Spectral distribution of NMR IN mixed valence complexes. ElectrIC-dipole absorption
L.D. Falkovskaya, A.Ya. Fishman, V.Ya. Mitrofanov, B.S. Tsukerblat
Institute of Metallurgy, Ural Division of the Russian Academy of Sciences, Ekaterinburg, Russia
Ben-Gurion University of the Negev, Beer-Sheva, Israel
In this study we reveal the specific character of the frequency spectral distribution of NMR in mixed valence (MV) complexes in magneto-ordered crystals with cubic structure. Chalcogenide chromium spinels can be exemplified in wich Cr2+ or Cr4+ ions exist in the octahedral sub-lattice along with Cr3+ ions. It was shown that due to the presence of essential dipole momentum and unquenched orbital angular momentum in such MV centers one can expect a considerable amplification of NMR signal induced by a high intensity electric-dipole transition between components of E-term .
Introduction
In crystals with small concentration of non-isovalent substitutions, vacancies in cation or anion sub-lattices, the extra electrons or holes are localized by the Coulomb interactions near the lattice defects [1-3]. The extra electrons can be delocalized over the metal ions in the vicinity of the defect giving rise to a mixed-valence clusters containing metal ions in different oxidation degrees like manganese and chromium dimers, Mn4+ - Mn3+, Cr4+ - Cr3+.
One of the constituent ions of these MV complexes, either dn or dn1, usually has orbitally degenerate ground state so that the MV clusters entire can possess some peculiarities that are intrinsic to the isolated Jahn-Teller (JT) systems. Low symmetry local fields created by the excess charge remove this degeneracy. Nevertheless the ground state of the whole MV complex can be degenerate due to excess electron transfer among 3d-ions of the complex. Due to the external perturbation or cooperative interactions the orbital degeneracy proves to be removed and the system is expected to possess some new properties resulting from the redistribution or reorientation of the density of the excess electron. This originate a dipole moment of a MV cluster essentially depending upon the character of its orbital state. Close interrelation between magnetic and electric properties of these compounds can be understood on the basis of the model of the double exchange [4,5].
Being extremely sensitive to the local fields the nuclear magnetic resonance (NMR) provides an important information regarding the peculiarities of the magnetic states and gives an opportunity to analyze the local charge distribution in MV centers. The lack of the theory of NMR spectral distributions in systems containing transition metal MV clusters in magnets essentially complicates the interpretation of observable spectrums. To a large extent this is connected with the absence of adequate models of electronic structure of the impurity complexes. This task is just more complicated for systems in which some types of defects in cation and anion sub-lattices take place. Such situation is typical for NMR spectrums of 55Mn [6-8] in (LaMn)1-2xO3 crystals containing MV clusters of manganese ions. The corresponding NMR spectrum of 55Mn [6] represents a series of non- uniformly broadened lines. A drastic amplification of the NMR signal was observed in the whole frequency range with the increase of the defect concentration. Till now these results have not received any adequate interpretation in terms of current theories.
The present paper is devoted to the theoretical treatment of the peculiarities of the spectral distribution in NMR spectra closely related to the essential dipole momentum of MV complexes. As an example we consider MV centers in the octahedral positions of the crystals with spinel structure.Analysis of MNR spectra was carried out for the case of orbitally degenerate ground state of MV complex (E-term). In this case the main mechanisms determining splitting of the lowest E-term are the spin-orbit interaction and random crystal fields. The peculiarities of NMR spectra of such centers arise from the existence of the anomalously strong direct interaction of the type of A1(I) between orbital and nuclear I momenta of 3d-ion in one of two possible charge configurations, either 3dn or 3dn1 [9,10]. We show that due to the essential dipole momentum of MV center the intensity of NMR transitions under consideration can be significantly amplified.
Hamiltonian of mixed valence center. COUPLING between orbital and nuclear sub-systems
Let us consider a MV center consisting of a triad of exchange-coupled 3d-ions and an excess t2g-hole (or t2g-electron) delocalized over these ions. Such center in a cubic crystal can appear in the presence of cation or anion vacancies or nonisovalent substitutions. For the sake of definiteness we shall consider a MV center consisting of a cluster with a point defect in cation sublattice of crystal with spinel structure and t2g-hole coupled to this system (see Fig. 1). The basis set of the system involves three localized states corresponding to three configuration of the magnetic ions each containing the excess hole. The chalcogenide chromium spinels where along with Cr3+ (3d3) ions also Cr4+(3d2) ions exist in octa-positions [9, 10] could be mentioned as an example.
/ Fig.1. Mixed valence center of formed by three cations in octahedral spinel sub-lattice with the delocalized excess t2g-holePD - point defect in cation sublattice, A - anion, C - cation, p – extra hole.
It is supposed that the lowest energy state for the ions with configuration is two-fold degenerate with respect to orbital quantum number (trigonal E-term). The ground electronic state of two other magnetic ions with configuration 3dn is assumed to be orbitally non-degenerate.
We shall represent the Hamiltonian of a triad by
(1)
where Hres is the Hamiltonian describing the transfer of a t2g – hole in a triad (double exchange), and Hk is the Hamiltonian of the cluster when a t2g hole is localized at center к (к= 1,2,3), c+kand cl are the creation and annihilation operators for t2g holes whose spin projection is at centers k and l in orbital states and , correspondingly, bkl is the transfer integral for a t2g hole between the mentioned states.
Let us confine our treatment to the states with the maximum projection of the total spin of the triad along the quantization axis. Since the maximum energy gain is achieved when the spin of the dn triad and the moving t2g hole are parallel the ground state of the system is contained precisely in the mentioned group of the levels. Six corresponding lowest states of a triad (MV center) are analyzed. The low-lying group of the levels arise from three localized states dn-1-dn-dn, dn- dn-1-dn, dn-dn-dn-1,each state being double degenerate and corresponds to localization of the excess t2g hole on one of the constituent ions. The delocalized states transform accordingly to the irreducible representations of the symmetry group C3v of the whole system: А1, А2 and 2E. Leaving aside spin-orbit interaction one can find the following expressions for the energy levels [11]:
(2)
In this case the energies are counted of from the ground ferromagnetic state of the magnetic material with an excess t2g-hole; the energy levels E(E) belong to the repeated E representations, h0 is the parameter of tetragonal crystal field caused by the source of excess charge. The transfer integrals are equal to [11]
(3)
Further on we shall focus on the situation when a doublet state turns out to be the lowest state of the triad (this is valid when 0>0 and b1/b21/2 [11]). The main interactions determining splitting of the ground Е–term, and hence also properties of MV centers are the spin-orbit interaction and low symmetry fields. They can be represented as:
, (4)
where orbital operators are defined in space trigonal triad basis
(5)
is the spin-orbit interaction parameter, q is the reduction parameter for the orbital angular momentum in the ground triad state comparatively to that in a free 3dn-1ion, index enumerates the projection on the trigonal symmetry axis
, (6)
Smaxis the maximum projection of system total spin, m is the unit vector along the trigonal axis of cluster, n is the unit vectorparallel to magnetization, hиh are components of low-symmetry field acting on the JT cluster. The low-symmetry fields can include both the random crystal fields and magnetic anisotropy fields caused, for example, by the second order effects of spin-orbit interaction [11,12].
The functions of the constituent ions can be involved with different weights in the wave functions of the lowest (0) and excited (ex) complex states with the maximum total spin projection
(7)
where functions k() are the antisymmetrized products of wave functions of three ions with t2g hole on a site к (к = 1,2,3) in one of states E( = +, ).
While describing the hyperfine interactions in impurity complex let us for simplicity confine ourselves (like in [10]) by taking into account of only isotropic term for ions with configuration 3dn
(8)
For state 3dn-1 we shall take into account the isotropic hyperfine interaction and anisotropic direct interaction of the orbital angular momentum with nuclear spin
(9)
where I is the nuclear spin of 3d ion, S', S, A'0, A0 are spins and parameters of isotropic hyperfine interaction in configurations 3dnand 3dn-1, is the orbital operator defined in the trigonal basis of an isolated center.
It is convenient to transform the Hamiltonian of hyperfine interaction using the electronic functions 0 and ex diagonalizing Hamiltonian (4) as basis functions. Respectively, under this rotation the orbital operators , , are transformed into the new orbital operators , and . As a result the Hamiltonian doublet state of the triad that includes spin-orbit interaction, low-symmetry fields and hyperfine interaction takes the following form
, (10)
where Eel is the energy splitting of ground E-term of MV center, kare NMR frequencies of transitions associated with the ions k = 1, 2, 3, belonging to MV center. One can see that direct interaction mixing the orbital and nuclear excitations on the MV center takes place. Such interactions can induce a new electric-dipole mechanism of NMR absorption, a new contribution to NMR signal amplification and occurrence of the “hybrid” orbital-nuclear states in the region of crossing of corresponding excitation energies.
The following notations where used in expressions (10):
(11)
,
where mk is a unit vector directed along local trigonal axis of triad ion with number k.
The obtained expressions allow one to proceed to the analysis of spectral distribution for frequencies of nuclear transitions in 3d ions of MV centers.
Dipole momentum of MV center. Coefficient of electric-dipole absorption of electromagnetic radiation
The coefficient of electric-dipole absorption in the triad under consideration can be presented in the following form
(12)
where P is dipole momentum of the MV complex, N0 is the number of MV centers, с is the light velocity, n is medium refraction coefficient. For the centers with double degenerate ground state under investigation such electric dipole momentum is described by the following expressions
,
(13)
Here indexes and = x, y, z enumerate tetragonal coordinate axes, = 0 (z), 2/3(x), -2/3 (y). Parameter p0 characterizes the absolute value of the dipole momentum, which can be attained in MV center,
(14)
where q0 is the value of the excess charge, R0 is the distance between the source of excess charge and nearest 3d-ions of cluster .
After some transformations the expression for coefficient of electric-dipole absorption, eq. (12), can be presented as:
(15)
Green-functions that are contained in expression (15) can be obtained in the second order of perturbation theory with respect to the Hamiltonian of hyperfine interaction. In this case one can obtain for absorption coefficient the following expression that is accurate in the region far from crossing of orbital and nuclear excitation energies:
(16)
The following notations are adopted here
(17)
The NMR frequencies k (k = 1,2,3) of the triad can be presented in the form
(18)
As long as we are interested in the absorption coefficient within the range of NMR frequencies which are considerably smaller than electron energies
, (19)
we can neglect in comparison with Eel in the square brackets of expression (16). Then it turns out to be that
, (20)
.
Taking into consideration the well-known formula
(21)
one can obtain the following expression for the absorption coefficient in the range of NMR frequencies:
(22)
When one summarizes contributions to the absorption from different MV centers it is necessary to consider not only the MV clusters with trigonal axis [-111] as represented in Fig.1, but also the remaining three types of clusters in the lattice with the following trigonal symmetry axes: [1-11], [-1-1-1] and [11-1].
In the case of systems where random crystal fields introduce the dominating contribution to low symmetry fields removing degeneracy , expression (22) should be averaged over different configurations of the random fields, h, h , using for example the Gaussian distribution function
(23)
where is the dispersion of random crystal fields.
Results of the calculations the NMR spectral distribution
1) First let us consider NMR spectrum of MV centers in the case when external magnetic field n is directed along tetragonal crystal axis [001]. Since four space diagonals of a cube make the same angle with cubic axis [001] all four types of clusters give the same contribution to the absorption coefficient. The typical frequency dependence of NMR absorption was shown in Fig.2. The following values of parameters were used:
One can see, that the values of all energy parameters are given in the relative units , i.e. presented as the ratio of these parameters to the maximum value of spin-orbit splitting 3qSmax). It can be easily cleared up that the absorption maximum at the frequency /3qSmax = 0.047 in Fig.2 is related to the ions k = 1,2, while the peak at frequency /3qSmax = 0.069 is related to the ion 3. While integrating over the orientation of the random fields the main contribution to the absorption coefficient comes from the range of values Fig.2 represents the results for the case of relatively weak random fields /3qSmax=0.1, therefore the absorption peaks are observed at the frequencies 1,2/3qSmax = 0.047 and 3/3qSmax = 0.069.
Fig.2.Frequency dependence of NMR spectral distribution on MV centers at electro-dipole absorption mechanism. Magnetic field is directed along axis of [001]-type.2) Let’snow analyze NMR spectral distribution in the case when magnetic field n is directed along axis [110].
Frequency dependence of the absorption coefficient in this case is presented in Fig.3 for the following set of parameters:
Fig. 3.Frequency dependence of NMR spectral distribution of MV centers, electric-dipole absorption mechanism is involved. Magnetic field is parallel to axis of [110]-type. a. MV centers with trigonal axes [-1-1-1] and [11-1],b. MV centers with axes [-111] and [1-11].
At such direction of magnetization MV clusters are divided into two groups - clusters with the trigonal axes [-111] and [1-11], which are perpendicular to the selected direction of magnetization, and clusters with the axes [-1-1-1] and [11-1], which are not perpendicular to magnetization. For clusters of the first type the combination
(24)
is equal to zero so that spin-orbit interaction does not contribute to the splitting of degenerate ground cluster state. As a result NMR frequencies depend only upon random fields
. (25)
After configuration averaging the spectral distribution of NMR for clusters of this type does not depend upon number k and it is characterized by a broad absorption band centered at frequency = (2A0S + A0S)/3.
The second group of clusters for which ()1/2, gives weak peaks at frequencies /3qSmax=0.06 and /3qSmax=0.072. If the value of random field is h, the first frequency coincides with the NMR frequency for the constituent ions with numbers k = 1 and 2 while the second one – for the ion with number k = 3. One can see that the intensity of absorption by the clusters of the first type is three orders larger than that for the second type of clusters. Analytical estimations show that at relatively weak random fields the ratio of the corresponding intensities turns out to be of the order of ()4. Thus one can expect an appreciable amplification of the NMR signals in the fields that are perpendicular to trigonal axis of MV clusters.
Conclusion
1. The existence of electric-dipole contribution to the NMR spectral distribution of MV centers was determined. It was shown that due to the unquenched orbital angular momentum an essential enhancement of NMR signal could be achieved that is expected to be promising for applications. The effect is attained by means of high intensity electro-dipole transitions between split components of MV center E-term and of mixing orbital and nuclear states by hyperfine interaction. The comparison of electric-dipole, eq. (16), and magnetic-dipole (see for example [9,10]) contributions shows that the role of the first mechanism can be rather important in a wide range of low-symmetry fields values. The magnetic-dipole contribution of MV centers to the absorption of electromagnetic radiation in the range of NMR frequencies can be described by
(26)
where the first term is related to the excitation of the orbital sub-system, while the second one describes the absorption caused by the interaction of nuclear sub-system with spin waves at k = 0. The contribution of the direct absorption in the nuclear sub-system was dropped from equation (26) because it is (N/B)2 times less than the terms that have been taken into consideration.
The first term in md is (p0/B)2 times less than the corresponding electric-dipole contribution. The ratio of intensity of electromagnetic radiation absorption described by the second term in md ,eq.(26), to the intensity of electric-dipole absorption in the range of NMR frequencies turns out to be of the order of , where k=0 is the energy of magnons with k= 0.