Structures of Impurity Defects in Lithium Niobate and Tantalate Derived from Electron Paramagnetic and Electron Nuclear Double Resonance Data

Abstract

Point intrinsic and extrinsic defects, especially paramagnetic ions of transition metals and rare-earth elements, have essential influence on properties of lithium niobate, LN and tantalate, LT, and often determine their suitability for numerous applications. Discussions about structures of the defects in LN/LT have lasted for decades. Many experimental methods facilitate progress in determining the structures of impurity centers. This paper gives current bird’s eye view on contributions of Electron Paramagnetic Resonance (EPR), and Electron Nuclear Double Resonance (ENDOR) studies to the determination of impurity defect structures in LN and LT crystals for a broad audience of researchers and students. Symmetry and charge compensation considerations restrict a number of possible structures. Comparison of measured angular dependences of ENDOR frequencies with calculated ones for Li and Nb substitution using dipole–dipole approximation allows unambiguously to determine the exact location of paramagnetic impurities. Models with two lithium vacancies explain angular dependencies of EPR spectra for Me³⁺ ions substituting for Li⁺ like Cr, Er, Fe, Gd, Nd, and Yb. Self-compensation of excessive charges through equalization of concentrations of Me³⁺(Li⁺) and Me³⁺(Nb⁵⁺) and appearance of interstitial Li⁺ in the structural vacancy near Me³⁺(Nb⁵⁺) take place in stoichiometric LN/LT due to lack of intrinsic defects.

1. Introduction

Point intrinsic and extrinsic defects, especially paramagnetic ions of transition metals and rare-earth elements, belong to the most important defects in lithium niobate (LN, LiNbO₃) and tantalate (LT, LiTaO₃), because of their essential influence on properties of this material, such as domain structure, electro-optical coefficients, light absorption, refractive indices, birth and evolution of wave-front dislocations ([1,2] and references there), and their consequences for present and potential applications [3,4,5,6,7]. A lot of effort was spent to establish a correlation between the observable data and the crystal composition, and to develop experimentally supported models of the defects: ion charges, identities, and position of the ions in the lattice, their nearest surroundings, ways of charge compensation and recharge mechanisms. Discussions about structures of intrinsic and extrinsic defects in LN/LT have lasted for decades [1]. With time, the proposed structures were evolved and detailed. Some early models were rejected. For instance, after a supposition that effective net charges are about 2.0+ [8], or 1.59+ for Nb and 1.21+ for Ta [9], it was natural to assume that divalent and trivalent impurities should preferably substitute for Nb, but not for Li. Later, numerous investigations have shown that the real picture is more complicated and richer.

In the course of the investigation of defect structures in LN/LT the following difficulties take place:

• high quality conventional samples with crystal composition x_C are usually grown from the congruent melt with the composition x_m ≈ 48.4%, for LN and x_m ≈ 48.7% for LT (x = [Li]/([Li] + [Nb/Ta]); this means that the congruent crystals with x_m = x_C contain many intrinsic (non-stoichiometric) defects, causing a broadening of the observable spectral lines and ambiguities in their interpretation;

• the crystal composition x_C of the undoped samples depends on both melt composition and growth conditions;

• the most probable positions for impurity incorporation, the Li and Nb sites as well as the octahedral structural vacancy, have the same local symmetry C₃; this means that they are not distinguishable by many spectroscopic techniques.

Techniques for the defect study can be conditionally divided into two groups: direct (Mössbauer spectroscopy [10], Rutherford Backscattering Spectrometry (RBS) [11], Extended X-ray Absorption Fine Structure Analysis (EXAFS), Particle Induced X-ray Emission (PIXE) combined with channeling, X-ray standing wave (XSW) [12], Electron Paramagnetic Resonance (EPR), and Electron Nuclear Double Resonance (ENDOR) [13]) and indirect (optical absorption and luminescence, measurements of electro-optical coefficients, etc.).

Attempts to determine impurity positions by indirect methods often gave contradicting information. Direct methods also have some shortcomings. For instance,

• Mössbauer spectroscopy demands the presence of special nuclei,

• channeling investigations are more successful in the case of heavy, many-electron ions,

• channeling methods are not sensitive to the charge state of the impurity and do not distinguish centers with C₃ and C₁ symmetry,

• due to the relatively low sensitivity the EXAFS needs high levels of crystal dopants (about 3–5 mol.%), for which clustering and occupation of both Li and Nb positions become very probable,

• EPR/ENDOR techniques are applicable to systems with non-zero spins only.

Investigations of impurity defects were carried out by all direct and indirect techniques in parallel and together, and made invaluable contributions to the problem solution. Let’s mention briefly a few of them. The positions of many impurity ions were determined by the EXAFS, PIXE, XSW, and channeling methods. Lithium substitution was found for most elements including Sc, Ti, Mn, Ni, In, Pr, Gd, Ho, Er, Tm, Yb, Lu ([11,14,15,16,17,18], and references there), Nd [19,20], Co, Fe [21,22,23,24,25,26,27,28]. A few ions substitute for Nb in LN (Hf [29], Er [30]) and in LN heavily co-doped by Mg or Zn (Sc, In, Nd, Lu [31]). Some of the ions can occupy both positions (Cr [32], Gd, Nd). Mössbauer spectroscopy [33,34] gave indirect evidence of Fe²⁺ and Fe³⁺ substitution for Li⁺. Raman spectroscopy data [35] were interpreted on the base of Fe substitution for Li. Magneto-optical and luminescence studies [36,37] identified non-paramagnetic pairs Cr_Li-Cr_Nb and established a correlation between optical bands and Cr³⁺ positions.

Convincing arguments of Fe³⁺ substitution for Li⁺ were obtained by ENDOR [38,39,40]: it was shown that spectrum angular dependencies calculated on the base of dipole-dipole interactions of Fe³⁺ electrons with surrounding Li nuclei are qualitatively different for Li and Nb substitution; quantitative agreement calculated and experimental data was achieved for Li substitution.

The huge amount of nonstoichiometric intrinsic defects in congruent LN/LT grown by the conventional Czochralski method [41,42] (we shall call them cLN and cLT) has often undesirable effects on crystal properties. Several techniques were invented in order to reduce the concentration of these defects. Some decrease of their concentration was achieved by using melts with Li excess (with x_m up to 60%) [43,44]. Thin LN/LT samples were enriched with Li by vapor transport equilibrium treatment (VTE [45,46,47,48,49,50], we shall call them vLN and vLT). Using double crucible growth [51,52,53,54,55] with excess of Li allowed one to obtain nearly stoichiometric samples (nsLN). Crystals grown by the Czochralski method from a melt to which potassium oxide K₂O has been added ([56,57,58,59,60,61,62,63,64,65,66,67], and references there) have composition x_C ≈ 50% (LN_K, stoichiometric LN, sLN). The method is scientifically called the High Temperature Top Seeded Solution Growth (HTTSSG).

The defect elimination in sLN crystals has led to a tremendous narrowing of the resonance lines and a corresponding increase of spectral resolution, and resulted in improved precision of obtained spectroscopic characteristics. Discovery of new centers substituting for Nb (or Ta in LiTaO₃) and a change of charge compensation mechanism in sLN was accompanied by a significant progress in understanding the nature and structures of intrinsic and extrinsic defects [68,69,70,71,72,73,74,75,76]. Several methods of determination of real crystal composition were developed [68,77,78,79,80]. It should be mentioned that even crystals with x_C = 50% are not completely free of intrinsic defects. There is a class of intrinsic defects which break the regular order of the LN lattice without a change of x_C: the permutations Nb_Li and Li_Nb, and cyclical permutation of Nb in Li site, Li in structural vacancy, and vacancy of Nb. Such defects can cause broadening of spectral lines.

EPR is resonance absorption of microwave quanta by electron spins placed in a swept external magnetic field at definite value of magnetic field B_res. Depending on measurement conditions it has usual sensitivity about 10¹²–10¹⁵ spins/cm³ or higher. Equipment with wave lengths 3cm (X-band, microwave frequency ν ≈ 9.5 GHz) and 8 mm (Q-band, ν ≈ 34 GHz) are usually used. EPR lines have typical widths about 1–100 MHz. This defines the resolution ability of the EPR.

Nuclear magnetic resonance (NMR) is resonance absorption of radiofrequency quanta by nuclear spins. It has worse sensitivity due to thousand time smaller magnetic moments of nuclei. However, NMR has significantly higher resolution ability about 1–100 KHz or less. Only nuclei with large natural abundance of magnetic isotopes (like ⁷Li—92.5%, ⁹³Nb—100%, ¹⁸¹Ta—99.99%) can be studied by NMR. Due to interactions with electron spins, nuclei in the impurity neighborhood have energy levels and resonance frequencies different from those of bulk nuclei. As their concentrations, which are about impurity concentration, are significantly smaller than concentrations of bulk nuclei, their NMR signals are too weak to be registered directly.

ENDOR is the indirect way to register NMR of nuclei in impurity surrounding via a stimulated change of intensity of the EPR signal in magnetic field B_res. ENDOR has sensitivity that is worse than EPR (about one-two orders of magnitude), but is better than NMR. Its resolution is comparable with NMR resolution. Characteristics of all nuclei (natural abundance of isotopes, their nuclear spins, magnetic and quadrupole moments) are well known and tabulated. This allows to relate observed ENDOR frequencies to interactions of impurity electrons with own nucleus and surrounding nuclei.

For a paramagnetic impurity these techniques are capable of determining: its element, charge, spin state, nearest surrounding (i.e., its location), and many spectroscopic characteristics. Fortunately, most impurities which are useful for optoelectronic application (including transition and rare-earth elements) enter LN and LT in a paramagnetic state. Li, Nb and Ta nuclei have significant (93–100%) natural abundance of magnetic isotopes. The main isotope ¹⁶O has no magnetic moment and is invisible in EPR/ENDOR/NMR experiments.

The aim of this paper is to give current bird’s eye view on contributions of EPR/ENDOR studies to the determination of impurity defect structures in LN and LT crystals for a broad audience of researchers and students. The name “center” will be used for a complex consisting of the paramagnetic impurity and its surrounding that can have zero, one or more defects besides regular ions.

2. Materials and Methods

Most of our measurements presented below were carried out on samples grown by E.P.Kokanyan from Li-rich melt or from the melt with the addition of K₂O [56,57].

The EPR/ENDOR spectra were measured at microwave frequencies v ≈ 9.4 GHz (X-band) and v ≈ 34.4 GHz (Q-band) on the Bruker ELEXSYS EPR/ENDOR spectrometer at temperatures between 4.2 and 300 K at Montana State University (Bozeman, MT, USA).

The treatment and simulation of all spectra and their angular dependencies, as well as the preparation of corresponding figures were carried out with the help of the “Visual EPR” and “Visual ENDOR” program packages [81].

All lattice images were generated using CrystalMaker® program [82] and crystallographic data [83,84,85,86]. Sizes of oxygen balls were artificially reduced, since nonmagnetic O²⁻ ions are invisible for EPR/ENDOR. As LiNbO₃ and LiTaO₃ are isostructural crystals, all figures with lattice and structures of impurity centers are equally applicable to both LN and LT. However, dipole-dipole interactions should be calculated taking into account a difference of their lattice constants.

3. Basics of EPR and ENDOR Spectra Interpretation

3.1. Symmetry Considerations

The ideal lattice of LN and LT crystals (Figure 1, see also [87]) has two molecules in its rhombohedral elementary unit cell and the space group symmetry is R3c (C_3v⁶, 3m) at room temperature [8,88,89,90]. If ionic radii are taken into account, the LT/LN lattices look like consisting mainly of planar layers of closely packed oxygen ions with small voids between them, filled by Li and Nb/Ta. There are several electrically non-equivalent positions in the lattice. Some of them—the sites on the z (or optical c) axis of the crystal, including the sites of Li, Nb/Ta and the octahedral structural void (vacancy), v_oct—have the symmetry of the point group C₃. An isolated defect in any of these positions creates a C₃ (in the following also labeled “axial”) center. The tetrahedral structural void, oxygen sites and all other off axis positions have the lowest possible symmetry, C₁.

A complex of two defects has C₃ symmetry, if both are located on the crystal z axis, and C₁ symmetry in all other cases. If the difference in positions of the nearest oxygen ions is ignored, then each unit cell has the following site sequence along the z-axis: Li, Ta, v_oct, Li, Ta, v_oct. However, oxygen octahedrons and next neighbors for two positions of one type cation are not identical, therefore, the correct assignment of the sequence should be Li_L, Ta_L, v_oct,L, Li_R, Ta_R, v_oct,R. The surrounding of the “Right” (R) position can be transformed to the “Left” (L) one by a reflection x ⇔ −x and a shift by c/2, because zy is a glide mirror plane in the R3c lattice (Figure 1b). The same consideration is applicable to C₁ positions also. This means that each axial (C₃) or low-symmetry (C₁) L center has a corresponding R partner. The L and R partners are electrically identical and are not distinguishable by optical or channeling methods, but they are magnetically non-equivalent and can be resolved from each other in favorable cases (high spin value, small line width) by magnetic resonance techniques.

Each C₁ center in the R3c lattice has two additional magnetically nonequivalent partners, which can be transformed into each other by a rotation around the z-axis of the crystal by 120° and 240°. They can be distinguished by the EPR at arbitrary orientations of the magnetic field.

As any crystal characteristic, the EPR spectra must reflect the crystal symmetry and symmetry of all present paramagnetic centers. The best way to determine the defect symmetry is to study angular dependence of the spectra in external magnetic field B under rotation of the crystal sample around z-axis (shortly speaking, in xy plane). For a C₃ paramagnetic defect with electron spin S = 1/2 the EPR spectrum consists of one resonance at the same position B_res at any orientation of B with respect to crystallographic axes. A set of up to six lines can be observed at arbitrary orientation of B for C₁ centers; however, some of them coincide for directions of B along x, y, or z. If two or more centers present in the sample studied, the spectrum has two or more sets of the lines. If a center has electron spin S > 1/2 multiplets of 2S lines can be observed for every center.

For centers with S = 1/2 the resonance line positions can be described using spin-Hamiltonian

(1)$H = {\mu }_{B}BgS$

Here μ_B is the Bohr magneton, B is the vector of static magnetic field, g is the tensor of spectroscopic splitting, and S is the vector of electron spin.

As an example of the multi-center spectra let’s consider ion Nd³⁺ in LN [91,92,93] (Figure 2). Every spectrum consists of a single dominant line labeled with the number 1 and many satellite lines of smaller intensities. The positions of the dominant line have the same value of the resonance field for B||x and B||y (φ = 90 deg), whereas positions of other lines do not coincide. We have to suppose that the single line #1 belongs to the axial C₃ symmetry center, Nd₁.

To make correct labeling of all observed lines a detailed study of angular dependence of EPR line positions is required. For instance, we can rotate magnetic field in xy crystallographic plane, changing azimuthal angle φ between x-axis and B. Circular diagram of measured spectra (Figure 3) show that dominant line draw a circle, i.e., has no dependence of B_res(1) on φ. Angular patterns of other lines demonstrate the presence of all elements of the 3m group, namely, C₃ symmetry and the mirror with respect to y-axis at φ = 90 deg. Measurement of such a diagram is redundant, since xy plane angular patterns simply repeat themselves with a period 60 deg. Due to the glide mirror plane, the 30 degree dependence measured from x axis contains all information.

Usually, 90 deg of the measured dependence is plotted (Figure 4). If spectra are measured with small angular steps (said 1 deg), it is easy to trace every line. The line tracing allows to put labels on every resonance line on Figure 2. Without doubt, at least four groups of angular branches are clearly distinguished on Figure 4: one straight branch of axial Nd₁ and three sets of curved branches, which correspond to low-symmetry centers Nd₂, Nd₃, and Nd₄. Therefore, we can conclude that Nd³⁺ in Nd₁ center occupies one of three possible positions on the z-axis. If Nd³⁺ has additional defect or defects (charge compensators) in its neighborhood, the defect is located on the same axis. In the case of Nd₂, Nd₃, and Nd₄ centers with the C₁ symmetry, the additional defects are located off the z-axis. It is unlikely that Nd³⁺ ions occupy sites with C₁ symmetry (tetrahedral void or O²⁻) due to large charge misfit.

Symmetry of any center reflects the symmetry of lattice sites occupied by impurity and charge compensators (if any) and their relative locations. To distinguish the C₃ and C₁ symmetries the measurement of angular dependence in xy plane is sufficient. However, to obtain a full set of spectroscopic characteristics of the center (like six components of g-tensor) by fitting measured angular dependence a study of spectra under three rotations in perpendicular planes (road map) is required.

3.2. Isotropic and Dipole-Dipole Interactions

Calculations of the ENDOR frequencies and transition probabilities (i.e., ENDOR line positions and relative intensities) for i-th nucleus is usually based on nuclear spin-Hamiltonians H_i

(2)$H_i = -{\mu }_n{g}_n^{\left(i\right)}B{I}_{}^{\left(i\right)}+S{A}_{}^{\left(i\right)}{I}_{}^{\left(i\right)}+I_{}^{\left(i\right)}{Q}_{}^{\left(i\right)}{I}_{}^{\left(i\right)}$

where μ_n—nuclear magneton, $g_n^{\left(i\right)}$—nuclear g-factor; A⁽ⁱ⁾, Q⁽ⁱ⁾—tensors of hyperfine and quadrupole interactions.

In many cases there are two dominant contributions to the hyperfine tensor A⁽ⁱ⁾: isotropic (contact) hyperfine interaction, $a^{\left(i\right)}S{I}_{}^{\left(i\right)}$ and dipole-dipole interaction of vectors of electron ${\mu }_{B}gS$ and nuclear $-{\mu }_n{g}_n^{\left(i\right)}{I}_{}^{\left(i\right)}$ magnetic moments. The dipole–dipole interaction can be described by

(3)$A_{jm}^{\left(i\right)dd} = \frac{{\mu }_B{\mu }_n}{{\left(R^{\left(i\right)}\right)}^3}{g}_n^{\left(i\right)}\left[\frac{3}{{\left(R^{\left(i\right)}\right)}^2}\left({\displaystyle \sum }_{p = x,y,z}{g}_{jp}{R}_{p }^{\left(i\right)}\right)\left({\displaystyle \sum }_{l = x,y,z}{R}_{l }^{\left(i\right)}{\delta }_{lm}\right)-g_{jm}\right]$

where R⁽ⁱ⁾ is the vector from paramagnetic impurity to the i-th nucleus. Most characteristics in Equation (3) are known: ${\mu }_B, {\mu }_n, g_n^{\left(i\right)}$ are tabulated, $g_{jm}$ components of g-tensor are determined from EPR measurements. Therefore, comparison of measured angular dependences of ENDOR frequencies with calculated ones on the base of Equations (2) and (3) can be used for determination of R⁽ⁱ⁾, i.e., the position of the paramagnetic impurity relative to surrounding nuclei.

Nuclei at the same distance from impurity ions are usually called a shell. The first shell of an impurity in a Li site has six nearest Li nuclei off the z-axis at the distance R⁽¹⁾ (Figure 5a). Three of them are located above the impurity (the subshell 1a), other three—below the impurity (the sub-shell 1b); if the impurity is shifted from regular Li site then R^(1a) ≠ R^(1b). The second shell consist of six Li nuclei in the xy plane of the impurity at the distance R⁽²⁾ (subshells 2a, 2b).The first shell of an impurity in a Nb site has one Li nucleus on z-axis, the second shell has three Li nuclei (the nuclei are labelled 1 and 2 on Figure 5b). As directions from the impurity to the nuclei in a shell are different, their hyperfine interactions are magnetically non-equivalent, i.e., produce different branches in angular dependences of ENDOR spectra.

The simplest way to determine the impurity lattice site is:

• to calculate $A_{jm}^{\left(i\right)dd}$ for the several nearest nuclei around the impurity taking lattice distances from X-ray data and g_jm from EPR measurements,

• to calculate ENDOR frequencies, to plot patterns of their angular dependencies, and

• to compare the patterns with measured angular dependencies of ENDOR for a definite EPR line [13,38,39].

Due to different surroundings for impurities in the Li and Nb sites, the calculated patterns are completely different (Figure 6). ENDOR frequencies for the first shell of Li nuclei for the Li site vary with rotation of magnetic field in xy plane(Figure 6a), whereas for the single Li nucleus of the first shell for Nb site two straight branches should be observed in angular dependence (Figure 6c). As Li nuclei of the all corresponding shells are closer to the impurity ion in the Nb site than in the Li site, the range of angular variation for Nb site is larger than for Li site. The branches of the 2nd and 3rd shells for Li site practically coincide with measured angular dependencies for Nd₁ center (Figure 6b), whereas no branch for Nb site is close to observed one. The branches of the 1st shell for Li substitution also agree with observed ones after small correction due to isotropic hyperfine interaction (Figure 6b). Based on clear agreement of hundred measured values of ENDOR frequencies for the dominant EPR line #1 with calculated frequencies for Li site and obvious disagreement with calculated ones for Nb site, we can definitely conclude that the Nd₁ line in EPR spectra (Figure 2) belongs to Nd³⁺ ion substituted for Li.

The simulation of dipole-dipole interactions for the Li and Nb sites has no fitting parameter. Therefore, the described qualitative and quantitative approach for the determination of impurity positions gives reliable results. Note that hyperfine interactions for Nb/Ta nuclei have additional contributions due to a polarization of inner electron shells of oxygen and niobium ions (transferred hyperfine interaction).

The characteristic of isotropic hyperfine interaction $a^{\left(i\right)}$ is related to a density of electron cloud at the location of i-nucleus R⁽ⁱ⁾. For the isotropic g-factor:

(4)$a^{\left(i\right)} = \frac{8\pi }{3}g{\mu }_B{g}_n^{\left(i\right)}{\mu }_n{\left|\psi \left(R^{\left(i\right)}\right)\right|}^2$

The isotropic hyperfine interaction often (but not always) exponentially decreases with the distance from impurity. The largest value $a^{\left(i\right)}/g_n^{\left(i\right)}$ should correspond to nuclei nearest to the impurity. It allows to determine nuclei of the nearest surrounding by comparison of $a^{\left(i\right)}/g_n^{\left(i\right)}$ for Li and Nb nuclei, and therefore, to find the impurity location. Impurities substituted for Li have Nb nucleus on z-axis in the nearest surrounding, whereas ions substituted for Nb have Li nucleus (Figure 6, shell #1).

Another way to use contact and dipole-dipole interactions is:

• to measure the road map of angular dependencies in three perpendicular planes (Figure 4a),

• to determine all components of $A_{}^{\left(i\right)}$ tensors by fitting observed angular dependencies,

• to separate isotropic and anisotropic parts,

• to find principal values of the anisotropic part, and finally,

• to compare these principal values with calculated ones on the base of Equation (3).

3.3. Charge Compensation and Intrinsic Defects

The structure of a center, in which a lattice site is occupied by an extrinsic impurity ion having a charge different from that of the respective lattice ion (in the following labeled “non-isocharged replacement”), depends on the charge of the impurity and the mechanism of charge compensation. Because of non-stoichiometry, the real lattice of conventional congruent LN contains many intrinsic defects, the relative concentrations of which have not yet been determined reliably. The following entities have been considered (their charges with respect to the lattice being given by Kröger-Vink notation in brackets): Nb⁵⁺_Li⁺ (Nb_Li^4•) antisite defect, v_Li⁺ (v_Li^’) lithium vacancy, v_Nb⁵⁺ (v_Nb^5’) niobium vacancy, Nb⁵⁺_v (Nb_v^5•) niobium on structural vacancy, Li⁺_v (Li_v^•) lithium on structural vacancy, v_O oxygen vacancy. During recent years the existence of the following charge compensated complexes has been postulated most often: Nb_Li + 4v_Li [90,94,95] (including three-dimensional complexes 3v_Li + Nb_Li + v_Li [96,97]), 5Nb_Li + 4v_Nb [98,99], 2Nb_Li + 2Nb_v + 3v_Li + 3v_Nb [96] and some others [100]. Some features accompanying the crystal growth (Li₂O evaporation, variation of oxygen deficiency in Nb₂O_5-x [101]) and specific changes of some crystal properties after thermal oxidation and/or reduction definitely indicate that the oxygen sub-lattice is not always perfect and stable as well; therefore the oxygen non-stoichiometry has been also discussed for a long time [102,103,104,105]. The intrinsic defects by themselves or complexes of them, which are not charge compensated, can furthermore serve as local or distant charge compensators for non-isocharged extrinsic or radiation defects. Due to the high concentration of the intrinsic defects the congruent LN and LT crystals are very tolerant to substitutional or interstitial impurities, including non-controlled ones, because the necessary charge compensators (local or distant) can be easily found among the non-stoichiometric defects. Real crystals often contain H, Cu, Co, Mn, Fe, etc. in concentrations about 0.00X–0.0X.

Non-isovalent cation Meⁿ⁺ (n > 1) substituted for Li⁺ requires negative charge compensator like lithium vacancy or interstitial O²⁻. If Meⁿ⁺ (n < 5) substituted for Nb⁵⁺ or Ta⁵⁺ the n − 5 negative charge can be compensated by antisite ions, oxygen vacancies, v_O or interstitial H⁺ and Li⁺.

In some cases, a self-compensation of impurity charges takes place. For instance, no additional charge is required if two Me³⁺ ions substitute for both Li⁺ and Nb⁵⁺ in nearest neighbor or next neighbor sites (local self-compensation) or even relatively far one from another (distant self-compensation). There are two critical parameters, which stimulate the self-compensation: total concentration of possible charge compensation defects, [D] and the total concentration of Me³⁺ ions in the sample, [Me]. The [D] includes a dominant contribution due to a deviation of the crystal composition from stoichiometry, concentrations of H⁺ ions and v_O, as well as contributions due to non-controlled impurities (Mg, C, Si, Cl, …), which are always present in real crystal in concentrations about 1–500 ppm. If [D] >> [Me] the isolated centers with additional intrinsic defects are dominating. If [D] << [Me] then the self-compensation is preferable due to lack of charge compensators.

Distant charge compensators produce small distortions of the crystal field at the impurity site, what normally causes EPR line broadening, and an asymmetry of their shapes, but do not change the center symmetry revealed in the observed positions and splitting of EPR lines. Defects in the immediate neighborhood (local charge compensation) cause strong changes of the center characteristics (g and A tensors in the case of Nd³⁺) and lowering of center symmetry.

3.4. Di-Vacancy Models for Trivalent Impurities

Many impurities in LN/LT crystals create a family of paramagnetic centers that includes dominant axial center and several satellite low-symmetry centers with EPR lines of smaller intensities (like presented on Figure 2). It would be reasonable to look for center models which are able to describe the whole family.

Low-symmetry C₁ centers appear in three cases:

• Impurity ion like Nd³⁺ has an off-axis lattice defect (charge compensator) in the immediate neighborhood.

• Impurity ion substitutes for O²⁻ (this is very improbable for cations).

• Impurity ion incorporates into a small tetrahedral void (this is possible, but often unlikely due to larger charge misfit than for Li substitution).

For axial centers, the directions of principal axes of the g-tensor are dictated by crystal symmetry: the 3rd axis of the center coincides with the crystal axis and the directions of the 1st and 2nd axes are arbitrary. For low-symmetry centers, there are no symmetry restrictions on the orientation of principal axes. However, if the symmetry lowering is related to an off-axis lattice defect, it is reasonable to expect that directions of the center axes are related to the distortion created by the off-axis defect. The 3rd axis will have some inclination from the z-axis to the defect, and projections of the 1st or 2nd axis will be close to the projection of line from the impurity ion to the defect. The distortion should decrease with the distance from the impurity ion to the defect.

ENDOR data for the dominant EPR line have confirmed that Nd³⁺ substitutes for Li⁺ and only Li and Nb nuclei were found in the neighborhood. Therefore, intrinsic defects without nuclear spin should be considered for the required 2-charge compensation. The size O²⁻ is comparatively large to be placed into small octahedral or tetrahedral voids. Nearly stoichiometric crystals have significantly reduced concentration of v_Li. Nevertheless, their concentration often exceeds the impurity concentration. Sufficient concentration of v_Li can be also created in the process of growth of samples with non-isocharged impurities.

A key to the identification of models which describe all varieties of Nd³⁺ centers with minimal assumptions was obtained from the analysis of angular dependencies of the Nd₄ center. The extrema in the xy plane and, correspondingly, projections of one principal axis of the g-tensor, are close to ϕ values of 15, 45, and 75 degrees (Figure 4b). In the projection of the LN lattice onto the xy plane (Figure 7, after [93]) there are no similar values of ϕ from Nd³⁺ to any ion in the lattice: directions from the Nd³⁺ site to all cation sites have azimuthal angles equal to n × 30°. Therefore, the simplest model, that has a single lattice defect, like a Li⁺ vacancy, v_Li, cannot explain the observed angular dependencies of Nd₄ centers. However, if two Li⁺ vacancies are located in the first and second shells of the surroundings of an Nd³⁺ ion substituted for a Li⁺, the three defects are organized into a triangle. The longest side of the triangle connected two v_Li has a perpendicular oriented in the required direction (Figure 7).

The hypothesis that Nd³⁺ substituted for Li⁺ has two v_Li as charge compensators allows for consistent models for the whole family of Nd³⁺ centers in LN crystals with a low concentration of neodymium doping. After analysis of principal values and axes of g-tensors it was concluded [92,93] that Nd³⁺ centers have lithium vacancies in sites described in Table 1. Some of possible structures are presented on Figure 7 and Figure 8.

The di-vacancy models can explain EPR spectra of many other (but not all) trivalent impurities Me³⁺_Li.

4. Structures of Impurity Centers

4.1. Monovalent Cations

The most probable incorporations of Me⁺ is the substitution for Li⁺. No charge compensation is required in this case. The Me_Li⁺ center should have axial C₃ symmetry.

Proton H⁺. This is an example of off-site position: protons occupy positions between two oxygen ions in an oxygen plane. The non-paramagnetic OH⁻ centers were studied by infrared and NMR spectroscopies [80,106,107,108].

A hydrogen associated paramagnetic center (g = 2.0028, A = 3 mT at 77 K) was identified as an OH²⁻ ion, produced because of an electron capture by a diamagnetic OH⁻ ion, substituting the O²⁻ ion in LN [109].

Ni⁺ (3d⁹, S = 1/2). Observed EPR spectra of the Ni⁺ have axial symmetry at room temperature [110,111]. That excludes interstitial position in tetrahedral structural vacancies. However, at low temperatures the Ni⁺ center has C₁ symmetry, as up to six lines were observed at arbitrary orientation of magnetic field (Figure 9a, after [111]).

The center was characterized by anisotropic g-tensor with principal values g₁ = 2.246, g₂ = 2.217 and g₃ = 2.061; principal axes of the g-tensor are rotated with respect to crystallographic axes by Euler angles α = γ ≈ 0, β ≈ 55°. As the 3rd principal axis is directed approximately to one of the nearest oxygen ions, the reason for the low symmetry is a static Jahn-Teller effect for 3d⁹ ions in Ni⁺O₆²⁻ complexes (Figure 9b, [111,112]), and not a presence of an intrinsic defect in the neighborhood. Dynamic averaging due to center reorientation leads to the axial symmetry of observed EPR spectra at room temperatures [111].

Mg⁺ (3s¹, S = 1/2). Following vacuum reduction at 1000 °C, LN crystals heavily-doped with Mg exhibit an optical absorption spectrum that can be decomposed into two bands peaking near 760 and 1200 nm, and a broad EPR spectrum with g_c = 1.82 [113]. The 1200-nm band and ESR signal are associated with an electron trap (identical to the one produced during the irradiations). This electron trap is suggested to be a Mg⁺ complex. There is an alternative interpretation of this spectrum [114].

4.2. Divalent Cations

For divalent Me²⁺ impurities a substitution for Li⁺ ions and incorporation in structural vacancies has essentially less charge misfit than a substitution for Nb⁵⁺. Since the Li-Li distance is much larger than Li-Nb or v_oct-Li, v_oct-Nb, the Me²⁺_Li + v_Li centers should be slightly distorted by the presence of a local charge compensator. For v_Li located on C₃ axis and for distant cation vacancies (local and distant charge compensation) axial centers should be observed.

Co²⁺ (3d⁷, S = 1/2, I = 7/2). Dominant axial Co²⁺ center with g_‖ = 2.6 and g_┴ = 4.96 ÷ 5.04, A_‖ ≈ 0, |A_┴| = 0.0154 cm⁻¹, as well as low-intensity low-symmetry satellite centers were observed in cLN [115,116] and vLN [117]. Similar g- and A-values were reported for LiTaO₃:Co²⁺ [116]. The picture agrees with Co²⁺ substitution for Li⁺ in the dominant axial center (Co_Li) and excess charge compensation by v_Li (the Co²⁺ ⇔ 2Li⁺ substitution mechanism). A small EPR line of axially symmetric cluster of Co²⁺ ions appeared in sLN [118] (Figure 10a). To explain it the substitution mechanism 4Co²⁺ ⇔ 3Li⁺ + Nb⁵⁺ [119] was considered. The four Co²⁺ ions can occupy nearest possible cation sites by occupying one Nb site and three neighbor Li sites, creating a trigonal pyramid with C₃ symmetry (Figure 10b).

From comparison of measured and calculated characteristics it was found that Co²⁺ does not occupy exactly the host Li⁺ site but undergoes an off-center displacement 0.006 nm away from the oxygen octahedron center in LiNbO₃ (or LiTaO₃) [120,121].

Cu²⁺ (3d⁹, S = 1/2). Copper nuclei have two isotopes ⁶³Cu (I = 3/2, g_n = 1.484, natural abundance 69.2%) and ⁶⁵Cu (I = 3/2, g_n = 1.588, 30.8%). Hyperfine interaction of Cu²⁺ electrons with their own nucleus leads to splitting of its single EPR line into a quartet ([110,111], and references there; [122,123,124]). As magnetic moments of ⁶³Cu and ⁶⁵Cu are very close, the quartets from the two isotopes overlap in cLN (Figure 11a). The Cu²⁺ center in LN was characterized by anisotropic g-tensor (g₁ = 2.095, g₂ = 2.111 and g₃ = 2.428; α = γ ≈ 0, β ≈ 51°) [111]. At low temperatures a static Jahn-Teller effect for 3d⁹ ions in Cu²⁺O₆²⁻ complexes reduces the center symmetry to C₁(Figure 11b). At room temperature, the Cu²⁺ center has axial symmetry due to center reorientation and motional averaging. The EPR parameters of the impurity Ni⁺, Cu²⁺, and Ni³⁺ in LiNbO₃ were theoretically studied from the perturbation formulas for 3d⁹ ions [112,122].

Ni²⁺ (3d⁸, S = 1). Several additional terms, which describe zero field splitting (ZFS) of energy levels, should be added to the spin-Hamiltonian (1) for paramagnetic centers with S > 1/2:

(5)$H_{ZFS} = {{\displaystyle \sum }}_{k = 2,4,6}^{}{f}_k\left[{{\displaystyle \sum }}_{q = 0}^k{b}_k^q{O}_k^q\left(S\right)+{{\displaystyle \sum }}_{q = 1}^k{c}_k^q{\mathsf{\Omega }}_k^q\left(S\right)\right]$

Here f₂ = 1/3, f₄ = 1/60, f₆ = 1/1260; $O_k^q, {\mathsf{\Omega }}_k^q\left(S\right)$—Stevens operators, which are non-zero for k ≥ 2S. For C₃ symmetry only q equal to 0, 3, and 6 are allowed. For S = 1 only terms with k = 2 are present in Equation (5).

The Ni²⁺ centers in LN exhibit the EPR spectra of C₃ symmetry. Therefore, the sum in (5) turns into one term $b_2^0{O}_2^0/3.$ It was found that $b_2^0 =$−5.31 cm⁻¹ and Δg = g_‖ − g_┴ = 0.04 [115,125]. Since EXAFS data supports Ni²⁺ substitution for Li⁺ [24], the reasonable choice for the charge compensator is one v_Li that is located on the C₃ axis or very far of Ni²⁺.

A cluster substitution 4Ni²⁺ ⇔ Ta⁵⁺ + 3Li⁺ was considered for LT [126]. Note that an agreement of measured and calculated spin-Hamiltonian parameters [127] was obtained for Ni²⁺ substitution for Nb⁵⁺ in LN without a charge compensator.

The charge excess of one interstitial Me²⁺ or two Me²⁺_Li could be exactly compensated by an additional O²⁻ ion. However, such a compensation looks unlikely, as the ionic radius of O²⁻ (about 1.4 Å) is larger than the sizes of octahedral or tetrahedral vacancies.

Mn²⁺ (3d⁵, S = 5/2, I = 5/2). For S = 5/2 the ZFS causes splitting into 2S = 5 components of fine structure, each of them additionally splits into 2I + 1 = 6 lines due to the hyperfine interaction of Mn²⁺ electrons with their own nucleus (Figure 12).

Angular dependences of Mn²⁺ in both LN and LT are described by spin-Hamiltonian of axial symmetry with g ≈ 2.0, and A ≈ −0.008, $b_2^0$ = 0.0731 (LN), $b_2^0$ = 0.1694 (LT), all in cm⁻¹, [128,129,130,131,132,133,134,135,136,137,138]. Our spectra (Figure 12) were fitted with $b_2^0$ = 0.074 (LN), $b_2^0$ = 0.175 (LT).

Hyperfine interactions with four Li and two Nb shells of surrounding nuclei were determined by ENDOR study in LN [139]. For isotropic g-tensor, the principal values of dipole-dipole interaction (3) can be described by:

(6)$b_{dd}^{\left(i\right)} = \frac{g{\mu }_B{g}_n^{\left(i\right)}{\mu }_n}{{\left(R^{\left(i\right)}\right)}^3}$

Comparison of measured values of $b_{dd}^{\left(i\right)}$ with values calculated by Equation (6) for Li and Nb substitution has definitely shown that Mn²⁺ ions occupy Li site [139].

4.3. Trivalent Cations

Most transition metals (including iron, titanium, and chromium) and rare-earth elements enter LN in this valence. If Me³⁺ substitutes Li⁺ there are three possibilities to compensate its 2+ excess charge: v_Li, v_Nb and self-compensation with Me³⁺_Nb⁵⁺ in the nearest or distant neighborhood. Every Me³⁺_Li can be compensated by two v_Li; every five Me³⁺_Li⁺ ions—by two niobium vacancies. The positive antisite defect Nb_Li can serve as a charge compensator for Me³⁺ replacing Nb⁵⁺ (but not Me³⁺_Li⁺ or Me³⁺_v). It is remarkable that one Nb⁵⁺_Li⁺ exactly compensates the excess charge of two Me³⁺_Nb⁵⁺ ions. Since v_Li⁺ has only one negative charge relative to the ideal lattice, it produces a 4–5 times weaker perturbation of the crystal field than Nb⁵⁺_Li⁺ or v_Nb⁵⁺. Therefore the centers with v_Li⁺ at distances of about 6 Å should probably are not distinguishable from axial centers with non-local charge compensation.

Interstitial Li⁺ ions should be considered as charge compensators for Me³⁺_Nb⁵⁺ in Li-rich, VTE treated, and stoichiometric crystals. The association of Me³⁺_Nb⁵⁺ with one Li⁺ ion in the nearest vacancy (partial local charge compensation) leads to an axial center, the second Li⁺ in the next vacancies can decrease symmetry to C₁, if located off center axis and near the impurity. Mg²⁺ or Zn²⁺ ions substituted for Li⁺ can be also suitable compensators for Me³⁺_Nb.

Distances between the replaced ion and shells of possible location of compensators, various configurations of Me and charge compensators, as well as the symmetries of the corresponding complexes are given in References [87,140,141,142].

Cr³⁺ (3d³, S = 3/2). In congruent and Li-rich LN samples, the EPR lines of dominant axial Cr³⁺ center, characterized with ZFS $b_2^0$≈ 0.39 cm⁻¹, are accompanied with small satellite lines (Figure 13). Initial discussion with plausible but contradictory arguments about Li⁺ or Nb⁵⁺ substitution [134,143,144,145,146,147,148] should be ended after PIXE [32] and detailed ENDOR [140] studies have shown that Cr³⁺ substitutes for Li⁺ and slightly shifted from regular Li site. ENDOR measurements confirmed that Cr³⁺ substitutes for Li⁺ also in all satellite centers. Therefore, the whole family of these Cr³⁺ centers can be described as Cr³⁺_Li with location of charge compensator on C₃ axis for axial or off it for low-symmetry centers.

The ENDOR measurements [140] found that hyperfine interactions with Nb nuclei of the 2nd and 3rd shells (Figure 5a) are stronger than with Li nuclei (Figure 14, bottom). However, lines of Nb nuclei on the center axis (1st and 4th shells) were not identified. It can be caused by unfortunate conditions of their observation, petal distribution of electron density for the 3d³ ion or absence of Nb ion in one of these sites, i.e., v_Nb. Two v_Nb can serves as the charge compensator for five Cr³⁺_Li. Although the presence of v_Nb in undoped LN looks unlikely, the charge compensation defects in doped crystals (especially, if dopant concentrations exceed 0.X%) can differ from dominant intrinsic defects in undoped LN or LT. During the growth process, the required compensators can organize themselves around impurities or enter from air in order to minimize the creation energy for the impurity center. This is why structures with v_Nb were proposed for satellite centers of Cr³⁺_Li [87].

On the other hand, v_Li are considered as dominant intrinsic defects in LN and LT. Angular dependencies of EPR spectra for the dominant (C₃) and satellite (C₁ symmetry) Cr³⁺ centers (Figure 15) are pretty similar to observed patterns for Nd³⁺ (Figure 4). Therefore, two vacancy models for trivalent impurities (Table 1) and structures on Figure 7 and Figure 8 can be viable alternatives for Cr³⁺ family.

A lack of intrinsic defects in stoichiometric samples leads unavoidably to a change of charge compensation mechanism for trivalent impurities, and substitution for Nb⁵⁺ becomes possible. An axial Cr³⁺ center with significantly smaller ZFS $b_2^0$ = 0.0215 cm⁻¹ (Figure 16a) was found in LN_K samples [142]. ENDOR study has shown that hyperfine interactions with Li nuclei significantly larger than with Nb nuclei for this center, i.e., the nearest surrounding consist of Li nuclei. This means that Cr³⁺ substitutes for Nb⁵⁺ in this center. As lines of protons, H⁺, were found in the ENDOR spectra (Figure 14, top), they compensate the negative charge of Cr³⁺_Nb⁵⁺ (Figure 16b).

Exchange interaction S^AJS^B between spins of Cr³⁺ ions (S^A = S^B = 3/2) leads to gaps between states with values of total spin S = S^A + S^B equal to 0, 1, 2, and 3. The state with S = 0 is non paramagnetic. For pairs at a close distance the gaps can exceed energies of microwave quantum (36 GHz ≈ 1.2 cm⁻¹). It was found by magneto-optical study that for Cr³⁺_Li–Cr³⁺_Nb substituted for nearest Li and Nb sites (at the distance about 0.3 nm) the exchange interaction is antiferromagnetic and J ≈ 480 cm⁻¹. As 1 K × k_B ≈ 0.7 cm⁻¹ the upper states of such pairs with non-zero S are not populated even at room temperatures, and the pairs are EPR silent. However, the pairs of the next orders with the isotropic exchange coupling parameter J ≈ 1.5 cm⁻¹ were observed by EPR at relatively low concentration of chromium in LN (less than 0.1 at.%) [87,134,148,149,150].

PIXE/channeling study [32] revealed that chromium ions occupy both regular cation sites (60% on Li sites and 40% on Nb sites) in congruent LN doped with 0.1 mol.% of Cr. This means that the majority of chromium ions enter cLN as non-paramagnetic pair centers. The EPR observes only a top of iceberg in cLN: dominant Cr³⁺_Li and small signals of non-nearest paramagnetic pairs.

ENDOR study of Cr³⁺ centers in LN heavily doped with Mg and co-doped with Cr has unambiguously shown that Cr³⁺ in the dominant center has ZFS $b_2^0$ close to zero (nearly isotropic case) and substitute for Nb [151,152,153,154,155]. Measured anisotropic hyperfine interactions of Cr³⁺ with for four Li shells were close to $b_{dd}^{\left(i\right)}$ for Nb site. From comparison of data obtained by EPR, ENDOR, optical absorption, fluorescence, fluorescence line narrowing, selective excitation and radiative lifetime measurements [156,157,158,159,160,161,162] it has been concluded that the addition of Mg²⁺ ions to LN does not create new Cr³⁺ complexes, but changes the relative concentrations of the Cr³⁺_Li and Cr³⁺_Nb centers.

The measured value of ZFS for dominant Cr³⁺ in cLT $b_2^0$ ≈ 0.444 cm⁻¹ [144] is very close to the value for Cr³⁺_Li in LN. EPR spectra of this center together with signals of weaker intensities of a second center [163,164] were explained in a supposition that they originate from Cr³⁺ ions located at Li⁺ sites and that two v_Li play the role of a divalent charge compensator for both centers. EPR study of Cr³⁺ in nsLT and superposition model analysis [165] are in good agreement with the Cr³⁺ substitution for Li. The temperature dependence of b₂⁰ term showed a non-monotonic behavior in the region of 40 K.

Finally, a lot of studies were devoted to various properties of Cr³⁺ in LN and LT crystals of different compositions in order to clarify relations of optical characteristics with structures of chromium centers [166,167,168,169,170,171,172,173,174,175] etc.

Dy³⁺ (4f⁹). The observed Zeeman splitting [143] was described with g-tensor of axial symmetry: g_‖ = 8.7 and g_┴ = 1.3. The single EPR line had width about 8–10 mT at B||z and became broader at B┴z. Such a behavior can be related to unresolved splitting due to satellite centers, if Dy³⁺ occupies Li position and its charge is compensated by v_Li. Two Dy³⁺ centers with g_xx(1) = 2.56(1), g_zz(1) = 4.43(1), and g_xx(2) = 6.67(1), g_zz(2) = 1.23(1) and linewidth about 20 mT were registered in LN after γ-irradiation [176]. Both centers were attributed to Dy³⁺_Li. A broad line of the third center with g_zz(3) ≈ 1.2 appeared only at B close to the z-axis. Weak hyperfine lines due to isotopes ¹⁶¹Dy (natural abundance 19%), ¹⁶³Dy (2.49%) was observed in single crystal of LN [177].

Er³⁺ (4f¹¹). Due to fast spin-lattice relaxation, the EPR signals of Er³⁺ are observable at low-temperatures only. Earlier studies claimed that Er³⁺ ions in cLN create an axial center with g_‖ ≈ 15.1–15.4 and g_┴ ≈ 2.1 [143,178] or g_zz ≈ 15.5 and g_xx ≈ g_yy ≈ 0.8 [179]. A proposed model with lithium vacancies statistically distributed around Er³⁺_Li [179] supposed that the center with no vacancies in surrounding, i.e., the center with axial C₃ symmetry, should give a dominant (54%) line in the EPR spectra. Note that angular dependences of EPR spectra in xy-plane were not measured in these studies. Later measurements in all three principal planes [180,181,182,183] have shown that there is no line without angular dependence in the xy-plane, i.e., dominant Er³⁺ center in cLN has C₁ symmetry. This does not agree with statistically distributed v_Li around Er³⁺_Li [179].

Significant narrowing of EPR lines in LN_K (Figure 17a) allowed us to trace two different Er₁³⁺ and Er₂³⁺ centers with extrema at about 15, 45 and 75 degrees in xy plane (Figure 17b) [184,185]. The divacancy model (Figure 7) gives a possible explanation if a shift of Er³⁺_Li from regular Li site is taken into account: RBS, XSW and ion-beam/channeling studies have determined that Er occupies Li sites, but is shifted from the ferroelectric Li position by 0.03 [186], 0.046 [187], and 0.02 nm [188]. In this case, distances from Er_Li to the v_Li in the 1a and 1b shells (Figure 5a and Figure 7b) are completely different, and these centers have one charge compensating v_Li in the nearest neighborhood (the shell 1a for Er₁, and 1b for Er₂), and the second v_Li in the next nearest neighborhood (shells 2a, 2b, Figure 7).

Magnetic moments ${\mu }_{B}gS$ for both Er³⁺ centers are very large and strongly anisotropic. At low temperatures their interactions lead to magnetic ordering for Er concentration about 0.5 at.% in sLN [189].

Various models with two v_Li were also extensively discussed in papers devoted to site-selective spectroscopy [190,191,192,193,194,195,196] and references there. Note that models with one of two v_Li on z-axis (Figure 8) do not agree with the EPR spectra for dominant lines and hyperfine satellites on Figure 17a. However, such centers can probably be associated with weaker lines or may have no EPR lines at all, if they are non-paramagnetic.

EPR spectra detected in cLN heavily doped with Mg or Zn and co-doped with Er were described with g_‖ = 4.3, g_┴ = 7.6 for the Mg-doped samples and g_‖ = 4.26, g_┴ = 7.8 for the Zn-doped ones [197]. The spectra can be attributed to Er³⁺ located at the Nb⁵⁺ site of LN, as they are compared to additional centers observed for some trivalent transition metal ions (particularly Cr³⁺) in LN: Mg or LN:Zn.

Fe³⁺ (3d⁵, S = 5/2). EPR studies have shown that the dominant iron center, Fe₁ in cLN has axial symmetry with the ZFS $b_2^0$ about 0.1680 cm⁻¹ [131,133,134,198,199,200,201,202,203,204,205,206,207,208]. Calculations of optical and EPR characteristics based on the superposition models or generalized crystal field theory gave a preference for Nb substitution [209], no definite conclusion for Li or Nb substitution [210,211,212,213] or some preference for Li site [214,215]. Mössbauer [33,34] and Raman [35] spectroscopy data, Stark effect [216] were interpreted on the base of Fe substitution for Li. Iron clusters were observed at high Fe concentration [33]. Comparison of calculated dipole-dipole interactions of Fe³⁺ electrons with surrounding Li nuclei for Li and Nb substitution (Equation (6)) with measured ENDOR spectra [38,39,40] has shown that Fe³⁺ is undoubtedly located in Li site in both cLN and cLT (Figure 5a). Independently, XSW [12,28], EXAFS [21,22,23,24,25,26], and ion beam/channeling [18,27] confirmed Fe³⁺ substitution for Li⁺. It was concluded also that Fe³⁺ ions can be shifted from regular Li-site; however, the obtained values for the shift were very different: 0.006 nm [21] or 0.05 nm from octahedron center [24], less than 0.012 nm [25], about 0.01 nm [26], 0.018 ± 0.007 nm [28], less than 0.005 nm [39]. Fitting observed ENDOR data in LN crystals grown from Li-rich melt (x_m ≈ 55%) was obtained with the shift 0.009 nm from Li site [217].

Fe³⁺ center with small ZFS, Fe₂ observed in congruent LN doped with Mg [104,218,219,220,221,222], In [223], and Zn [224] was assigned to Fe³⁺_Nb.

Comparison of EPR spectra of congruent (Figure 18a, 1), near-stoichiometric grown from Li enriched melts (Figure 18a, 2) and grown from congruent melt with the addition of K₂O, LN_K samples (Figure 18a, 3) shows^a:

• Lines of allowed transitions in LN_K become symmetric—intensities of left (up) and right (down) wings become equal.

• Lines of forbidden transitions (see yellow box on Figure 18) disappear in LN_K.

In LN_K the EPR lines become narrower (up to dozen times at some magnetic field orientation). The narrowing strongly increases spectral resolution. This allows to register even trace impurities in undoped (nominally pure) LN_K samples (see lines of Fe³⁺ and Mn²⁺ on Figure 18a, 3).

^aA brief glance on history of sLN for curios researchers. Looking for better crystals, G. I. Malovichko asked crystal growers for different samples of LN and LT. Dr. E. P. Kokanyan (at that time, one of the engineers in the group of Dr. V. T. Gabrielyan) was interested to grow crystals under various conditions (melt composition x_m, growth under applied electric field, variation of temperature, etc.). He has prepared for her a set of non-stoichiometric LN samples grown from melts with x_m from 43% to 60%. During her PhD research of LN and LT of different compositions, Malovichko found that EPR lines become narrower and more symmetric in samples grown with larger x_m. Dr. V. G. Grachev has simulated EPR spectra with random distributions of non-axial components of ZFS, $b_2^q$ (q ≠ 0) for Cr³⁺ and Fe³⁺ and confirmed that line width and asymmetry, as well as intensities of forbidden transitions are related to intrinsic defects in non-stoichiometric samples [43]. In 1991, on Malovichko’s request Dr. Kokanyan has grown several congruent samples with 2, 4, and 6% of K₂O in the melt (LN_K). First EPR measurement has surprisingly shown narrow symmetrical lines in LN_K with traces of Fe³⁺ (Figure 18a, 3). Based on the experience of studies of Li-rich samples, it was concluded that the concentration of intrinsic defects in LN_K is smaller than in LN grown from the melt with x_m = 60%, and that K₂O may serve as a catalyst in electrochemical reaction of crystal growth [56]. As abilities of research techniques in Ukraine were limited, Dr. Malovichko asked Prof. O. F. Schirmer (Osnabrueck University, Germany) for an international collaboration. Prof. Schirmer was very enthusiastic and has quickly managed to involve many of his colleagues in the investigation of LN_K properties. K. Betzler, B. Faust, B. Gather, F. Jermann, S. Klauer, U. Schlarb, M. Wesselmann, M. Woehlecke and others participated in the LN_K study by different techniques resulting in publications [57,69,77].

All these features are the result of significant reduction of intrinsic defects in LN_K samples. Estimations of crystal composition x_C made by different methods [59,68,77,79] have shown that x_C can exceed 49.8–49.9%, i.e., LN_K is really a stoichiometric crystal.

As for other impurities, the lack of intrinsic defects in LN_K:Fe has led to the appearance of centers where impurities substitutes for Nb. Two additional axial Fe³⁺ centers named Fe₃ and Fe₄ were observed in LN_K [69]. Their ZFS are $b_2^0$ = 0.0495 and $b_2^0$ = 0.0688 cm⁻¹. Compared with Fe₂, the Fe₃ and Fe₄ centers were also assigned to Fe³⁺ in Nb sites with different charge compensation. The Fe³⁺ center with $b_2^0$ = 0.0656 cm⁻¹ (that is very close to $b_2^0$ for Fe₄) was observed in VTE treated stoichiometric LN [76]. Therefore, Fe₄, Fe₃, and Fe₂ were attributed to Fe_Nb with different charge compensator ions in nearest structural vacancy v_oct: Fe₄—non-regular, interstitial Li⁺_v (Figure 19a), Fe₃–K⁺_v, and Fe₂–Mg²⁺_v. Two Li⁺_v (Figure 19b), or interstitial Mg²⁺_v or two Mg²⁺_Li⁺ are required for full charge compensation of Fe³⁺_Nb⁵⁺. Our measurements of EPR in LN:Mg show angular patterns with extrema at 15, 45 and 75 degrees in xy plane. It is why we think that models with two differently located Mg²⁺_Li⁺ (Figure 19c,d) are more suitable for Fe_Nb (similar to models with two v_Li for Fe_Li). ENDOR measurements could confirm some of these reasonable assignments.

Dominant Fe₁³⁺ center in cLT has $b_2^0$ = 0.33 cm⁻¹ [38,225] (Figure 18b, 1), and according to ENDOR data [38] it is definitely Fe³⁺_Li. Line narrowing in nsLT grown by double crucible Czochralski method from an Li rich melt composition (about 60 mol.% Li₂O) allowed to determined b₂⁰ more accurately [226].

The main features of the spectra in sLT obtained by VTE treatment are strong line narrowing, higher resolution, and a clear presence of the second axial Fe₂³⁺ center with $b_2^0$ = 0.205 cm⁻¹ (Figure 18b, 2) [75]. Comparison of calculated angular dependences of ENDOR frequencies for Li and Nb substitution using Equation (6) with measured ones (Figure 20) obviously shows that in the case of Fe₂ center the Fe³⁺ ion substitutes for Nb. Both Fe₁ and Fe₂ centers in sLT have C₃ symmetry. No foreign nuclei in the nearest neighborhood were detected for Fe₁. As sLT still have some residual concentration of v_Li, the charge compensators for Fe₁ centers are one v_Li on center axis (shells 5a, 5b on Figure 7) and probably one other distant v_Li (any vacancy cannot be directly detected by ENDOR).

A pair of ENDOR lines for Fe₂ (fuchsia line on Figure 20b) were attributed to the additional Li⁺_v in the nearest v_oct at the distance about 0.277 nm from Fe³⁺_Nb. It is difficult to make a choice between two models on Figure 19a,b as lines of the second Li⁺_i in the next or next-next v_oct are not identified yet. The ratio of concentrations of Fe₂ and Fe₁ centers changed from less than 0.2 for Fe concentrations 1.1 × 10¹⁹ cm⁻³ to about 1 for 6.7 × 10¹⁹ cm⁻³ in the crystals. Therefore, there are three different mechanisms for excessive charge compensation: distant v_Li for small concentrations of Fe³⁺_Li⁺, partial self-compensation of charges of Fe³⁺_Li⁺ by Fe³⁺_Nb⁵⁺ and partial compensation of Fe³⁺_Nb⁵⁺ by Li⁺ in v_oct for Fe³⁺ concentrations, which exceed the concentration of v_Li.

Gd³⁺ (4f⁷, S = 7/2). The EPR spectrum for every Gd³⁺ center consists of 2S + 1 = 8 strong lines of fine structure and many low intensity lines of so called forbidden transitions due to nearly equal values of Zeeman splitting and ZFS. Results of the first EPR study [227], were interpreted as a presence of two axial Gd₁ and Gd₂ centers with $b_2^0$ = 0.118 and $b_2^0$ = 0.126 cm⁻¹. From the viewpoint of charge compensation, a preference was given to Gd³⁺ substitution for Nb⁵⁺. A small rhombic distortion ($b_2^2$ ≈ 0.004) determined for Gd₂ [228] was attributed to an off axis charge compensator, while for Gd₁ the charge compensator may either be absent or must lie on the same threefold axis as Gd³⁺. However, no decision on whether Gd³⁺ substitutes for Li⁺ or Nb⁵⁺ or for both was made. Similar results with slightly smaller $b_2^2$ ≈ 0.002 were also reported [229].

At least four different Gd³⁺ centers were identified in our study of Li-rich LN doped with 1 wt.% Gd₂O₃ in the melt. As patterns for every EPR transition in xy-plane (Figure 21) are very similar to presented on Figure 4b and Figure 15, the divacancy model is suitable for Gd³⁺ in LN. Gd³⁺ substitutes for Li⁺ in all centers. The dominant axial Gd₁ center and low-symmetry Gd₂, Gd₃, and Gd₄ centers have v_Li in positions described in Table 1. This assignment agrees with the Gd substitution for Li found by RBS and PIXE channeling [31].

Nd³⁺ (4f³) First EPR spectra have shown that in cLN Nd³⁺ creates an axial center with g_‖ ≈ 1.43 and g_┴ ≈ 2.95 (Nd₁) [143,144,230] and second center with g_‖ ≈ 1.33 and g_┴ ≈ 2.95 (Nd₂) [143]. EPR/ENDOR studies in sLN [91,92,93] has established that Nd³⁺ substitutes for Li⁺, and resolved eight different Nd³⁺ centers. Divacancy models explain angular dependencies of EPR spectra for the whole family of Nd³⁺ centers [93], and Section 3.1 of this paper.

It is supposed that Nd³⁺ center found in LN:Mg and LN:Zn belong to Nd³⁺ substituted for Nb [231].

Ti³⁺ (3d¹, S = 1/2) EPR spectrum of Ti³⁺ consist of one line with g_‖≈1.961 and g_┴ ≈ 1.840 in LN [232,233,234], and g_‖ = 1.948, and g_┴ = 1.827 in reduced LT [235]. An axial EPR signal observed in vacuum annealed LiNbO₃ single crystals doped with 8 mol.% Mg and 0.05 mol.% Ti has g_‖ = 1.760 and g_┴ = 1.786 for T = 5 K and g_‖ = g_┴ = 1.893 for T = 74 K. The signal has been attributed to Ti³⁺ on Nb site [236,237,238,239,240]. The g tensor components of these centers were explained by a model calculation involving a dynamic pseudo Jahn-Teller effect. Spin-orbit coupling, lattice vibration, pseudo Jahn-Teller interaction, and the Zeeman term were treated on equal footing. Electron transfer from the observed Ti³⁺_Nb center to lattice niobiums, resulting in Nb⁴⁺ trapped polarons, has been stimulated by illumination in the near UV region.

Yb³⁺ (4f¹³) The EPR lines of Yb³⁺ in congruent LN are very broad (Figure 22a,c) [241]. The axial center Yb₁ with g_‖ ≈ 4.7–4.86 and g_┴ ≈ 2.7 was observed in cLN with 0.5–1.2 wt.% Yb₂O₃ in the melt [144,230,241]. The second Yb₂³⁺ center with g_‖≈1.9 and g_┴ ≈ 2.8 was found in LN:Mg [241]. By comparison of these observations with EPR data for Cr³⁺, Er³⁺, Fe³⁺, and Ti³⁺ in LN and LN:Mg the Yb₁ center was tentatively assigned to Yb³⁺_Li and Yb₂ center to Yb³⁺_Nb. A variation of lattice parameters found in cLN:Yb supported Yb³⁺substitution for Li compensated by Li⁺ vacancy [242].

Complicated EPR spectra without 60 deg repetitions in xy plane were registered in cLN doped with 1wt.% Yb, and cLN doubly doped with 0.8 wt.% of Yb and 0.1 wt.% of Pr [243,244,245,246,247]. Some of the spectra were attributed to Yb³⁺ pairs with the parameter of isotropic exchange interaction J = −0.0283 cm⁻¹.

Our study of LN_K crystals doped with 0.02 wt.% Yb₂O₃ has shown that the broad line observed in cLN belongs in reality to a family of at least 8 different centers (Figure 22b,d) [92,248,249]. Hyperfine structures from isotopes ¹⁷¹Yb (I = 1/2, natural abundance 14.4%) and 173 (I = 5/2, 16.6%) that are barely distinguished in cLN, were well resolved in LN_K (see Figure 22 and Figure 23). The dominant Yb₁ line that represents even isotopes with I = 0 (^evenYb, 69%) was described with g_‖ = 4.46, g_┴ = 2.706. Intensities of the Yb³⁺ lines were proportional to natural abundances of isotopes and their spins: ℐ(even):ℐ(171):ℐ(173) = 0.69:0.144/2:0.166/6 = 0.69:0.072:0.028. It was found for Yb₁ that ¹⁷¹A_‖ = 0.119 cm⁻¹, and ¹⁷¹A_┴ = 0.0715 cm⁻¹. Branches of angular dependencies of the Yb₅ center (fuchsia lines on Figure 22) were described with spin-Hamiltonian for S = 1 and anisotropic S^AJS^B interaction (J_ik ≈ 0.012–0.066 cm⁻¹). They were assigned to low-symmetry Yb³⁺–Yb³⁺ pairs. The presence of additional lines of hyperfine structure with similar angular dependence (indicated by red arrows in Figure 23) supports the assignment, as their intensities are proportional to probabilities to meet two ^evenYb, or one ^evenYb and one ¹⁷¹Yb, or one ^evenYb and one ¹⁷³Yb in such pairs. A self-compensated pair consisted of Me³⁺_Li and Me³⁺_Nb in the nearest sites (Nb shell 1 at a distance 0.3 nm on the crystal axis, Figure 5a) creates an axial center with rather strong exchange interaction (Jiso > 300 cm⁻¹). Therefore, the observed low-symmetry pairs were attributed to the Yb³⁺–Yb³⁺ ions in next neighbor or next-next neighbor positions (shells 2 and 3 on Figure 5a).

One of the satellite centers (Yb₆) has axial symmetry; all others are low-symmetry centers. ENDOR measurements has revealed that Yb³⁺ substitutes for Li: at the first, comparison of measured angular dependencies with calculated ones on the base of dipole-dipole interactions (Equation (3)) gave undisputable preference for Li site, and at the second, the strongest axial hyperfine interaction was found for ⁹³Nb on the z-axis (the first shell on Figure 5a). The most reasonable explanation for the existence of the whole family of ytterbium centers is that Yb³⁺_Li is compensated by one or two v_Li in different configurations.

4.4. Tetra-, Penta- and Heptavalent Cations

The non-paramagnetic tetravalent impurities (C and Si) are always present in LN in rather high concentrations (about 50–500 ppm). Since determined concentrations of chlorine Cl (50–500 ppm) and manganese Mg (1–100 ppm) have the same order of magnitudes, no other charge compensators are necessary if C or Si substitutes for Nb creating C⁴⁺_Nb – Cl⁻_O²⁻ or C⁴⁺_Nb – Mg²⁺_Li. The additional possibilities for the charge compensation supply H⁺ and Li⁺_v ions. It is supposed that Ti⁴⁺ substitutes for Nb⁵⁺; however, at present a mechanism of its charge compensation is not well established.

Due to the lithium deficiency of congruent lithium niobate crystals, v_Li have been considered as possible charge compensators for Nb⁵⁺_Li⁺ or Ta⁵⁺_Li⁺ antisites for a long time. Models with plane and space configurations of v_Li for non-stoichiometric defects were proposed [96,97] and used for calculations of stability of intrinsic defects and defect clusters in LN [250,251,252,253] and LN:Mg [254]. The antisites become paramagnetic ions Nb⁴⁺_Li⁺ or Ta⁴⁺_Li⁺ after irradiation (see Section 4.5 below).

The Ta⁵⁺ substitution for Nb⁵⁺ in LN causes minor lattice distortions only. The heptavalent ions (Mo, W) probably substitute for Nb⁵⁺ having lithium vacancies as charge compensators.

U⁵⁺ (5f¹). EPR spectra of U⁵⁺ were studied in LiNbO₃ powders doped with natural U₃O₈ and ²³³U₃O₈. A hyperfine sextet of EPR line for ²³³U (I = 5/2) was described with A_‖ = 0.0145 and A_┴ = 0.0128 cm⁻¹. g_‖ = 0.71, g_┴ = 0.724 were determined for the line of natural U⁵⁺ (even isotopes with I = 0 have total natural abundance about 99%). It was observed that U⁵⁺ takes part in photoinduced valence change which is the basic mechanism for photorefraction.

4.5. Radiation and Reduction Defects

A radiation usually recharges of regular lattice, interstitial and impurity ions or produces interstitial ions. Two kinds of recharged defects were observed in LN: electron traps like Nb⁴⁺ and hole traps like O⁻ ions [236,255,256].

Nb⁴⁺ (4d¹). Hyperfine interaction of the unpaired 4d¹ electron with the ⁹³Nb nuclear spin I = 9/2 splits its EPR line into ten components. The ten-line EPR spectrum (g_‖ = 1.90 and g_┴ = 1.72) has been described for congruent LiNbO₃ after ionizing radiation [257]. Later this spectrum has been reproduced in vacuum-reduced and UV bleached crystals [258,259,260,261] (Figure 24, 1) and ascribed to antisite Nb [255]). It is remarkable that at least part of the Nb⁴⁺ centers has C₁ symmetry [262], although the main possible positions for Nb (regular Nb and Li site, v_oct) have C₃ symmetry. It means that compensating defects for Nb⁴⁺, most probably lithium vacancies, are located in the nearest neighborhood (Figure 25a,b).

Similar spectra were observed in LiNbO₃ doped with 6 mol.% Mg (Figure 24, 2) after X-irradiation or vacuum reduction treatments and were related to Nb⁴⁺ centers on niobium sites [104,113,114], obviously with nearby defects. The Nb⁴⁺ center in LiNbO₃ doped with 10 mol.% Mg belongs to Nb in regular position, but with Mg²⁺_Li in neighborhood (Figure 24, 3) [238,263]. The neighboring Mg²⁺ ion redistributes a cloud of Nb⁴⁺ electrons and changes the hyperfine splitting.

Ta⁴⁺ (5d¹). The isotope ¹⁸¹Ta has I = 7/2 and 100% natural abundance. The eight-line axial EPR spectrum with g_‖ = 1.503 and g_┴ = 1.172, A_‖ = 0.0023, and A_┴ = 0.0234 cm⁻¹ has been observed in LiTaO₃ after reduction in argon and attributed to axial Ta⁴⁺_Li [257]. Other ways to obtain Ta⁴⁺ are an irradiation of as-grown crystals with X-rays or optical bleaching of crystals that had been previously reduced [264]. A possible model for axial Ta⁴⁺ center is similar to the presented on Figure 25a.

Tb⁴⁺ (4f⁷). EPR study at 15 K revealed a signal of g ≈ 2.0 appearing after UV irradiation with a simultaneous decrease in the Fe³⁺ signal intensity in near-stoichiometric LiNbO₃:Tb and LiNbO₃:Tb:Fe [265]. This implies that the Fe³⁺ ions act as electron traps. Irradiation by UV light induced an absorption band extending from λ ≈ 650 nm to the absorption edge caused by the charge transfer from UV-sensitive absorption centers to Fe³⁺ ions via the conduction band.

Other electron and hole traps. A trapped-hole center with S = 1/2 was produced in LiNbO₃ by ionizing radiation [266]. Its ESR spectrum contains at least 26 equally spaced lines with 1.54 mT separation at B||z. This hyperfine pattern was explained as one “hole” interacting equally with three ⁹³Nb nuclei (I = 9/2 and 100% abundant). The hole is equally shared by three equivalent oxygen ions adjacent to a cation vacancy. A center with 25 lines of hyperfine interaction with two ⁷Li and two ⁹³Nb nuclei was ascribed to O⁻ in regular O²⁻ site with unclear stabilizing factor for the hole trap [236].

An OH²⁻ ion was identified in undoped and weakly doped with Mg LN samples after γ-irradiation and subsequent partial UV bleaching [109]. Specific hyperfine structure with ⁹³Nb was observed for a hole trap in LN doped with 6–8 mol.% Mg. The center was attributed to O⁻ near Mg²⁺ ion substituted for Nb⁵⁺ (Figure 25c) [263].

5. Impurity Identification

At an investigation of a sample grown from a melt with an addition of a paramagnetic spice in concentration 0.00X–0.X at.%, which warrants that EPR signal significantly exceeds noise, it would be reasonable to expect EPR signal from this spice. Often, studied samples (especially, commercial ones) contain so-called non-controlled or trace impurities like Cr, Mn, Fe, Cu etc. in a slightly smaller or comparable concentration. Fortunately, most paramagnetic impurities are already investigated by EPR/ENDOR techniques and their characteristics—electron and nuclear spins, zero-field splitting, hyperfine and quadrupole interactions, kinds of charge compensators and their locations, i.e., “passports” of the impurities—are known. Comparison of published and observed spectra (see, for instance, figures with EPR spectra and their angular dependencies above) allows to identify the impurities in films, epitaxial layers, and fibers, or to evaluate profile of impurity distribution in bulk samples. Spectra of LN/LT powders and ceramics can be simulated using determined spin-Hamiltonian parameters.

6. Conclusions

Very detailed information about structures of impurity defects (charge state, point symmetry, hyperfine interactions with neighbor nuclei, charge compensation mechanisms etc.) was obtained with the help of EPR and ENDOR.

The necessity of a charge compensation for non-isovalent substitution usually leads to the creation of families of electrically and magnetically non-equivalent impurity centers. The families of satellite centers exist due to the different relative locations of the impurity ion and its charge compensator. Two or more different centers were observed for Co²⁺, Cr³⁺, Cu²⁺, Er³⁺, Fe³⁺, Gd³⁺, Mn²⁺, Nd³⁺, Yb³⁺, and other ions. Since the relative concentrations of satellite centers are comparable with the concentration of the main center, both kinds of centers generally are equally responsible for many of the properties of LN/LT crystals, and they should both be taken into consideration, especially in non-stoichiometric crystals.

Several different mechanisms for a compensation of excessive charge of Me impurity were found:

• the compensation by nonstoichiometric defects (v_Li, v_Nb) for small concentrations of Me_Li in crystals,

• the partial self-compensation of charges of Me_Li by Me_Nb mainly for the Me_Li concentration that exceeds the concentration of v_Li,

• the partial compensation of Me_Nb by interstitial Li⁺ in v_oct,

• the compensation by other impurities (H, co-dopants like Mg, Zn, etc.).

The presence of non-stoichiometric defects is one of the reasons why LN tolerates a strong incorporation of dopants non-isovalent to Li⁺ and Nb⁵⁺. As long as the impurity concentration [Me] is smaller than δx_C = |50% − x_C|, the number of intrinsic defects is large enough to compensate the corresponding charge excess. However, for stoichiometric or nearly stoichiometric samples with high impurity concentrations (when [Me] > δx_C) and with the lack of charge compensators a decrease of the distribution coefficient of impurities is observed in comparison with congruent material. A further increase of the [Me]/δx_C ratio up to [Me] >> δx_C can result in a change of the charge compensation mechanism. This can reveal itself in the appearance of new impurity centers.

The charge compensation by v_Li works well for most of the non-isovalent ions. Vacancies in the nearest neighborhood of Me_Li decrease symmetry of centers from C₃ to C₁, whereas distant vacancies cause spectral line broadening. For Me³⁺_Nb/Ta in stoichiometric LN/LT the charge compensation by interstitial H⁺ and Li⁺_v ions was found. The interstitial Li⁺_v should be considered as a concurrent charge compensation mechanism in VTE treated samples. The compensation of Me_Nb by Mg_Li, Zn_Li and other impurities occurs when the co-dopant concentration exceeds some threshold that depends on δx_C. Typically, the threshold is about 6–8% for congruent samples, but it is lower than 1% for nearly stoichiometric samples.

The use of stoichiometric or nearly stoichiometric crystals with δx_C ≈ 0 presents many advantages for the investigation of impurity centers by spectroscopic techniques. The decreased concentration of intrinsic defects causes a tremendous narrowing of the spectral lines. This is accompanied with the increase of spectral resolution and sensitivity, facilitates the analysis of the spectra, and simplifies the interpretation of the data. However, together with the disappearance of intrinsic defects the satellite centers disappear also. For a detailed investigation of such additional centers, the crystals with high x_C ≈ 49.5–49.85% are more suitable than others: they have more narrow spectral lines than congruent samples, but the satellite centers are still present. Further study of nearly stoichiometric and stoichiometric LN and LT samples should help to eliminate some disagreements in published data and to clarify intimate details of structures of impurity defects in materials, which are important for both physics and applications.

Derived structures and obtained by EPR/ENDOR characteristics of Zeeman and zero-field splitting, quadrupole and hyperfine interactions of impurity electrons with own and surrounding nuclei can be reliable corner stones for modelling of the structures.

Author Contributions

V.G.G.—theorist (yin), and G.I.M.—experimentalist (yang) made inseparable complementary contributions. All authors have read and agreed to the published version of the manuscript.

Funding

The publishing fee is covered by V.G.Grachev, SimuMag, Bozeman, MT, USA.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The data presented in this study are available on request from the corresponding author.

Conflicts of Interest

The authors declare no conflict of interest.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Figures and Table

Figure 1. (a) Ideal lattice of lithium tantalite (LT). To simplify the representation of the lattice and defect structures in it, the sizes of “balls” imitating ions were made intentionally different. The L (Left) and R (Right) positions are distinguished by their different oxygen surroundings. (b) An illustration of local C3 symmetry for the nearest surroundings of cation sites and glide mirror zy plane that transforms the L center to the R partner; a projection of one Li, Ta layer between two oxygen layers on the xy plane is shown.

Figure 2. Fragments of electron paramagnetic resonance (EPR) spectra of Nd3+ impurity ions in nearly stoichiometric lithium niobate (LN) for microwave frequency ν = 34.445 GHz, T = 19 K. The numbers 1–4 correspond to four different electrically non-equivalent defects. Lines of magnetically nonequivalent partners have the same number.

Figure 3. (a) Projection of ideal lithium niobate lattice with Nd3+ impurity ion on xy plane (artificial sizes of balls are used). (b) Circular diagram of measured angular dependence of electron paramagnetic resonance (EPR) lines of Nd3+ ions in Figure 3. Magnetic field was rotated in xy plane from 0 to 360 deg. ν = 34.445 GHz, T = 19 K. Symbol sizes reflect line intensities.

Figure 4. (a) The road map rotation corresponds to the following changes of the θ, ϕ angles: θ = 0–90° at ϕ = 0°; θ = 90°, ϕ = 0−90°; θ = 0−90° at ϕ = 90°. (b) Angular dependence of the EPR spectra in xy plane for nearly stoichiometric LN:Nd3+. Rhombs represent experimental line positions for Q-band measurements (ν = 34.445 GHz), their sizes reflect line intensities. Horizontal whiskers near rhombs represent line widths. Cyan, lime, fuchsia, and blue curves represent simulated dependencies for axial Nd1, and low-symmetry Nd2, Nd3, and Nd4 centers, respectively. Curves for isotopes with nuclear spin I ≠ 0 are not shown.

Figure 5. Surroundings of impurity ions substituted for Li (a) and Nb (b) in LN crystal lattice with corresponding shell numbers. Large blue balls represent Li+ ions, medium green balls—Nb5+ ions, small red balls—O2− ions, and the largest magenta ball—the impurity ion.

Figure 6. Angular dependence of electron nuclear double resistance (ENDOR) frequencies of Li nuclei in xy plane for the Nd13+ center in LiNbO3. The green, red and blue curves—calculated frequencies for Li nuclei of the 1st, 2nd and 3rd shells based on dipole-dipole interaction for Nd3+Li (a,b), and Nd3+Nb (c). ENDOR frequencies represented on the part (b) were measured at magnetic fields corresponding to the position of dominant EPR line #1 on Figure 2 (B = 820 mT for ν = 34.445 GHz, rhombs, and B = 228 mT for ν = 9.500 GHz, triangles). νLarmor is Larmor frequency of 7Li nuclei.

Figure 7. A projection of lithium niobate crystal lattice on xy plane (a) and 3D model (b) with shell numbers for an impurity substituted for Li+ ion (sizes of all ions were artificially changed in order to make clear positions of cation ions). Hatched circles represent two vacancies of Li+ ions for Nd4 center.

Figure 8. Divacancy models for an impurity substituted for Li+ ion in LN crystal lattice. Hatched circles represent two vLi. (a) Axial center, (b) nearly axial center, (c) low-symmetry center.

Figure 9. (a) X-band EPR spectra of Ni+ in LiNbO3 at θ = 25 deg in zy plane at various temperatures T: 1–40 K, 2–90 K, 3–151 K, 4–206 K, 5–244 K, 6–300 K. The spectrum 7 was measured at θ = 0 deg, T = 300 K. (b) Distortion of oxygen octahedron due to the Jahn-Teller effect.

Figure 10. (a) Measured EPR spectra of Co2+, X-band, T = 5 K. 1—sLN, B||z; 2—sLN, B||x; 3—cLN, B||x. The red arrow indicates an additional line assigned to clusters of Co2+ ions. (b) Possible model for a cluster of four Co2+ ions substituted for one Nb5+ and three Li+ ions.

Figure 11. (a) Measured EPR spectrum of Cu2+ in LN, X-band, T = 7 K. (b) Distortion of oxygen octahedron due to Jahn-Teller effect.

Figure 12. (a) Measured EPR spectrum of Mn2+ in LNK, ν = 34.45 GHz. (b) Model of axial Mn2+Li center with vLi.

Figure 13. Measured EPR spectrum of Cr3+ in LN grown from the melt with xm ≈ 60% and 0.1 wt.% of Cr (room temperature, ν = 9.84 GHz). Black braces indicate lines of dominant Cr3+ center, red arrows—lines of satellite centers.

Figure 14. The ENDOR spectra of Cr3+Li (high-field EPR transition) and Cr3+Nb (central EPR transition) at B||x, T = 5 K. To facilitate a comparison of the spectra, they were shifted to the 93Nb Larmor frequencies.

Figure 15. The angular dependencies of EPR spectra in the xy plane for LN:Cr3+ grown from the melt with xm ≈ 60%, ν = 9.26 GHz. Rhombs represent line positions; their sizes are proportional to line intensities. Horizontal whiskers near rhombs represent line widths. Cyan, lime, fuchsia, and blue curves represent simulated dependencies for axial Cr1, and low-symmetry Cr2, Cr3, and Cr4 centers, respectively.

Figure 16. (a) Measured EPR spectra of Cr3+ in LN grown from the melt with addition of K2O and 1 wt.% of Cr; room temperature, ν = 9.445 GHz. (b) Model of Cr3+Nb center.

Figure 17. (a) EPR spectra of Er3+ in congruent and stoichiometric LN at B||z, X-band. (b) Calculated angular dependence of EPR spectra in xy plane for the Er13+ (green lines) and Er23+ (fuchsia lines) centers in sLN. Symbols with horizontal whiskers represent line position and widths.

Figure 18. Normalized spectra of Fe3+ in LN (a) and LT (b), X-band. (a) Congruent (1), grown with xm = 0.545 (2), undoped sample grown from congruent the melt with 6 wt% of K2O (3), B||z. (b) Congruent (1), vapor transport equilibrium (VTE) treated sample (2), B||x.

Figure 19. Possible structures of Me3+Nb5+ (large magenta ball) centers in LN. Structure with one (a) and two (b) additional Li ions in voct (marked with red crosses). Axial (c) and low-symmetry (d) structures with two Mg2+Li+ ions.

Figure 20. Angular dependence of ENDOR frequencies of Li nuclei in xy plane for the Fe23+ center in LiTaO3. The green, magenta, blue and lime curves—calculated frequencies for Li nuclei of the 1st, 2nd, 3rd and 4th shells based on dipole-dipole interaction for Fe3+Li (a), and Fe3+Nb (b). Fuchsia lines on (b)—calculated frequencies for the additional Li nucleus in the nearest voct on the axis of Fe3+Nb center. ENDOR frequencies represented by rhombs were measured on the EPR line at B = 1162 mT (ν ≈ 34.5 GHz).

Figure 21. The angular dependencies of EPR spectra in xy plane for Li-rich LN:Gd3+, ν = 9.26 GHz. Rhombs represent line positions; horizontal whiskers—line widths. Cyan, lime, fuchsia, and blue curves are simulated dependencies for axial Gd1, and low-symmetry Gd2, Gd3, and Gd4 centers, respectively.

Figure 22. EPR spectra of Yb3+ in congruent (a, c) and stoichiometric (b, d) LN.

Figure 23. Angular dependence of EPR spectra of Yb3+ in stoichiometric LN, xy plane, ν = 9.86 GHz. Red arrows—lines of hyperfine structure for Yb5 centers.

Figure 24. (a) EPR spectra of Nb4+ centers in cLN (1) and cLN doped with 6 mol.% Mg (2) and 10 mol.% Mg (3), X band. (b) Model for axial Nb4+Nb – Mg2+Li center.

Figure 25. Models for Nb4+Li (large green ball) centers with three vLi in LN. Hatched circles represent lithium vacancies. (a) Axial center. (b) Low-symmetry center. (c) The center with O− near Mg2+ substituted for Nb5+.

Table 1

Sites for lithium vacancies, v_Li in the surroundings of Me³⁺_Li centers in LiNbO₃.

CentervLi Site	Me1	Me2	Me3	Me4	Me5	Me6	Me7	Me8
First	5a/5b	1a/1b	2a/2b	1a	1b	1a/1b	2a/2b	5a
Second	distant	5a/5b	5b/5b	2a	2b	3a/3b	3a/3b	5b