Ising exchange interaction in lanthanides and actinides

The Ising exchange interaction is a limiting case of strong exchange anisotropy and represents a key property of many magnetic materials. Here we find necessary and sufficient conditions to achieve Ising exchange interaction for metal sites with unquenched orbital moments. Contrary to current views, the rules established here narrow much the range of lanthanide and actinide ions which can exhibit Ising exchange interaction. It is shown that the arising Ising interaction can be of two distinct types: (i) coaxial, with magnetic moments directed along the anisotropy axes on the metal sites and (ii) non-coaxial, with arbitrary orientation of one of magnetic moments. These findings will contribute to purposeful design of lanthanide and actinide based materials.


Introduction
Strong magnetic anisotropy on metal sites gives rise to novel magnetic properties, such as single-molecule magnet behavior [1,2], magnetic multipole ordering [3,4], and various exotic electronic phases [5,6]. In lanthanides and actinides the spin-orbit coupling exceeds the crystal-field splitting of the ground ionic LS term leading to unquenched orbital momentum L [7]. As a result the low-lying spectrum on these metal ions is well described as crystal-field split eigenstates of the total angular momentum J L S, = + where S is the spin of the corresponding term. If the symmetry of the metallic sites is lower than cubic, the ground crystal-field multiplet is either a Kramers doublet or a quasi doublet, for odd and even numbers of electrons on the metal ion, respectively. The presence of unquenched orbital momentum in these doublets makes them strongly anisotropic, which is also reflected in the strong anisotropy of their exchange interaction. The limit of this anisotropy is the exchange interaction of Ising type, which in the case of two interacting doublets or interacting doublet and isotropic spin has the form: where s ĩ is the pseudospin 1/2 describing the doublet state of site i, S 2 is the isotropic spin of site 2, and z i denotes the projection of pseudospin or isotropic spin along the main magnetic axis of doublet on site i. Common beliefs concerning the Ising exchange coupling in lanthanides and actinides can be summarized with the following rules: (1) The interaction involving quasi doublets of metal ions with an even number of electrons (the so-called Ising ions [3]) will always be the Ising type. However, in the case of metal ions with an odd number of electrons the Ising interaction is only achieved when the corresponding Kramers doublets have a zero magnetic moment in the transversal directions with respect to the main magnetic axis (i.e., are perfectly axial [8]) [3].
(2) In both of the above cases the Ising interaction is the coaxial type (1), when the local magnetic moments in the exchange states are directed either along the main magnetic axes z i of the corresponding metal sites or, in the case of isotropic spin, along the main magnetic axis of the neighbor site (figure 1(a)) [9].
Rule (1) appears to be weakly restrictive predicting, in particular, Ising exchange interaction to all metal ions with even number of electrons. As discussed below, both these rules are based on an oversimplified treatment of the exchange interaction for metal ions with unquenched orbital momentum. Given the increasing interest in strongly anisotropic magnetic materials, the knowledge of precise conditions for the realization of the Ising exchange interaction between strongly anisotropic metal ions would be highly desirable.
In this work, the general conditions for achieving the Ising exchange interaction for metal sites with unquenched orbital momentum are found via analysis of the microscopic exchange Hamiltonian. These conditions are specified for interacting anisotropic metal ions in their doublet states and for such ions interacting with isotropic spins. Despite the overwhelming complexity of exchange interaction between centers with unquenched orbital moments, the realization of Ising exchange interaction has been shown to depend in most cases solely on the structure of doublet wave functions on the metal sites and not on the details of intersite interaction. This reduces the task of designing magnetic materials with Ising exchange interaction to create an appropriate crystal field on the metal sites.
Despite the lack of justification, this form is often used for the description of the interaction between lanthanides or actinides (or a similar form, J S , 1 2 µ · in the case of their interaction with an isotropic spin) [11][12][13][14][15][16][17][18]. Direct calculations show that with this form of exchange interaction the rules (1)-(2) hold true 1 . This is easily seen when we choose a particular form of wave functions (2) for strongly axial doublets:  achievable either at high axial symmetry [8] or at strong axial component of the crystal field on sites [20]. Indeed, it can be checked directly that for odd numbers of electrons the transverse components of any angular momentum is zero for m 1 2, i > while in the case of even number of electrons the smallest m i for a quasi doublet (4) is m i = 1 (the same is true for equation (2)) . Under such circumstances rules 1-2) and the conditions for zero transverse magnetization on metal ions (perfect axiality) hold true simultaneously 2 .
We should note that the magnetic dipolar interaction between magnetic sites is also bilinear in total angular momentum operators albeit in a form different from equation (3): where g L is the Landé g factor, B m is the Bohr magneton, R 12 is the distance between magnetic sites, and n 12 is the normalized direction vector from site 1 to site 2. Therefore, it will also reduce to equation (1) when downfolded on the axial doublets (4) (m (1) is a simple expression of the relative orientations of the local main magnetic axes and the vector connecting magnetic sites [9]. This interaction is considered dominant for pairs of lanthanides ions, which justifies the description of their interactions via bilinear operators (3) [11][12][13][14][15][16][17][18], as well as the validity of rules 1-2). Such a view is, however, ungrounded for simple reasons that (i) dipolar magnetic coupling can be diminished arbitrarily on geometric grounds and (ii) the exchange interaction can significantly exceed the dipolar coupling, especially when it is mediated by strongly covalent bridging groups [22]. In the cases of lanthanides interacting with isotropic spins (transition metals or radicals) and of interacting actinides (between themselves or with isotropic spins), the magnetic dipolar interaction is significantly smaller than the exchange interaction. The latter, however, is not described by the simple form (3) but represents a much more involved expression. In the case of two interacting ions with unquenched orbital momenta it has the form [4,23]: ¢ ( ) are Stevens operators 3 whose ranks k and k′ have to obey the relation k k + ¢=even due to the invariance of the Hamiltonian with respect to time inversion [10]. In equation (6), the terms such that one of k and k′ is zero are not included because they are not exchange but crystal field. A metal ion with unquenched orbital momentum interacts with an isotropic spin as follows [23]: where k is odd due to the time reversal symmetry. The highest rank k of Stevens operators entering equations (6) and (7) is found from the relation [23]: where l 1 is the atomic orbital momentum and J 1 is the total angular momentum of the ground atomic multiplet on site 1 (a similar relation holds for k max ¢ on site 2). In the case of f N ions, k l 2 1 7 max = + = for N = 2-4 and 7-13, k J 2 5 max = = for N = 1,5, and k 0 max = for N = 6. The maximal component q ( 0 > ) in equations (6) and (7) is generally determined by the details of intersite interaction [23]: is the maximal difference of orbital momentum projections on site 1, m and n, entering as indices in the parameters V mm n n ¢ ¢ and t t , mm n n 12 21 ¢ ¢ defining the direct and kinetic intersite exchange interaction, respectively (V is the bielectronic integral and t , 12 t 21 are electron transfer parameters). For low-symmetric systems, all these parameters will be non-zero, leading to l 2 .

D =
Therefore, in such cases q max is determined by intrasite properties only (as k max ). The maximal value of q′ at the second site, q , max ¢ is determined in a similar way.
With the above information on the exchange interaction for the entire J multiplet, the effective Hamiltonian is derived. For more information on the exchange interaction, such as the form of kqk q  ¢ ¢ and explanations of k max and q , max see [23]. 2 A simplified treatment of kinetic exchange interaction within the so-called 1/U approximation [21] gives two interacting doublets (4) on Dy 3+ ions an Ising exchange interaction similar to the first equation of (1). At the same time the interaction of one such Dy 3+ doublet with an isotropic spin results in an Ising exchange, as in the second equation of (1), only for m J 15 2 1 1 = = (unpublished results). 3 The Stevens operator is defined so that it transforms as spherical harmonics under rotation, which enables us to apply the Wigner-Eckart theorem (see, e.g., [24]).

Derivation of the effective exchange Hamiltonian
In this section, we derive the effective exchange Hamiltonian H eff between the ground doublets of metal sites and between the ground doublet of the metal site and isotropic spin. This is done by projecting the complete form of the exchange Hamiltonian for the entire J-multiplet, equations (6) and (7), into the one within the truncated Hilbert space, and then describe the doublets by the s 1 2 = pseudospin operators. We consider first doublets of type (4), for which the derivation is the simplest.

Interacting axial doublets
Within the truncated Hilbert space, J J m : , ) is the projector into the truncated space. The matrix element of the exchange Hamiltonian (6) is Here, the property of the Clebsch-Gordan coefficient related to the time inversion is used [25]: The two doublet functions (4) on site i can be put in correspondence to the eigenfunctions 1 2, 4 We follow the convention of the spherical harmonics and the Clebsch-Gordan coefficients of [25].
where i = 1, 2, I ĩ is the two-dimensional unit matrix, s s is Finally, the general exchange interaction (6) Here, the parameters  and  are combinations of exchange parameters kqk q  ¢ ¢ from equation (6) and products of the Clebsch-Gordan coefficients:  The effective Hamiltonian (19) contains the constant part (the first term), the zero-field splitting (ZFS) part (terms 3-6), and the exchange part (the rest of the terms). With above equations we are able to identify the situations when  and  become zero 5 . The overview of such situations as a function of the parity of the number of electrons on sites (N 1 and N 2 ) and of the relation between 2m i (i = 1, 2) and q q , max max ¢ (9) is given in appendix A.2. This information is sufficient to find the general conditions under which the exchange Hamiltonian for two interacting doublets (19) becomes the Ising type.

Interacting axial doublet and isotropic spin
In the case of a doublet (4) interacting with an isotropic spin, the derivations are similar. Now we only need to consider the matrix elements of operators referring to the first site in equation (7): The matrix elements of Ĥ (7) are given as As in the previous case, equation (19), the exchange Hamiltonian (32) can become the Ising type under certain conditions.

Two types of Ising exchange interaction
where all terms contain s . This drawback disappears when the description of such ions is done with integer pseudospins, e.g., s 1 = [26]. This is not necessary here because the exchange part (which is only of interest here) is described correctly already within such a simplified treatment. the basis of two states (4), we can still define its rotation via induced transformations of the doublet wave functions [27]. Then the eigenfunctions of the new pseudospin operator s z 2 ¢ will be the same functions (4) defined with respect to rotated quantization axis z′, i.e., J m .
which has the form of an Ising Hamiltonian of non-coaxial type because the magnetization at the second site will not be directed along the corresponding main magnetic axis z 2 but along a different axis z′ (left plot in figure 1(b)). This makes it qualitatively different from the coaxial Ising interaction described by equations (1) and (  is a rotated spin S z 2 from direction z 1 to z′. This means the Hamiltonian (39) is the non-coaxial Ising type, in which the magnetic moment on site 2 points to a direction different from the main magnetic axis on site 1, as indicated in the right plot of figure 1(b) (cf figure 1(a)). Note that the Ising interaction of type I is only achieved when z 1   in equation (40) are both zero, which requires special symmetry of the exchange bridge. The classification for all cases is shown in appendix B and table 1 summarizes our main results. It shows that rule 2) (Ising interaction type I) is not satisfied in most cases. We can also see that the type of resulting exchange Hamiltonian merely depends on the relative values of m 1 , m 2 , and q 2,  Table 1 shows that rule 1) does not hold either. Instead, the realization of the Ising exchange interaction depends on the value of q max (q max ¢ ), which is determined by the smaller of the quantities 1 max D + and k max (equation (9)). D on sites become sufficiently large, q max (q max ¢ ) is determined by k max (k max ¢ ), equation (8), i.e., the smaller of the quantities l 2 1 i + (= 7 for f orbitals) and 2J i on corresponding sites. Designing a strongly axial crystal field on sites we can achieve m i values as high as J i [9]. Then such ions with J 7 2 i > will a priori provide Ising exchange interaction with any other magnetic site according to table 1. Contrary to what is stated by rule 1), these are precisely the ones that can be called Ising ions. They are listed in table 2. Table 1. Exchange interaction between two doublets and between a doublet and an isotropic spin. I and II denote the two types of Ising exchange interaction, equations (1), (35) and equations (38), (39), respectively; N stands for a non-Ising exchange interaction.

Conclusions
In this work, we find necessary and sufficient conditions to achieve Ising exchange interaction in materials involving lanthanide and actinide metal ions. Unless the symmetry of the exchange bridge is not very high, these conditions are determined solely by the electronic properties of individual metal ions. According to these conditions, by far not all metal ions can display Ising exchange interaction (table 2). We also find that two types of Ising exchange interaction can arise in these systems: the coaxial and non-coaxial ones. In particular, it is established that the Ising interaction between an anisotropic doublet and an isotropic spin is the second type. The basic property of Ising coupling is the lack of dynamics in the resulting exchange doublets, which is also in the applied magnetic field. This condition is indispensable for achieving single-molecule magnets since it ensures the quenching of quantum tunneling of magnetization in individual exchange doublets. The insight gained in this work will contribute to purposeful design of lanthanide and actinide based materials.  Table 2. Ising metal ions. Only valencies of metal ions in known complexes are listed [7,29]. The relations between exchange parameters, (20)- (29), are derived here. In the derivation, we use the property of the exchange parameter under time inversion [23]: and the property of the Clebsch-Gordan coefficients [25]: and we obtain equation (21). Similarly, we obtain equation (22).
Combining these relations, and we obtain equation (24). Here, we used that k′ is odd. Therefore, we obtain and equation (26). Similarly, we obtain equation (27).

A.2. Symmetry properties
Some of  and  become zero due to (i) time reversal symmetry and (ii) the structure of the exchange interaction.
(i) Time reversal symmetry: When the magnetic site has an odd (even) number of electrons, the total angular momentum and its projection m i are half-integers (integers), and thus, 1 1 Since the maximum of k is fixed, we have to further consider two cases regarding J:  The case of N = 6 (J = 0) is not considered since the ion is nonmagnetic. The above discussion is summarized as follows: