groups. Within each layer, the CuO₄ square planes containing the x²-y² orbitals are arranged as presented in Figure 61a [95], so the pattern of the Cu²⁺ ion arrangement has interconnected chains of edge-sharing hexagons composed of six triangles (hereafter, the hexagon chains, for short) as depicted in Figure 61b. This magnet undergoes a ferrimagnetic ordering at T_C = 4.5 K, and its magnetization evidences a rapid increase below about 0.01 T. The latter is followed by a much slower increase, eventually reaching a 1/5-magnetization plateau at 8 T (Figure 61c) [96]. There are two puzzling observations to note: the M vs. H curve is smooth and resembles that observed for a paramagnet of S = 1/2 ions, and only ~23.6% of the spins participate in the ferrimagnetic ordering, which led to the suggestion that the remaining spins are still fluctuating [96].

To probe the cause for these observations, it is necessary to know the spin exchanges in each layer of interlinked hexagon chains (Figure 61b). What matters for spin exchanges is not the geometrical arrangement of the magnetic ions but that of their magnetic orbitals. To see if the spin lattice of Cu₅(VO₄)₂(OH)₄ was spin-frustrated, we examined the seven spin exchanges defined in Figure 62a. Note that due to the absence of a vertical mirror plane of symmetry in each hexagon chain, the exchanges J₅ and J₆ were treated as different (see Figure 62b), and so were the spin exchanges J₃ and J₄. The spin exchanges of J₁–J₇ determined using DFT+U calculations are summarized in Figure 62c (see Section S15 of the Supplementary Materials), from which we observed the following:

(1)

Within each hexagon chain, the AFM exchanges J₁ and J₅ dominated over the FM exchanges J₂, J₄ and J₆ so that there was effectively no spin frustration in all spin triangles of the hexagon chains.

(2)

The exchanges J₁ and J₅ formed zigzag pentamer ferrimagnetic fragments of (3↑2↓) spin configuration with M = M_sat/5 (Figure 62d).

(3)

Since J₆ is more strongly FM than J₄, each hexagon chain preferred to have adjacent (3↑2↓) ferrimagnetic fragments to have an AFM coupling rather than an FM coupling within each hexagon chain (Figure 63a,b).

(4)

Since the interchain exchange J₇ is AFM, an AFM coupling was preferred to an FM coupling between adjacent (3↑2↓) ferrimagnetic fragments between hexagon chains.

(5)

Thus, the most stable arrangement between adjacent (3↑2↓) ferrimagnetic fragments was AFM both within and between hexagon chains (Figure 63c), and the least stable arrangement was an FM both within and between hexagon chains (Figure 63d).

Though seemingly paradoxical, the above theoretical analysis is entirely consistent with the experimental observations for Cu₅(VO₄)₂(OH)₄. The key point to understand is that the AFM exchanges J₅ and J₁ leading to a (3↑2↓) ferrimagnetic fragment are very strong, so that each (3↑2↓) ferrimagnetic fragment acts as an effective S = 1/2 unit. Indeed, the observed magnetization curve is similar to the one found for a paramagnet of S = 1/2 ions (see the red curve in Figure 61c). This realization explains the low-temperature magnetic properties of Cu₅(VO₄)₂(OH)₄, namely, why only one out of five spins appears to participate in the magnetic ordering below T_C = 4.5 K and why the magnetization behavior resembles that expected for a paramagnet of S = 1/2 ions. The magnetic susceptibility of Cu₅(VO₄)₂(OH)₄ reveals the presence of five spins per formula unit at high temperature, because thermal agitation would break the AFM coupling leading to the (3↑2↓) ferrimagnetic fragment. As already discussed in Section 4.1.3, such a phenomenon of reduced spin moments due to strong AFM coupling was also found for volborthite, which consists of two-leg spin ladders with rungs made up of linear trimers of Cu²⁺ ions. In this case, the AFM coupling between adjacent Cu²⁺ ions was so strong that each linear trimer acted as an effective S = 1/2 unit.

6.4. Cu₇ Cluster of Corner-Sharing Tetrahedra for the 3/7-Plateau in Pb₂Cu₁₀O₄(SeO₃)₄Cl₇ and Na₂Cu₇(SeO₃)₄O₂Cl₄

Pb₂Cu₁₀O₄(SeO₃)₄Cl₇ has a complex crystal structure consisting of one nonmagnetic Cu⁺ (S = 0) ion and nine Cu²⁺ (S = 1/2) ions per formula unit [97]. Of the nine Cu²⁺ ions, two were found in a dimer unit and seven in a heptamer unit made up of two corner-sharing (Cu²⁺)₄ tetramers (Figure 64a). Pb₂Cu₁₀O₄(SeO₃)₄Cl₇ orders antiferromagnetically at T_N = 10.2 K and below this temperature exhibited a sequence of spin-flop transition at 1.3 T and a 1/3-plateau at 4.4 T, which persisted at least up to 53.5 T (Figure 64b). The spin exchanges of Pb₂Cu₁₀O₄(SeO₃)₄Cl₇ evaluated using DFT+U calculations showed that the magnetic properties of Pb₂Cu₁₀O₄(SeO₃)₄Cl₇ were governed by the ferrimagnetic heptamer (Cu²⁺)₇ with (5↑2↓) spin configuration (Figure 64c). The heptamers formed chains (Figure 64d) with interchain AFM coupling, so the magnetic ground state of the chain was an AFM state (Figure 64e). Under the magnetic field, the inter-cluster bonds became broken, eventually reaching the ferrimagnetic state in which the ferrimagnetic clusters were ferromagnetically coupled (Figure 64f). A further increase in magnetization required a high magnetic field because it was necessary to break the magnetic bonds within a ferrimagnetic cluster, hence leading to the 3/7-plateau.

Note that this discussion is based solely on the seven Cu²⁺ ions of a heptamer (Cu²⁺)₇. The two Cu²⁺ ions, strongly coupled antiferromagnetically in a dimer, were magnetically “silent”. If we include these two magnetic ions in our analysis, the 3/7-plateau discussed above becomes equivalent to a 1/3-plateau. A similar 3/7-plateau was found for Na₂Cu₇(SeO₃)₄O₂Cl₄ (Figure 65a) [98], which also consisted of (Cu²⁺)₇ heptamers made up of two corner-sharing (Cu²⁺)₄ tetramers (Figure 65b). Na₂Cu₇(SeO₃)₄O₂Cl₄ differs from Pb₂Cu₁₀O₄(SeO₃)₄Cl₇ in the bridging mode between adjacent heptamers (Figure 65c), but the composition of the heptamers is identical, namely, it is composed of five square planar and two trigonal bipyramid units (Figure 66a). Our DFT+U calculations summarized in Figure 65d (see Section S16 of the Supplementary Materials) show that each (Cu²⁺)₇ heptamer had a (5↑2↓) spin configuration (Figure 64e), thereby explaining why Na₂Cu₇(SeO₃)₄O₂Cl₄ exhibits a 3/7-plateau as does Pb₂Cu₁₀O₄(SeO₃)₄Cl₇.

Finally, we comment on why the (Cu²⁺)₇ heptamer adopts the (5↑2↓) spin configuration. From one trigonal bipyramid (TBP) to the central square plane (SP) to another trigonal bipyramid (TBP) in a heptamer, the three Cu²⁺ ions form a linear path (Figure 66a), and the two nearest-neighbor spin exchanges are of the Cu-O-Cu type. These two spin exchanges are strongly AFM in this linear path because the atoms associated with the Cu-O-Cu-O-Cu exchange paths are coplanar, so that the a₁ magnetic orbitals of the two TBPs [99] and the x²-y² magnetic orbital of the central SP are coplanar (Figure 66b). This makes the in-plane 2p orbitals of the two bridging O atoms intact efficiently with the a₁ and x²-y² magnetic orbitals, leading to a strong ↑↓↑ coupling of the three Cu²⁺ spins along the TBP-SP-TBP linear path. Note that the central SP is nearly orthogonal to every SP at both ends. This makes their x²-y² magnetic orbitals nearly orthogonal to each other, so the associated spin exchange becomes FM. Consequently, the heptamer adopts the (5↑2↓) spin configuration shown in Figure 66c.

7. Concluding Remarks

In an effort to find a conceptual picture describing the magnetization plateau phenomenon, we surveyed the crystal structures, the spin exchanges and the spin lattices of numerous magnets exhibiting magnetic plateaus. Our analyses show that an important key to understanding this phenomenon is the realization that a magnet under field absorbs Zeeman energy in accordance with Le Chartlier’s principle, which occurs by breaking its magnetic bonds. For a magnet with spin lattice defined by several spin exchanges of different strengths, its weakest bonds are broken preferentially to partition the spin lattice into either antiferromagnetic or ferrimagnetic fragments, which fill the whole spin lattice without overlapping each other. For a magnet with a spin-frustrated spin lattice defined by a few spin exchanges of comparable strengths, the weaker magnetic bonds are broken to partition the spin lattice into small ferrimagnetic fragments filling the whole spin lattice without overlapping each other. Such field-induced fragmentation is influenced by the spin-lattice interactions brought about by the fragmentation.

As illustrated in this survey, the conceptual aspects of the magnetization plateau phenomenon in any magnet can be readily explained or predicted once its crystal structure and its spin exchanges are known. Given the crystal structure of a magnet, it is straightforward to evaluate the spin exchanges for various exchange paths between its magnetic ions, and hence the relative strengths of its magnetic bonds, using the energy-mapping analysis based on DFT calculations. It goes without saying that this approach is not designed to provide quantitative descriptions. The latter lie in the realm of quantitative calculations using model Hamiltonians with a minimal number of adjustable parameters (e.g., spin exchanges) to generate the magnetic energy spectrum of a magnet under investigation. Even with powerful computers currently available, such quantitative analyses cannot be carried out for most magnets because their spin lattices are complex and low in symmetry. The conceptual picture of the magnetization plateau phenomenon, based on the supposition of field-induced partitioning of a spin lattice into magnetic fragments, is valid for all magnets regardless of whether their spin lattices are complex or not.

The magnetization plateau phenomenon can be highly anisotropic, as found for the Ising magnets Ca₃Co₂O₆ and CoGeO₃, in that their 1/3-magnetization plateaus observed with the field along the easy axis do not occur if the field is perpendicular to the easy axis. A strong plateau anisotropy, though weaker than those found for the Ising magnets, is also observed for Cs₂Cu₃(SeO₃)₄·2H₂O and azurite Cu₃(CO₃)₂(OH)₂. In Cs₂Cu₃(SeO₃)₄·2H₂O, the value of M = M_sat/3 depends on the field direction (i.e., H⊥c vs. H||c) because the Cu²⁺ ion has a higher spin moment when the magnetic field is perpendicular than parallel to the CuO₄ square plane. In azurite, the width of the 1/3-magnetization plateau depends on the field direction (H||b vs. H⊥b) for three reasons: (1) the Dzyaloshinskii–Moriya interactions between the Cu²⁺ ions depend on the relative orientations of their CuO₄ square planes, (2) the spin moment of a Cu²⁺ ion depends on the field direction with respect to the CuO₄ square plane, and (3) the spin lattice of azurite is three-dimensional with nonnegligible interlayer spin exchange. In both Cs₂Cu₃(SeO₃)₄·2H₂O and azurite, the strong anisotropy of their magnetization plateaus stems from the presence of near orthogonal arrangements of their CuO₄ square planes. This emphasizes once more the importance of analyzing the structural chemistry associated with magnetic ion arrangements.

Our supposition that the spin lattice of a magnet exhibiting one or more magnetic plateaus is partitioned into magnetic fragments is supported by the experimental observation that an Ising magnet can exhibit a magnetization plateau when the applied field is parallel to the easy axis of the magnet, but this magnetization plateau disappears when the field is perpendicular to the easy axis. This reflects that Zeeman energy, being a dot product between the magnetic field and the spin moment, is nonzero for the parallel field but zero for the perpendicular field. More support comes from the observation that the highly anisotropic width of the 1/3-magnetization plateau (H||b vs. H⊥b) in azurite Cu₃(CO₃)₂(OH)₂ arises from the field-dependent Zeeman energy available for the magnet.

This survey reflects our efforts to comprehend the magnetization plateau phenomenon based on the relative strengths of magnetic bonds. Thus, our discussion focused on the arrangement of magnetic ions and their spin exchanges leading to the spin lattices responsible for magnetization plateaus. It is our hope that the conceptual picture of the magnetization plateau phenomenon presented in this survey will promote further developments in this and related research fields.

Near the completion stage of this survey, new magnets were reported to exhibit magnetization plateaus at low temperatures. They include CsCo₂Br₄ [100], YCu₃(OD)_6+xBr_3−x (x ≈ 0.5) [79], Ni₂V₂O₇ [101], TbTi₃Bi₄ [102], TbRh₆Ge₄ [103], GdInO₃ [104], Cu₅(PO₄)₂(OH)₄ [105], Ba₂Cu₃(SeO₃)₄F₂ [106], Sr₂CoTeO₆ [107], Cu₃Bi(TeO₃)₂O₂Cl [108] and Na₃Ni₂BiO₆ [109]. We expect that the magnetic plateau phenomena of these new magnets can be readily explained using the concept of the field-induced partitioning of their spin lattices once their spin exchanges are determined.

In this survey, we focused on the “classical” magnetization plateaus that are readily described by the field-induced partitioning of spin lattices into spin superstructures. It should be pointed out that there may be cases that require a more sophisticated description beyond this classical picture [110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125]. For instance, as recently reviewed by Yoshida [125], quantum plateau phases may emerge from quantum spin liquids.

Note added in proof: The essential results of our paper were recently summarized and commented in arXivpp [126].