8677
2007, 111, 8677-8679 Published on Web 07/07/2007
Effective Spin Hamiltonian Model for Superexchange Interaction between Rare Earth Ions in Rare Earth Elpasolite Crystals Xianju Zhou Institute of Modern Physics, Chongqing UniVersity of Post and Telecommunications, Chongqing 400065, People’s Republic of China
Shangda Xia Department of Physics, Structure Research Laboratory, Academica Sinica, UniVersity of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China
Peter A. Tanner* Department of Biology and Chemistry, City UniVersity of Hong Kong, Tat Chee AVenue, Kowloon, Hong Kong S.A.R., People’s Republic of China ReceiVed: March 16, 2007; In Final Form: May 8, 2007
The fine structure of zero phonon lines in the electronic absorption spectra of lanthanide ion systems is interpreted by an effective spin Hamiltonian model. The splittings of several cm-1 observed for lanthanide ions with interionic separations near 1 nm in elpasolite lattices are attributed to superexchange interactions.
The highest symmetry that occurs for rare earth ions in crystals is octahedral site symmetry, as exemplified by the elpasolites Cs2NaLnCl6, Figure 1. Each Ln3+ ion is surrounded by 12 others, with the Ln3+ ion-ion separation being large, in the region of 0.7-0.8 nm. The electronic spectra of Ln3+ in these systems have been widely studied, and the energy levels and electronic and vibronic intensities are quite well understood and calculated.1 The fine structure of the pure electronic transitions has not previously been considered, however, and it comprises splittings on the order of a few cm-1 for the zero phonon lines. The splittings disappear in diluted crystals; therefore they are due to interactions between Ln3+ neighbors.2 As pointed out on page 4165 of ref 3, these splittings are due only to the exchange interaction and magnetic dipole-dipole interaction. However, our calculations, and those of others,4 show that magnetic dipole-magnetic dipole interactions would produce splittings of at least an order of magnitude smaller due to the large interionic separations. Especially in the elpasolite lattice, the effective magnetic field Hdip of this interaction would be zero based on the formula (19) of ref 3 and the highsymmetry distribution of the surrounding 12 Ln3+ ions. Therefore, the splitting is due only to exchange interaction. This Letter provides a description of the splittings in the electronic spectra of these neat materials Cs2NaLnX6 (X ) halogen), and a simple model interpretation is provided. It is well-known that the isotropic exchange interaction is proportional to the scalar product S(A)‚S(B) of the spins of the interacting ions A and B. However, in the general case, the anisotropic nature of the exchange interaction is very important.3,5,6 In the classical works,3,5 some orbital (space) tensors are introduced to consider the dependence of the anisotropic exchange interaction on the orbital states. Furthermore, follow10.1021/jp072107f CCC: $37.00
Figure 1. Unit cell and octahedral ligand coordination in the hexahalide elpasolite crystal lattice M2ALnX6.
ing Van Vleck’s idea, by coupling the orbital and spin variables, the total angular momentum J(Ln), together with S(Fe), was used to express the rare earth (4fN)-iron (3d5) anisotropic exchange interaction in garnets by Levy (ref 5, page A156). Recently, in the study of Guillot-Noe¨l et al.,4 an effective spin (Seff ) 1/2) Hamiltonian was utilized to describe the simple Kramers doublets of the Nd3+ ion in a crystal field and the electron exchange interaction of a Nd3+ ion pair in dilute Nd3+-doped crystals of LiYF4 and YVO4, to explain the splittings in the observed optical emission spectra. In their situation, both ions A and B are excited to have different Seff(A © 2007 American Chemical Society
8678 J. Phys. Chem. B, Vol. 111, No. 30, 2007
Letters
Figure 2. The 10 K absorption spectrum of polycrystalline Cs2NaErCl6 between 6480 and 7000 cm-1. The inset shows the lowest energy region in more detail. Refer to the text for explanation of the assignments.
+ B)2 values. Therefore, the studied system is a Nd3+ ion pair, and since the symmetry of the system is the point group D∞h, there are many parameters involved if using both spin and orbital tensors explicitly. For our objectives, we may reasonably consider that the studied rare earth elpasolite system at the temperature of 10 K is (at least approximately) a magnetically ordered system induced by the superexchange interaction between Ln3+ ions. Therefore, the theoretical models mentioned above can be referred to. Instead of spin S and orbital tensors, we introduce an effective spin operator Seff of the 4fN electronic states of rare earth ions located at Oh symmetry sites. The magnitudes of Seff are 3/2, 1/2, and 1/2 for Γ8, Γ7, and Γ6 states, respectively. These magnitudes of Seff for Γ8 and Γ6 are based on their same transformation properties under the 24 O-transformation operators as those of the set of states whose total angular momentum is J ) Seff under the spherical symmetry environment. Actually, on the basis of the approximate crystal field wave functions of Er3+ ions obtained from crystal field fits,7 we calculated the expected values of Jz ) Sz + Lz in the states of the groundlevel Γ8, and the results are 3.32((1/2) and 3.36((3/2), respectively, which are very close to (10/3)((1/2, (3/2); similarly, we obtained the exact values (7/3)((1/2) and 3((1/ 2) for the 2F7/2 ground-level Γ6 and the 2F7/2 level Γ7 of Yb3+, respectively. Therefore, apart from a common factor, the Seff we introduced is really an effective “angular momentum” for Ln3+ states of a crystal field level and arises because (S + L) is quenched partly by the crystal field interaction. We consider first the ion pair interaction between two rare earth ions A and B, which is given by
HAB ) c(AB)Seff(B)‚Seff(A) ) c(AB)
(1) (-1)m Seff(B)-m Seff(A)(1) ∑ m m
(1)
where the components m are defined as usual for a rank-1 spherical tensor operator. The reason why HAB depends upon
Figure 3. The 10 K absorption spectrum of polycrystalline Cs2NaYbCl6 between 10230 and 10550 cm-1. The inset shows the region of the lowest energy zero phonon line (ZPL).
the scalar product of effective spin is that the energy E(A + B) of the coupled system (A + B) must depend upon Seff(A + B)2 ) S(A)2 + S(B)2 + 2Seff(A)‚Seff(B), while S(A)2 and S(B)2 are fixed for the studied pair of ion states. The proportionality constant c(AB) depends upon the nature of the pair of ion states and on the relative position of the two ions. We now consider the interaction between rare earth ions in the concentrated system Cs2NaErCl6, where the two Er3+ ions in all of the neighboring ion pairs Er3+-Er3+ occupy the same sites having the same environments. Therefore, the directions of Seff of the studied Er3+(A) and the other 12 Er3+ neighbors are the same and are taken as the z axis. Then
HAB ) c(AB)Seff(B)‚Seff(A) ) c(AB)Seff,z(B) Seff,z(A) (2)
Letters
J. Phys. Chem. B, Vol. 111, No. 30, 2007 8679
Since there are 13 Er3+ ions in the studied neighbor interaction system and the simultaneous excitation of them is too difficult, we have 12
Hint )
∑ c(AB)〈GB|Seff,z(B)|GB〉Seff,z(A)
(3)
B)1
which gives
E(A) ) aSeff,z(A)
(4)
where 12
a)
∑ c(AB)〈GB|Seff,z(B)|GB〉
(5)
B)1
in which |GB〉 is the ground state of the Er3+(B) ion. Of course, the summation in (5) can be extended to contain the contributions from all of the other Er3+ ions. Therefore, under the interaction Hint, a Kramers quartet level Γ8 should be split into four levels with equal energy gaps a and the levels Γ6 and Γ7 split into two levels each (with respective separations, say, b and c). Such splittings occur for the two states involved in the optical transition so that by analysis of the spectral fine structure of the 0-0 transition, it should be possible to determine the parameters a, b, and so forth. As an illustration, Figure 2 shows the 4I15/2 f 4I13/2 absorption spectrum of Cs2NaErCl6 at 10 K, between 6480 and 7000 cm-1, with a spectral resolution of 2 cm-1. The electronic ground state is 4I15/2 aΓ8, and the excited 4I13/2 crystal field states are in the order of increasing energy, Γ6 < aΓ8 < aΓ7 < bΓ8 < bΓ7.7 Thus, the lowest energy transition (below 6500 cm-1) corresponds to aΓ8 f Γ6 (shown in red); then, the next highest is aΓ8 f aΓ8 (shown in blue). Other zero phonon lines correspond to sharp features, whereas broader bands represent vibronic structure, which has previously been assigned.8 Considering the interactions between Er3+ ions, the splittings of the aΓ8 ground state may be represented by a and those of the
excited aΓ8 and Γ6 states by b and c, respectively. The inset of Figure 2 enlarges the region of the aΓ8 f Γ6 and aΓ8 transitions and gives the band assignments, which are also depicted in the energy level diagram above the figure. The derived energies of a, b, and c are 3, 3.5, and 6 cm-1, respectively. Analysis of the next transition, aΓ8 f aΓ7 also gives a ) 3 cm-1. Further examples and interpretations of the zero phonon line splittings are illustrated by the 1 µm absorption spectrum of Yb3+ in Cs2NaYbCl6. The zero phonon line of the 2F7/2 Γ6 f 2F -1 5/2 Γ8 transition (Figure 3) is centered at 10246 cm , and 9 higher energy bands correspond to vibronic structure. In this case, the Γ6 splitting, a, is larger at 7.9 cm-1, and the Γ8 splittings are 4 cm-1 so that the spectral features are more clearly resolved. Naturally, the same value of a is deduced for the electric quadrupole-allowed transition 2F7/2 Γ6 f 2F5/2 Γ7, where the Γ7 splitting, b, is 11.8 cm-1. Similar zero phonon line splittings are observed in the optical spectra of other Cs2NaLnCl6 systems and are analogously assigned. Analogous splittings are also observed in the electronic spectra of the lanthanide hexafluoroelpasolites. For example, the splittings a and b in the 10 K 2F7/2 Γ6 f 2F5/2 Γ8 absorption spectrum of Yb3+ in Cs2NaYbF6 are found to be larger (a ) 11 cm-1; b ) 5.5 cm-1) than those in the corresponding spectrum of Cs2NaYbCl6. In conclusion, a simple spin Hamiltonian model has accounted for the multiple structure of the zero phonon lines of Ln3+ in elpasolite crystals. The Ln3+-Ln3+ separation is on the order of 1 nm so that the dominant long-range mechanism of interaction is attributed to superexchange. References and Notes (1) Tanner, P. A. Top. Curr. Chem. 2004, 241, 167. (2) Tanner, P. A. J. Mol. Struct. 1997, 405, 103. (3) Cone, R. L.; Wolf, W. P. Phys. ReV. B 1978, 17, 4162. (4) Guillot-Noe¨l, O.; Mehta, V.; Viana, B.; Gourier, D.; Boukhris, M.; Gourier, D. Phys. ReV. B 2000, 61, 15338. (5) Levy, P. M. Phys. ReV. 1964, 135, A155. (6) Mironov, V. S.; Chibotaru, L. F.; Ceulemans, A. Phys. ReV. B 2003, 67, 014424. (7) Faucher, M. D.; Tanner, P. A. Mol. Phys. 2003, 101, 983. (8) Tanner, P. A. Mol. Phys. 1986, 57, 737. (9) Tanner, P. A. Mol. Phys. 1986, 58, 317.
